MA005 Study Guide

Site: Saylor Academy
Course: MA005: Calculus I
Book: MA005 Study Guide
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Date: Friday, March 29, 2024, 4:26 AM

Navigating the Study Guide


Study Guide Structure

In this study guide, the sections in each unit (1a., 1b., etc.) are the learning outcomes of that unit. 

Beneath each learning outcome are:

  • questions for you to answer independently;
  • a brief summary of the learning outcome topic;
  • and resources related to the learning outcome. 

At the end of each unit, there is also a list of suggested vocabulary words.

 

How to Use the Study Guide

  1. Review the entire course by reading the learning outcome summaries and suggested resources.
  2. Test your understanding of the course information by answering questions related to each unit learning outcome and defining and memorizing the vocabulary words at the end of each unit.

By clicking on the gear button on the top right of the screen, you can print the study guide. Then you can make notes, highlight, and underline as you work.

Through reviewing and completing the study guide, you should gain a deeper understanding of each learning outcome in the course and be better prepared for the final exam!

Unit 1: Preview and Review

1a. Approximate a slope of a tangent line from a function given as a graph

  • Explain the difference between a secant line and a tangent line.
  • How do you calculate the slope of a line?
  • What method do you use to estimate the slope of the tangent line from the slope of the secant line?

In geometry, a tangent is a straight line that touches a curve at one point. At the place where they touch, the line and the curve both have the same slope (they are both going in the same direction). For this reason, a tangent line is a good approximation of the curve near that point. In Figure 1a.1., the tangent line is the red (or straight diagonal) line that just grazes the curve.

Figure 1a.1.
Figure 1a.1.

A secant is a line that intersects a curve or circle at exactly two points. In Figure 1a.2., the secant line is the red (or straight diagonal) line that passes through two points on the curve.

Figure 1a.2.
Figure 1a.2.

We call the steepness of a hill its slope. The same goes for the steepness of a line. In math, we define the slope as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run.

Finding the slope of the secant line is the same as finding the average rate of change between two points on a graph, but if you want the instantaneous rate of change at a single point, you need the slope of the tangent line. 

The slope of the tangent line is a fundamental concept in calculus, and we can estimate its value from the slope of the secant line by moving the two points on the graph closer and closer together.

To review, see Preview of Calculus.

 

1b. Approximate the area of an irregular figure by counting inside squares

  • How can you estimate the area of an irregular shape using squares?
  • How do smaller squares help improve the estimate?

There are formulas to calculate the area of common geometric shapes, but what if the shape is curved or is complex or irregular? Calculus allows you to calculate the area of the shape precisely, but you can get a good estimate of the area by approximating the shape with geometric shapes we know.

Take a look at Figure 1b.1., for example. We can put the shape on a grid, and count the number of squares (or rectangles or triangles, etc.) to get a good estimate, and if the squares are smaller, then the estimate will get better and better because the smaller squares will fit the shape more precisely. 

Figure 1b.1.a.Figure 1b.1.b.
Figure 1b.1.

To review, see Preview of Calculus.

 

1c. Calculate the slope of the line through two points

  • What is the equation for calculating the slope through two points?
  • What is the slope of a horizontal line?
  • When is the slope of a line undefined?
  • What does the graph of a line look like when the slope is negative?
  • What is the Cartesian Plane?
  • Define parallel and perpendicular lines.

Calculating the slope of a line from two points is often described by the mnemonic rise over run. This is just a helpful way to remember that the rise is the difference in the y-coordinates, and the run is the difference in the x-coordinates, and we divide them. Take a look at Figure 1c.1.

Figure 1c.1

Figure 1c.1.

A Cartesian plane is a graph with one x-axis and one y-axis. These two axes are perpendicular to each other. The origin (O) is in the exact center of the graph. Parallel lines are lines in a plane that are always the same distance apart (equidistant). Parallel lines never intersect. Perpendicular lines are at right angles (90°) to each other.

When the rise is 0, the line is constant or horizontal. We say the line has zero slope. When the run is 0, the line is vertical. Division by 0 is undefined, so we say the line has no slope. The slope of the line tells us how quickly the values are increasing (if the slope is positive), or decreasing (if the slope is negative).

To review, see Lines in the Plane.

 

1d. Write the equation of the line through two points using both slope-intercept and point-slope forms

  • What is the slope-intercept form of the equation of a line?
  • When should you use the slope-intercept form to find the equation of a line?
  • What is the point-slope form of the equation of a line?
  • When is it better to use the point-slope form a line?

Any two points can define a line, but we do not usually work with them in this form. Instead, we want to deal with them in the form of equations so that we can calculate any point on the line. This is a fundamental task in calculus.

There are several ways to represent the equation of a line and each is important for different tasks. From the slope of a line, we can determine whether the graph is increasing or decreasing.

The intercept tells us initial conditions. There are also special cases such as horizontal and vertical lines that will be important as we progress through calculus.

To review, see Lines in the Plane.

 

1e. Write the equation of a circle with a given center and radius

  • What is the equation of a circle centered at the origin?
  • How do you modify the equation to center it at a point other than the origin?
  • Given the endpoints of a diameter, how do you find the radius of the circle?
  • Given the center and a point on the circle, how do you find the radius of the circle?

A circle is defined by two points, the center and the radius. See Figure 1e.1.

If you have a point on the circle and the center, you can find the radius by finding the distance between the two points. If you have two points on a diameter, the center is the midpoint of that line. 

Figure 1e.1

Figure 1e.1.

To review, see Lines in the Plane.

 

1f. Write the equation of a line through a point given the slope1f. Write the equation of a line through a point given the slope

  • Plan your problem: Do you have a y-intercept, or a general point?
  • What is the slope-intercept form of the line?
  • What is the point-slope form of the line?
  • What is the standard form of the line?
  • What are the special cases?

To write the equation of a line, given a slope and point, best practice is to use the form of the line that matches your problem. If you have a y-intercept as the point, of the form (0,b), then use the slope-intercept form and you can form the equation of the line y=mx+b directly, by replacing m with the slope value and b with the intercept value. If you have a point which is not the y-intercept, you can use the point-slope form of the equation y-y_1 = m(x-x_1) where m is the slope and (x_1,y_1) is the point. Solve for y to obtain the slope-intercept form.

If you are asked for the standard form of the equation Ax+By=C, use one of the other methods first. Then multiply your equation to eliminate any fractions. Then move the x and y terms to the same side of the equation with the constant on the other side. In general, you also want to make your x-coefficient positive.

If you have a special case – a horizontal line (slope: 0) or a vertical line (slope: undefined) – the final equation will have the form y=b (horizontal line) or x=a (vertical line) going through the point (a,b).

To review, see Lines in the Plane.

 

1g. Write the equations of lines that are parallel or perpendicular to a given line

  • How are the slopes of parallel lines related?
  • How are the slopes of perpendicular lines related?

If two lines are parallel, their slopes are the same. Use the given line to obtain the required slope. You may need to solve the equation for y (slope-intercept form) to get the slope. Follow the process for writing the equation of the line using the slope you obtained and using the given point.

If two lines are perpendicular, their slopes are negative reciprocals of each other. This can be represented as m_2 = -\frac{1}{m_1} or m_1 \times m_2 = -1. Use the given line to obtain the slope of the given line. You may need to solve the equation for y (slope-intercept form) to get the slope: this is m_1. To find the slope of the new line, flip the fraction, and change the sign, eg. if m_1 = \frac{3}{4} then m_2 = -\frac{4}{3}. Finally, follow the process for writing the equation of the line using the slope m_2 you obtained and using the given point.

To review, see Lines in the Plane.

 

1h. Evaluate a function at a point, given by a formula, graph, table, or words

  • Define independent and dependent variables.
  • What does the function notation y=f(x) mean? Use the point (1,2) in your explanation.
  • Give an example of a function in words. How would you show it is a function?
  • Given a function like f(x)=x2+1, what does this function look like a graph?
  • How would you convert the function {(1,2), (4,3), (7,4), (8,8)} into a graph and a table?

Functions are a kind of equation where we can solve for y (the output variable) and have all the x-es (input variables) on the other side of the equation. We sometimes refer to these kinds of functions as explicit functions (but implicit functions or relations cannot be solved for one variable).

The independent variable is the input variable (the information that is known), and the dependent variable is the output variable, or the information to be calculated from the independent variable.

The notation is used to emphasize that the independent variable is x, and further allows us to express coordinate points in the notation such as is equivalent to the point (1,2). See Figure 1h.1.

Figure 1f.1.

Figure 1h.1.

To review, see Functions and Their Graphs.

 

1i. Evaluate a combination, or a composition, of functions when indicated by the symbols +, −, *, and /

  • What is the procedure for evaluating the expression (f+g)(x), (f−g)(x), (fg)(x), and (f/g)(x) if f(x)=x+2, and g(x)=x2+3x?
  • For the same functions, what is f(g(x))? What is f(g(3))?
  • Is f(g(x)) ever equal to g(f(x))? If not, why not? If it is, when?

Sometimes we want to combine the outcome of functions in various algebraic ways such as adding, subtracting, multiplying, or dividing the results. We can also apply more complex operations on functions such as applying the same function twice in succession or applying two different functions in sequence. See Figure 1i.1.

Properties of the resulting functions, both as graphs and as equations, are the result of the properties of both functions. Composition of functions will be especially important in calculus as we deal with complex functions.

Related to composition of functions is transformations of functions, which allow us to alter the shape and position of standard functions and can be a useful tool in graphing functions.

Figure 1g.1.

Figure 1i.1.

To review, see Combinations of Functions.

 

1j. Evaluate and graph the elementary functions as well as |x| and int(x)

  • How would you describe in words what the graphs of y=|x| and y=int(x) 
  • What is the value of |−2|, |4|?
  • What is the value of int(−0.2), int(1.4)?
  • How would you describe the operation of |x| and int(x) in words?
  • What are the other elementary functions? Describe their general shape in words.

The absolute value function and the greatest integer function are two examples of piecewise functions. The absolute value function is shaped like the letter V. See Figure 1j.1. The greatest integer function is a kind of piecewise function called a step function that is composed of only horizontal line segments.

Graphing these functions and working with their domains and ranges is an important skill in calculus for working with real-world problems and sketching graphs to understand their properties.

Figure 1h.1.

Figure 1j.1.


To review, see Combinations of Functions.

 

1k. State whether a given if-then statement is true or false, and justify the answer

  • Is the statement If x=2, and y=6, then xy=15 true or false?
  • Is the statement If xy=12, then x=2 and y=6 necessarily true?

A course like calculus is usually a student's first introduction to mathematical proofs which are based on logical statements and valid argument structures.

Logical AND (conjunction), OR (disjunction), NOT (negation), and IF-THEN (conditional) is defined. IF-THEN statements are especially important in mathematics, and altering the order of the statements in the conditional can change the meaning completely.

To review, see Mathematical Language.

 

1l. State which parts of a mathematical statement are assumptions, or hypotheses, and which are conclusions

  • In the statement IF y=4, THEN 3y+11=23, which portion is the hypothesis, and which is the conclusion?
  • What kind of assumptions must be made to determine if the equation has a solution?

Mathematics is built on a foundation of reasonable assumptions (sometimes called axioms) that cannot be proven absolutely, but appear to be reasonable (such as parallel lines do not intersect).

These assumptions are always in the background of every claim and are usually indicated in a proof by the term suppose. The hypothesis is the claim being made, and the conclusion is the final result.

To review, see Mathematical Language.

 

1m. State the contrapositive form of an if-then statement

  • Define equivalent statements.
  • What is the contrapositive of the statement IF x=3, THEN 3x=9?
  • What is the converse of the statement IF n=2, THEN 3x=9. Why is it not equivalent to the original statement? Give a counterexample.

Sometimes when writing proofs, the direct proof of an if-then statement is more difficult than proving an equivalent statement. Understanding what kind of proof is equivalent to proving the original statement is essential to ensure that the proof is not invalid.

Equivalent statements are statements that are written differently but hold the same logical equivalence (have the same logical value).

To review, see Mathematical Language.

 

1n. Write negative statements

  • What is the negative of the statement x=3?
  • What is the negative of the statement y>9?
  • What are some common ways that negation is expressed in mathematics?

Negation is a logical operator that applies to only one statement at a time, unlike other operators that express the relationship between two statements. To negate the statement x=1, we can write x\neq1. This means that x cannot be equal to 1, or equivalently, that x can be equal to anything except 1. For inequalities, when negating, the inequality changes and the inclusion of the equality sign changes. So the negation of x is written as x \nless 2 or x\geq2, because if x cannot be less than 2, then it can be 2, or greater than 2.

To review, see Mathematical Language.

 

Unit 1 Vocabulary

  • Absolute value
  • Assumptions
  • Average rate of change
  • Cartesian plane
  • Composition of functions
  • Conditional
  • Conjunction
  • Contrapositive
  • Converse
  • Counterexample
  • Dependent variable
  • Disjunction
  • Equivalent statements
  • Function
  • Function notation
  • Greatest integer function
  • Hypotheses
  • Independent variable
  • Intercept
  • Negation
  • Parallel lines
  • Perpendicular lines
  • Piecewise functions
  • Point-slope form
  • Proof
  • Radius
  • Secant line
  • Slope
  • Slope-intercept form
  • Standard form of a line
  • Tangent line
  • Transformations of functions

Unit 2: Functions, Graphs, Limits, and Continuity

2a. Explain the limit of a function

  • What is the definition of the limit of a function, in your own words?
  • When x gets close to some point c, what is happening to the function value?
  • What is the difference between a one-sided limit and a two-sided limit?

The limit (L) is a way of describing what happens to a function (f(x)) near a point (c). If the limit exists, then as x gets closer to c, the function will get closer to the limit value L, even if f(c) is not defined. If the limit does not exist, the value of the function may approach more than one value, or may increase (or decrease) without bound.

To review, see The Limit of a Function and Definition of a Limit.

 

2b. Determine the slope of the line tangent to the graph of a function at a point

  • What is the definition of a tangent line?
  • How is the tangent line related to the secant line?
  • How do you calculate the value of the slope of the tangent line, by definition?
  • What point is both on the graph of the function, and the tangent line?

The slope of the tangent line to the graph is defined by taking the limit of the difference quotient \frac{f(x+h)-f(x)}{h} that defines the slope of a secant line. As the difference between the two points (h) gets smaller and smaller (h goes to 0), the value of the expression for the secant line will approach the slope of the tangent line. Use the expression you obtain and evaluate that expression at the point on the graph.

To review, see Tangent Lines, Velocity, and Growth.


2c. Determine the values of one- or two-sided limits for a function given by a graph

  • What is the definition of a limit, in your own words?
  • What is the difference between a jump discontinuity and a point discontinuity?
  • What conditions must be satisfied for a function to be continuous at a point?
  • When does a two-sided limit not exist?
  • What are three methods for evaluating limits?

We can describe Calculus as the study of continuous change – the limit is the basic concept that allows us to describe and analyze this change. The limit of a function describes the behavior of the function when a limit is near. A limit (L) is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.

Limits are a fundamental concept in calculus that underpin many other concepts. For a limit to exist for a function, as x approaches a specific value c so that the difference between x and c is an arbitrarily small value, then the function value f(x) approaches some value that is arbitrarily close to the limiting value L. We can evaluate limits algebraically, numerically, or graphically. See Figure 2c.1.

Figure 2a.1.

Figure 2c.1.

Functions behave in a useful way for calculus wherever the function is continuous. There are two kinds of discontinuities for functions: jump discontinuities and point discontinuities. If the break in the function has a limit at that point (both the left- and right-handed limits exist and are the same value), then the discontinuity can be repaired by replacing a finite number of points with the values of those one-handed limits. If the one-sided limits are different, then the discontinuity is a jump discontinuity, and cannot be repaired. See Figure 2c.2.

Figure 2a.2.
Figure 2c.2.

To review, see The Limit of a Function.

 

2d. Use algebraic methods to determine the values of one- and two-sided limits for a function given by a formula or state that the limit does not exist

  • What are some methods for finding limits algebraically?
  • What are at least five properties of limits?
  • What are the properties of limits of composite functions?
  • What is the Squeeze Theorem?
  • What are the steps to showing that a limit does not exist?

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.

Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

Review this section of the textbook which discusses properties of limits that will help you find limits algebraically, such as by using algebraic reduction or rationalization. This chapter also introduces an important theorem called the Squeeze (Squeezing) Theorem, which allows you to compare limits of functions to determine a difficult limit. See Figure 2d.1.

Figure 2b.1.aFigure 2b.1.b

Figure 2d.1.

Graphing a function or exploring a table of values to determine a limit can be cumbersome and time-consuming. When possible, it is more efficient to use the properties of limits, which is a collection of theorems for finding limits.

Knowing the properties of limits allows us to compute limits directly. We can add, subtract, multiply, and divide the limits of functions as if we were performing the operations on the functions themselves to find the limit of the result. Similarly, we can find the limit of a function raised to a power by raising the limit to that power. We can also find the limit of the root of a function by taking the root of the limit. Using these operations on limits, we can find the limits of more complex functions by finding the limits of their simpler component functions.

Properties of Limits

Let  a ,  k ,  A , and  B represent real numbers, and  f and  g be functions, such that:

 \lim_{x\rightarrow a}f(x)=A and  \lim_{x\rightarrow a}g(x)=B .

For limits that exist and are finite, the properties of limits are summarized in this table:

Constant,  k  \lim_{x\rightarrow a}k=k
Constant times a function  \lim_{x\rightarrow a}\left [ k\cdot f(x) \right ]=k\lim_{x\rightarrow a}f(x)=kA
Sum of functions  \lim_{x\rightarrow a}\left [ f(x)+g(x) \right ]=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a} g(x)=A+B
Difference of functions  \lim_{x\rightarrow a}\left [ f(x)-g(x) \right ]=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a} g(x)=A-B
Product of functions  \lim_{x\rightarrow a}\left [ f(x)\cdot g(x) \right ]=\lim_{x\rightarrow a}f(x)\cdot \lim_{x\rightarrow a} g(x)=A\cdot B
Quotient of functions  \lim_{x\rightarrow a} \frac{f(x)}{g(x)} =\frac{\lim_{x\rightarrow a} f(x)}{\lim_{x\rightarrow a}g(x)}=\frac{A}{B},B\neq 0
Function raised to an exponent  \lim_{x\rightarrow a} \left [f(x) \right ] ^n\left [ \lim_{x\rightarrow a}f(x) \right ]^n=A^n , where  n is a positive integer
 n th root of a function,
where  n is a positive integer
 \lim_{x\rightarrow a}\sqrt[n]{f(x)}=\sqrt[n]{\lim_{x\rightarrow a}\left [ f(x) \right ]}=\sqrt[n]{A}
Polynomial function  \lim_{x\rightarrow a}p(x)=p(a)

To review, see Properties of Limits.

 

2e. State whether a given function is continuous at a point and use the properties of continuity to find limits and values of related functions

  • What are some reasons why continuity is important for calculus?
  • What are three examples of continuous functions?
  • What are two examples of functions that are not continuous at at least one point?

A continuous function can sometimes be described as a graph that you can draw without picking up your pencil. Continuous functions play an important role in calculus because they are one of the assumptions made in many theorems. Functions can have more than one type of discontinuity: a hole (point discontinuity) or a break (jump discontinuity). See Figure 2e.1.

Most functions students encounter in calculus will be continuous on some interval (having only a finite number of discontinuities), but some special types of functions can be discontinuous everywhere. 

Figure 2c.1.

Figure 2e.1.

To review, see Continuous Functions.

 

2f. Use the Intermediate Value Theorem to determine the number of times a function has a given value

  • What is the Intermediate Value Theorem?
  • Why does the Intermediate Value Theorem require that a function be continuous?

In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f and f at some point within the interval.

One of the important and useful consequences of continuity is that if you have a continuous function with values f(a) and f(b), then every value between f(a) and f(b) must also be a value of the function at some point between a and b. For example, if we know that f(a) is negative and f(b) is positive, then at some point between a and b, the function must be zero. We can apply this to any value between f(a) and f(b). See Figure 2f.1.

Figure 2d.1.

Figure 2f.1.

To review, see Continuous Functions.

 

2g. Approximate the roots of functions using the Bisection Algorithm

  • In your own words, what is the Bisection Algorithm and how does it work?
  • What are the steps to estimating a root using the Bisection Algorithm?
  • What is an example of a function for which the Bisection Algorithm will not work?

The Bisection Algorithm is a method of finding a root or zero of a function by estimation, using the Intermediate Value Theorem. See Figure 2g.1.

First, apply the intermediate value to the whole interval to test whether there is a sign change between the endpoints. Then split the interval and check the midpoint (the point of bisection). If the function value is positive, replace the previous positive endpoint; if it is negative, replace the previous negative endpoint. Then repeat. Each time the length of the interval will get shorter and shorter until you are close enough to estimate the root. If you need more accuracy, continue the process longer.

Figure 2e.1.

Figure 2g.1.

To review, see Continuous Functions.

 

2h. State the epsilon-delta definition of limit

  • What is the ε-δ definition of a limit?

As we noted above, limits are a fundamental concept in calculus that underpin many other concepts. For a limit to exist for a function, as x approaches a specific value c, so that the difference between x and c is an arbitrarily small value δ, then the function value f(x) approaches some value that is arbitrarily close with a distance ε to the limiting value L. See Figure 2f.1.

Figure 2f.1.

Figure 2h.1.

To review, see Definition of a Limit.

 

2i. For a given epsilon, find the required delta graphically and algebraically for linear and quadratic functions

  • If you are trying to prove the limit for a linear function f(x)=mx+b, what is the relationship between ε and δ?
  • Explain the process to estimate the relationship between ε and δ if the function is not linear?
  • Is there only one correct answer to that relationship for the process described above?

To estimate the relationship between ε and δ for a linear function, you must first know the limit at the desired point, which is easy since the function is continuous everywhere. Then place the limit into the expression |f(x)-L|=ε. Factor out the expression x−c for whatever value of c you are using and replace that with δ. The two will always differ by a factor of m (the slope). 

For nonlinear functions, you must first choose an interval for δ. Then estimate the value at the endpoint further away from the limiting value if you are estimating ε (use the closer value if you are estimating δ), in order to reduce your function to something linear.

To review, see Definition of a Limit.

 

Unit 2 Vocabulary

  • Average velocity
  • Bisection
  • Bisection algorithm
  • Break
  • Composition of functions
  • Continuity
  • Continuous function
  • Hole
  • Instantaneous velocity
  • Intermediate Value Theorem
  • Jump discontinuity
  • Limit
  • Linear function
  • Point discontinuity
  • Quadratic function
  • Rationalize
  • Root
  • Squeezing and the Squeeze Theorem

Unit 3: Derivatives

3a. Find the derivative of a function f(x) using the definition

  • What is the formal definition of the derivative?
  • How does this formal definition relate to the slope of the tangent line?

The formal definition of the derivative is related to the equation for the slope of a tangent line: it is just the difference quotient in the limit. See Figure 3a.1. It is derived from the slope of the secant line, but as the difference between the two points goes to zero. The derivative produces a function that can be evaluated at any point on the graph to find the slope of the tangent line.

Figure 3a.1.
Figure 3a.1.

To review, see The Definition of a Derivative.

 

3b. Recognize and use the common equivalent notations for the derivative

  • What are three notations for the first derivative?
  • How does each definition emphasize different meanings of differentiation?
  • Define magnification error.

There are three common notations for the first derivative.

  1. The prime notation (f'(x)) emphasizes the relationship to the function f(x) and is a commonly used formula because it requires the fewest additional characters.
  2. The differential notation (d(f)) emphasizes that the derivative is an operation on the function f, but it does not specify which variable the function is in.
  3. The Leibniz notation ( \frac{df}{dx}) emphasizes the relationship to the slope by representing the difference in the two function values divided by the difference in the two x values just as the slope formula.

Magnification factor f'(x) is the magnification factor of the function f for points close to x. If a and b are two points very close to x, then the distance between f(a) and f(b) will be close to f'(x) times the original distance between a and b: f(b) − f(a) ≈ f'(x)(b − a).

To review, see The Definition of a Derivative.

 

3c. State the graph and rate meanings of a derivative

  • What are some common interpretations of the first derivative?
  • Explain why it makes sense that the derivative of a constant function must logically be zero without using the formula?
  • Define horizontal tangent lines.

There are many common interpretations of the derivative depending on the area of application. In general, the derivative is the slope of the tangent line or the instantaneous rate of change. In a physical context, the derivative is the velocity of a particle moving on a path. The derivative of the velocity is the rate of change of the velocity, or the acceleration.

In a business context, rate of change is signaled by the term marginal. The derivative of a cost function is the rate of change of the cost, or marginal cost. The derivative of the profit function is the marginal profit.

Horizontal tangent lines exist where the derivative of the function equals zero. By definition, the derivative gives the slope of the tangent line. Since horizontal lines have a slope of zero, when the derivative is zero, the tangent line is horizontal.

To review, see Introduction to Derivatives.

 

3d. Estimate a tangent line slope and instantaneous rate of change from the graph of a function

  • How would one estimate the slope of a tangent line to a point on a graph?
  • Why is the slope equal to zero especially important for tangent lines?
  • What does it mean if the slope of the tangent line is equal to zero?

The slope of a tangent line to a point can be estimated from a graph by drawing a line that touches the graph at a single point and trends in the direction that the graph is going at that instant. See Figure 3d.1. You can also imagine drawing a secant line and drawing the two points closer and closer together. By extending the line, it is possible to estimate two points on the line in order to calculate an equation that approximates the tangent line.

Figure 3d.1.

Figure 3d.1.

To review, see Introduction to Derivatives.

 

3e. Write the equation of the line tangent to the graph of a function f(x)

  • What is the equation to find the slope of a tangent line at a given point?
  • What is the point-slope form of the line? The slope-intercept form?
  • How does the slope of the tangent line relate to the derivative of the function?
  • What are all the steps in the process to find the equation of the tangent line to a graph at a point?

The slope of the tangent line is the derivative of the function at the point. We can find the slope exactly at any point if we have the equation of the graph. Using the difference quotient equation, we can find the slope of the tangent line (the instantaneous rate of change), and with the slope and the ordered pair from the graph itself, we can write the equation of the tangent line.

To review, see Introduction to Derivatives.

 

3f. Calculate the derivatives of the elementary functions

  • What is the power rule?
  • What is the derivative for sin(x)? What is the derivative for cos(x)?
  • What are the circumstances that can produce a non-differentiable function?
  • What are the product and quotient rules?
  • What are some properties of derivatives?
  • What are the derivatives of other trigonometric functions?
  • What is the chain rule?
  • What is the derivative of an exponential function?
  • What is the derivative of a logarithmic function?

While the derivative of functions can be calculated from the definition, different functions produce different patterns when applying the definition. To quickly calculate derivatives, we learn the patterns, or short-cuts, rather than going through the longer process.

As functions become more complex, we can layer these rules on top of each other, from simple rules to more complex rules, that allow us to calculate the derivative of every type of differentiable function.

To review, watch Applying the Product Rule for Derivatives and Chain Rule Examples. Then, see Derivatives, Properties and FormulasDerivative Patterns, and Some Applications of the Chain Rule.

 

3g. Calculate second and higher derivatives and state what they measure

  • What is the process for calculating a second derivative? A third derivative?
  • Describe at least two notations for a second and third derivative.
  • What is the meaning of the second derivative of a position function?

Higher derivatives exist and can have many uses as we try to understand the behavior of functions in greater detail. We can find higher derivatives by taking derivatives of derivatives. To find the second derivative, take the derivative of the first derivative. To find the third derivative, take the derivative of the second derivative, and so on. Position functions give us the clearest common meaning of a second derivative: the second derivative of position is acceleration (how fast the rate of change, velocity, is changing).

To review, see Derivative Patterns.

 

3h. Differentiate compositions of functions using the chain rule

  • What is the chain rule?
  • What are some examples of functions that require the use of the chain rule to find their derivative?
  • How do you write the chain rule in Leibniz's notation?
  • What is the chain rule in composition of functions form?

The chain rule is a method of finding the derivatives for complex functions built up from function composition. The chain rule is also useful when using tables of derivatives for special functions because it allows you to recognize a pattern and apply it to a wide variety of similar functions with the same pattern.

To review, see The Chain Rule and Some Applications of the Chain Rule.

 

3i. Use the chain rule to solve applied questions

  • What are some applications where the chain rule can be applied?

To solve applied problems, first you need to identify the kind of problem. Set up the equation to model the scenario if it’s not given in the problem itself (you may need to apply some algebra skills here). Take the derivative using the chain rule as needed and answer the question. Problems may require one derivative (velocity, rate of change, etc.) or may require more than one derivative

To review, see The Chain Rule and Some Applications of the Chain Rule.

 

3j. Calculate the derivatives of functions given as parametric equations and interpret their meanings geometrically and physically

  • What is a parametric equation?
  • How do parametric equations relate to traditional functions?
  • How do you calculate  \frac{dy}{dx} for a set of parametric equations x(t) and y(t)?

Parametric equations can describe a path that is not a function using two equations, x(t) and y(t), that change with time and are independent functions. See Figure 3j.1. Because each component is a function, derivatives can be found using normal derivative rules and then recombined to find the slope of the tangent line to a curve at a specified point.

Figure 3i.1.

Figure 3j.1.

To review, see Some Applications of the Chain Rule.

 

3k. State whether a function, given by a graph or formula, is continuous or differentiable at a point or on an interval

  • What are three features of a graph that can make the function non-differentiable?
  • Draw examples of three functions that are not differentiable at x=0.
  • What are some examples of functions that are differentiable everywhere?
  • On what interval(s) is the function f(x)=1/x differentiable?

A function is considered differentiable on any interval where it is smooth, continuous, and contains no cusps or vertical tangent lines. A differentiable function has a continuous first derivative. While some common functions are not differentiable everywhere, they are differentiable on some interval, and so it is possible to apply our differentiation rules and properties as long as we avoid any non-differentiable points. See Figure 3k.1.

Figure 3j.1.

Figure 3k.1.

To review, see Derivatives, Properties, and Formulas.

 

3l. Solve related rate problems using derivatives

  • How are related rate problems related to an application of the chain rule?
  • What are the steps to solving a related rate problem?

Related rate problems are often based on geometric relationships for area or volume. The first step is to write an equation that relates the variables (not time) to each other. The next step is to take the derivative of your equation with respect to time using the chain rule.

Finally, substitute the available information into the derivative equation and solve for the missing piece. Related rate problems allow us to describe how one variable changes with respect to another variable, which is itself changing over time: such as the size of the circle as a ripple in a pond grows after dropping in a rock. See Figure 3l.1.

Figure 3k.1.

Figure 3l.1.

To review, see Related Rates.

 

3m. Approximate the solutions of equations by using derivatives and Newton's method

  • What are the steps of Newton's Method?
  • What are you trying to find when you apply Newton's Method?
  • What are some situations where Newton's Method will not work?

Newton's Method is a means of estimating the root (zero) of a function using derivatives. An initial point is guessed. The ratio of the original function divided by its derivative at the same point is subtracted from the initial guess to obtain a new guess. The process is repeated until the points remain sufficiently stable.

Newton's method can produce several errors such as when the derivative is zero. The initial guess may be a poor one, sending the point to infinity. Repeating loops or chaotic behavior can occur. It may be possible to correct these errors by guessing a different initial starting point or observing the graph of the function to make a guess closer to the correct value.

To review, see Newton's Method for Finding Roots.

 

3n. Approximate the values of difficult functions by using derivatives

  • What are some reasons why a linear approximation to a function may be sufficient?
  • What is the formula for a linear approximation?

Sometimes it is convenient to make a linear approximation to a complex function near a point. Linear functions are easy to work with and can be calculated by hand relatively easily. If you are estimating a nearby point, the difference between the estimate made from the tangent line and the true value will remain small. In these cases, we can estimate the change in the y-value as Δy as the derivative of the function at that point (the slope) times the change in the x-value as Δx, as long as Δx remains small.

To review, see Linear Approximation and Differentials.

 

3o. Calculate the differential of a function using derivatives and show what the differential represents on a graph

  • What is an example of a difficult function and a value that can be approximated by differentials?
  • What are some reasons why a linear approximation to a function may be sufficient?
  • How do the formulas for the differential differ from the formula for a linear approximation?

Sometimes it is convenient to make a linear approximation to a complex function near a point. Linear functions are easy to work with and can be calculated by hand relatively easily. If you are estimating a nearby point, the difference between the estimate made from the tangent line and the true value will remain small.

In these cases, we can estimate the change in the y-value as Δy as the derivative of the function at that point (the slope) times the change in the x-value as Δx, as long as Δx remains small. See Figure 3o.1.

Figure 3n.1.
Figure 3o.1.

To review, see Linear Approximation and Differentials.

 

3p. Calculate the derivatives of really difficult functions by using the methods of implicit differentiation and logarithmic differentiation

  • How does implicit differentiation relate to the chain rule?
  • What are the steps to performing logarithmic differentiation?

In implicit differentiation, we are unable to solve an equation explicitly, so we can apply more common differentiation rules. Instead, we assume that y is a function of x, and when we take the derivative term-by-term, we apply the chain rule any time we take the derivative of y and apply the product or quotient rule whenever x and y appear in the same term. We can then solve for  \frac{dy}{dx} to find an expression for the slope of the tangent line at any point on the curve that depends on both variables (because the function is not explicit, there may be points on the curve with the same x but different y values).

Logarithmic differentiation is a technique we can use for functions where the base of an exponential function contains x, as does the exponent of the function. Since we do not have any rules for a function like xx, this process allows us to apply logarithm rules to convert the exponent into a product, and then proceed with our more common rules. In implicit differentiation, we are unable to solve an equation explicitly for y so that we can apply more common differentiation rules.

Instead, we assume that y is a function of x, and when we take the derivative term-by-term, we apply the chain rule any time we take the derivative of y, and the product or quotient rule whenever and appear in the same term. We can then solve for  \frac{dy}{dx} to find an expression for the slope of the tangent line at any point on the curve that depends on both variables (because the function is not explicit, there may be points on the curve with the same x but different y values.

Logarithmic differentiation is a technique we can use for functions where the base of an exponential function contains x, as does the exponent of the function. Since we do not have any rules for a function like xx, this process allows us to apply logarithm rules to convert the exponent into a product, and then proceed with our more common rules.

To review, see Implicit and Logarithmic Differentiation.

 

Unit 3 Vocabulary

  • Acceleration
  • Chain rule
  • Composite function
  • Continuous function
  • Cusp (corner)
  • Derivative
  • Differentiable
  • Differential
  • Exponential function
  • First derivative
  • Horizontal tangent line
  • Implicit differentiation
  • Instantaneous rate of change
  • Logarithmic differentiation
  • Logarithmic functions
  • Magnification factor
  • Marginal cost
  • Marginal profit
  • Newton's Method
  • Parametric equations
  • Product rule
  • Quotient rule
  • Rate
  • Related rates
  • Second derivative
  • Slope
  • Smooth function
  • Tangent line
  • Trigonometric functions
  • Velocity
  • Vertical tangent line

Unit 4: Derivatives and Graphs

4a. State whether a given point on a graph is a global/local maximum/minimum

  • What distinguishes a global maximum from a local or relative maximum?
  • What distinguishes a global minimum from a local or relative minimum?

Global extreme values (extrema) are either larger (maximum) or smaller (minimum) than any value on a function. Local or relative extreme values are larger (maximum) or smaller (minimum) than and nearby value on a function. See Figure 4a.1. If the derivative of a function at a point is equal to some finite positive or negative value, then it is not an extreme value, either global or local.

Figure 4a.1.

Figure 4a.1.

To review, see Finding Maximums and Minimums.

 

4b. Find critical points and extreme values (max/min) of functions by using derivatives

  • What conditions must be met for a number to be a critical number (critical point)?
  • What does the Extreme Value Theorem say about finding global extreme values on a closed interval?

Critical numbers are values in the domain of a function where the derivative at that point is either equal to zero or undefined. These values are candidates for identifying possible locations of extrema. If the interval is open, then these are the only possible locations for extreme values, and it is possible that there are no extreme values.

If the interval is closed, then the endpoints are included in the list of critical numbers, and according to the Extreme Value Theorem, the function must attain its global maximum and minimum at some point in the closed interval.

To review, see Finding Maximums and Minimums.

 

4c. Determine the values of a function guaranteed to exist by Rolle's Theorem and by the Mean Value Theorem

  • What is Rolle's Theorem? What conditions must be satisfied?
  • What is the Mean Value Theorem? What conditions must be satisfied and what does it guarantee?

Rolle's Theorem says that if a function has the same value at both endpoints of an interval, and if the function is continuous, then there must be at least one point in the interval where the derivative is zero. In order to apply Rolle's Theorem, you must check both the values at the endpoints (to see that they agree) and test that the function is continuous on the given interval. See Figure 4c.1.

Figure 4c.1.
Figure 4c.1.

The Mean Value Theorem says that if a function is continuous on a closed interval and differentiable on the open interval, then there is at least one point on the open interval where the slope of the tangent line to the curve is the same as the slope of the secant line connecting the endpoints of the interval. See Figure 4c.2.

Figure 4c.2.

Figure 4c.2.

To review, see The Mean Value Theorem and Its Consequences.

 

4d. Use the graph of f(x) to sketch the shape of the graph of f'(x)

  • When the graph of a function is increasing, what do we know about the derivative of the function?
  • When the graph of a function is decreasing, what do we know about the derivative of the function?
  • When the graph of a function is constant, what is the derivative of the function?
  • If the graph of a function has a cusp, what do we know about the derivative of the function?
  • If the graph of a function has an extreme value, what do we know about the derivative of the function?
  • Define monotonic.

We can say important things about the derivative of a function and use that information to sketch the derivative function even if we do not have an equation. If the original function is increasing, then we know that the derivative is positive.

Likewise, if the original function is decreasing, then we know the derivative is negative. If the function is constant or has an extreme value, then the derivative is zero. If the derivative has a cusp, then there is a break in the derivative. See Figures 4d.1. and 4d.2.

Figure 4d.1.

Figure 4d.1.

Figure 4d.2.

Figure 4d.2.

A monotonic function is a function which is either entirely nonincreasing or nondecreasing. A function is monotonic if its first derivative (which need not be continuous) does not change sign.

To review, see The First Derivative and the Shape of a Function f(x).

 

4e. Use the values of f′(x) to sketch the graph of f(x) and state whether f(x) is increasing or decreasing at a point

  • When the derivative of a function is zero, what do we know about the original function?
  • When the derivative of a function is undefined, what do we know about the original function?
  • When the derivative of a function is positive, what do we know about the original function?
  • When the derivative of a function is negative, what do we know about the original function?
  • If the derivative of a function is zero at a point, how do we determine if the point is an extreme value or something else?

When the derivative of a function is zero, it corresponds to a critical point, which could indicate an extreme value. If it is zero over an interval, then the graph is constant over that interval.

If the derivative of the function is positive, then the function is increasing. If the derivative of the function is negative, then the original function is decreasing. At a minimum, the derivative is decreasing on the left and increasing on the right of the point. At a maximum, the derivative is increasing on the left and decreasing on the right. See Figure 4e.1.

Figure 4e.1.

Figure 4e.1.

To review, see The First Derivative and the Shape of a Function f(x).

 

4f. Use the values of f'(x) to determine the concavity of the graph of f(x)

  • What is concavity?
  • Draw a section of a graph that is concave down. Concave up?
  • What happens to the value of the second derivative when the graph is concave up? Down?
  • What is an inflection point?
  • What is the value of the second derivative at an inflection point?

Concavity is a way of describing the way in which a graph curves. The graph curves facing upward like a bowl when the second derivative is positive. The graph curves downward like a hill when the second derivative is negative. See Figure 4f.1. When the direction of the curve changes from up to down, this is called an inflection point, and at that point, the second derivative is zero.

Figure 4f.1.

Figure 4f.1.

To review, see The First Derivative and the Shape of a Function f(x).

 

4g. Use the graph of f(x) to determine if f'(x) is positive, negative, or zero

  • What is the sign of the second derivative on the interval (−∞, 0) on the graph of f(x)=1/x?
  • What is the sign of the second derivative on the interval (−∞, ∞) on the graph of f(x)=x2?
  • What is the value of the second derivative at the point x=0 on the graph of f(x)=x3? What is the name of this point?

When the graph curves upward like a bowl, the second derivative is positive. When the graph curves downward like a hill, the second derivative is negative. When the direction changes, the point at which the change happens is called an inflection point, and the second derivative there is zero. See Figure 4g.1.

Figure 4g.1.
Figure 4g.1.

To review, see The First Derivative and the Shape of a Function f(x).

 

4h. Solve maximum and minimum problems by using derivatives

  • What is the first derivative test?
  • What is the second derivative test?
  • If a point represents a local maximum, what do we know about the first and second derivatives at that point?
  • If a point represents a local minimum, what do we know about the first and second derivatives at that point?

In order to solve a problem that involves extreme values like maxima or minima, first set up the equation that models the problem. Find the first derivative and set it equal to zero to determine the critical values that could be candidate solutions. Determine if any solutions can be eliminated based on problem restrictions.

Then use the first derivative test, or the second derivative test to determine if the point represents a maximum or minimum. If the second derivative is zero, the test is inconclusive. Diagrams are often useful for setting up the initial equation to be solved.

To review, see Applied Maximum and Minimum Problems.

 

4i. Restate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer

  • What are some possible units of volume? How do you determine which to use from the problem?
  • What are some common geometry formulas that can guide you in setting up equations?
  • What are some useful techniques for organizing information in a problem?
  • What are some techniques for identifying the correct solution if the solution process produces multiple possible solutions?

When solving an applied problem, break down the verbal description of the problem one phrase at a time. It can be helpful to draw a diagram, particularly for geometry problems. It may also help to make a table of values and identify any formulas you know that seem to apply to the problem.

Test multiple solutions to determine which is the maximum or minimum and report that value that was requested by the problem. Check units. Do unit conversions before writing down your equations, so that all the values measuring similar things have the same units (do not mix inches with feet).

To review, see Applied Maximum and Minimum Problems.

 

4j. Determine asymptotes of a function by using limits

  • What kind of asymptote is associated with a limit as x goes to infinity if the limit is not itself infinity?
  • What kind of asymptote is associated with a limit as x goes to a finite value when the limit goes to infinity?
  • What are some of the algebraic tricks that can be used to find the value of a limit at infinity?
  • What are some examples of functions that don't have limits at infinity?

If the limit of a function goes to a finite value as x approaches infinity (or negative infinity), this limiting value represents a horizontal asymptote.

A function can have more than one horizontal asymptote. A function that has an infinite limit (or a one-sided infinite limit) at a point has a vertical asymptote. Asymptotes that are non-constant can be found through long division of rational functions, which may result in linear or non-linear asymptotes.

Linear asymptotes that are not vertical or horizontal may be referred to as slant or oblique asymptotes. Some functions, such as oscillating ones, do not converge to a single value as x increases (or decreases), and therefore have no limit at infinity.

One useful way of finding limits at infinity for a rational function is to divide every term in both the numerator and the denominator by the largest power term in the entire expression, then determine the limit of each term at infinity using properties of limits.

To review, see Infinite Limits and Asymptotes.

 

4k. Determine the values of indeterminate form limits by using derivatives and L'Hôpital's Rule

  • Give some examples of indeterminant forms for limits?
  • What kinds of indeterminant forms does L'Hôpital's Rule apply to?
  • What are some tricks to convert other indeterminant forms into one that L'Hôpital's can be applied to?

L'Hôpital's Rule states that you can find the limit of the form  \frac{\infty}{\infty} or  \frac{0}{0} by taking the derivative of the numerator and the denominator functions separately, and then re-evaluating the limit for  \frac{f'(x)}{g'(x)} at the same point.

The process can be repeated as often as needed until a determinant form is obtained. When another kind of indeterminant form is obtained, they must be rewritten into one of the above forms, such as turning a product into a ratio by moving the reciprocal of a product into the denominator, or when an exponent is involved, by taking the logarithm of the entire expression.

To review, see L'Hopital's Rule.

 

Unit 4 Vocabulary

  • Asymptote
  • Closed interval
  • Concave down
  • Concave up
  • Concavity
  • Critical number (or critical point)
  • Decreasing
  • Extreme Value Theorem
  • Extreme value and an extremum (or multiple extrema)
  • First derivative test
  • Global (absolute) maximum
  • Global (absolute) minimum
  • Horizontal asymptote
  • Increasing
  • Indeterminate form
  • Infinite limit
  • Inflection point
  • L'Hôpital's rule
  • Limit
  • Linear asymptote
  • Local maximum
  • Local minimum
  • Mean value theorem
  • Monotonic
  • Nonlinear asymptote
  • Oblique asymptote
  • Relative maximum
  • Relative minimum
  • Rolle's Theorem
  • Second derivative test
  • Slant asymptote
  • Vertical asymptote

Unit 5: The Integral

5a. Use sigma notation to represent sums

  • What is the purpose of using sigma notation?
  • What is an example of a summation with a lower limit of 1 and an upper limit of 5 using j as the index?
  • Define counter or index variable.
  • What are some properties of summations?
  • How would you write 12+23 +34 + 45 + ... +89 in sigma notation with k as the index?

Sigma notation, or summation notation, is a shorthand way of writing the sum of many terms in a shortened form. The bottom limit (under the sigma) states which value the index begins with and declares which variable is the index. The top limit (above the sigma) tells you what is the last index value to use in your sum. The expression that follows the sigma (the summand) expresses the pattern to be followed as values are replaced with different index values. See Figure 5a.1.

Figure 5a.1.
Figure 5a.1.

To review, see Sigma Notation and Riemann Sums.

 

5b. Approximate the areas of regions

  • What is the general idea behind approximating the area under a curve?
  • When is the process exact?
  • What happens to the accuracy as the number of rectangles increases?
  • How does the right-hand rule differ from the left-hand rule?
  • What is the procedure, in words, for the midpoint rule?

Areas of complex regions can be approximated by dividing up the region using shapes we already have geometric formulas for. In calculus, we generally use rectangles for a first approximation to estimate the area of the region. Divide up the x-axis into evenly sized segments \Delta x, and using the height of the function \( f(x_i) \) as the second dimension. Selecting the right side, the left side, or the middle of the region provides a slightly different estimate, but as the number of rectangles used increases, so does the accuracy of the estimate.

To review, see Introduction to Integration.

 

5c. Approximate areas by Riemann sums

  • How would you evenly partition the interval [3,7] for n=6?
  • What is the upper sum estimate for the area under the curve f(x)=1/x on the interval [3,7] using six rectangles?
  • What is the lower sum estimate for the area under the curve f(x)=1/x on the interval [3,7] using six rectangles?
  • What is the procedure for finding a Riemann sum for general n? Describe the steps?

Riemann sums are used to estimate the area under a curve by cutting up the region into small rectangles, a process that becomes more accurate with more rectangles. The upper sum for a region is an overestimate of the area, and the lower sum for a region is an underestimate.

As the number of rectangles gets larger, both calculations for area will converge to the same value, the true area. Procedurally, divide up the interval over which the area will be calculated into a partition with n subintervals. 

If you are calculating an upper sum, determine at which endpoint in the partition (right or left on each interval) will produce the larger area, and then evaluate the function curve at that point. Multiply the height of the function at each partition's upper value by the width of each partition. Then add up the areas to obtain the estimate. See Figure 5c.1.

Figure 5b.1.
Figure 5c.1.

To review, see Sigma Notation and Riemann Sums.

 

5d. Translate an area under a curve into a definite integral

  • What is the definite integral version of the shaded area in the figure? See Figure 5c.2.
  • How do definite integrals relate to Riemann sums? Give at least two ways.
  • What does it mean graphically if the area of a definite integral turns out to be negative?
  • How can we interpret definite integrals with some area positive and some negative in terms of total net change?
  • Define the integrand function.

Definite integrals are a way of representing area (and other concepts) instead of using Riemann sums. In general, the interval becomes the limits of integration on the integral symbol, with the smaller value on the bottom, and the larger one on the top.

The function that defines the height of the area goes after the integral sign. The expression is closed out by dx. See Figure 5d.1. When the function is above the axis, the area calculated will be positive. When the function is below the axis, the area calculated will be negative. 

Figure 5c.1.Figure 5d.1.

Figure 5c.2.

Figure 5d.2.

To find the geometric area, all components will need to be positive before adding. To find the total net change, add up the signed values. See Figure 5d.3.

Figure 5c.3.
Figure 5d.3.

To review, see The Definite Integral.

 

5e. Evaluate definite integrals geometrically using graphs of functions

  • What is the definite integral that represents the area under the curve shown in the figure? See Figure 5e.1.
  • How could we use geometry to find the area represented by the integral?
  • What are some geometric formulas we could apply to calculating areas? Draw an example of each.

Because one interpretation of integrals is that of area, we can use geometry to calculate integrals. For instance, a straight-line function might produce an area of a rectangle (if it is horizontal), a triangle (if it goes through zero on one endpoint), or a trapezoid. Or, we can combine several of these to find the area under a piecewise function. See Figure 5e.2.

Figure 5d.1.Figure 5e.1.

Figure 5d.2.
Figure 5e.2.

To review, see Areas, Integrals, and Antiderivatives and Properties of the Definite Integral.

 

5f. Determine whether a given function is integrable

  • What kinds of functions are integrable?
  • Give an example of a function that is not continuous but is integrable.

More functions are integrable than are differentiable. To be integrable, a function need only be piecewise continuous (with a finite number of pieces) and bounded (no part of it goes to infinity on the interval we are interested in). Using properties of integrals, we can break out each piece on which the function is continuous, and compute the integral separately on each piece, and then add the pieces back up. See Figure 5f.1.

Figure 5e.1.
Figure 5f.1.

To review, see Properties of the Definite IntegralAreas, Integrals, and Antiderivatives, and Finding Antiderivatives.

 

5g. Use antiderivatives to evaluate a function

  • What is the relationship between derivative rules and antiderivative rules?
  • What is the power rule for antiderivatives?
  • What is the exception to the power rule (for antiderivatives)?
  • What are the common trigonometric antiderivatives?
  • What is the antiderivative rule for exponential functions?
  • Why do we add a constant of integration to indefinite integrals?

If we want to evaluate \int x^2 dx, we apply the power rule for antiderivatives, which says that \int x^n dx = \frac{x^{n+1}}{n+1}+C. In this situation, n=2, and following the rule, we end up with \frac{x^3}{3}+C. The constant of integration is required because the derivative of any constant is zero, and so this is information we are unable to recover completely. If you took the derivative of \frac{x^3}{3}+1 or \frac{x^3}{3}-6 or \frac{x^3}{3}, they would all produce the same function, x^2.

Some common antiderivative rules:

  • \int x^n dx = \frac{x^{n+1}}{n+1}+C for x\neq -1.
  • \int \frac{1}{x} dx = ln|x| +C
  • \int \sin x dx = -\cos x +C
  • \int \cos x dx = \sin x +C
  • \int \sec^2 x dx = \tan x +C
  • \int \sec x \tan x dx = \sec x +C
  • \int e^x dx = e^x +C

For more expressions with more terms, separated by plus or minus signs, integrate term by term.

To review, see Areas, Integrals, and Antiderivatives and Finding Antiderivatives.

 

5h. Find antiderivatives by changing the variable and using tables

  • What is an example of an integral that requires substitution to integrate?
  • What is an example of an integral that uses change of variables, but where the inside function rule-of-thumb will not work?
  • How can substitution be used to assist us when using tables of integrals?

Many functions are built up from simple functions into more complicated functions. The change of variable technique permits us to simplify the functions again so that basic antiderivative rules can be applied. They also permit us to look up the general form of an integral in a table of integrals and match up the function we have with the form in the table.

To review, see Areas, Integrals, and Antiderivatives and Using Tables to Find Antiderivatives.

 

5i. Use the Fundamental Theorem of Calculus to evaluate definite integrals

  • What is the Fundamental Theorem of Calculus and why is it so important?

The Fundamental Theorem of Calculus allows us to relate the value of an integral to the antiderivative of the function being evaluated at the limits of integration. 

To review, see Areas, Integrals, and Antiderivatives and The Fundamental Theorem of Calculus.

 

5j. Differentiate integrals

  • What is the second part of the Fundamental Theorem of Calculus (Leibniz's Rule)?
  • Explain how to apply the second part of the Fundamental Theorem to find the derivative for  f(x)=\int_0^{sin(x)}{2 t^2} .

Area functions (or accumulation functions) and functions defined by integrals with variables in the limits. The second part of the Fundamental Theorem allows us to find derivatives for such functions, even when the limits of integration are functions themselves, rather than just a plain variable by incorporating the use of the chain rule.

To review, see The Fundamental Theorem of Calculus.

 

5k. Solve applied problems that involve generalized area, that is, distance, work, and so forth

  • What are some examples of problems that can be solved by using generalized area? Express the integrals in terms of units rather than specific functions.

To calculate area, the function inside the integral represents the height of the function. By integrating, we obtain area (accounting for the second dimension). Similarly, if the function inside the integral represents area, then integrating finds the volume of the region. Since work is force times distance, and integral with a function for force produces work.

To review, see First Application of Definite Integrals.

 

5l. Find an area between two curves

  • How do we find the area between two curves if there are no points where they cross?
  • How does the technique above have to be modified if the curves do cross? What if they cross more than three times or more?

To find the area between two curves, first, it is necessary to determine which function is the larger (higher on the y-axis), and to determine if the functions cross at any point. The area between the curves is the difference between the two functions integrated over an interval (which may be specified or may be determined by points of intersection).

If the orientation of the functions changes (their positions switch at a cross point inside the interval), then the integral must be broken up into pieces using properties of integrals in order to change the orientation to obtain the geometric area. See Figure 5l.1.

Figure 5j.1.
Figure 5l.1.

To review, see First Application of Definite Integrals.

 

5m. Find the average (mean) value of a function

  • What is a geometric interpretation of the mean (average) value of a function?
  • What is the formula used to calculate the average value of a function?

We calculate the average value of a function by finding the area of the region, divided by the length of the interval between the limits of integration. This gives the average height of the function. It is also the height a rectangle would need to be in order to be equal to the area under the curve on the same interval. See Figure 5m.1.

Figure 5k.1.
Figure 5m.1.

To review, see First Application of Definite Integrals.

 

Unit 5 Vocabulary

  • Accumulation function
  • Antiderivative
  • Area function
  • Counter
  • Definite integral
  • Differential calculus
  • Fundamental Theorem of Calculus
  • Indefinite integral
  • Index variable
  • Integral calculus
  • Integrand
  • Integration
  • Leibniz's Rule
  • Limits of integration
  • Limits of summation
  • Lower sum
  • Mean (average) Value Theorem (for integrals)
  • Partition
  • Riemann sum
  • Sigma notation
  • Summand
  • Upper sum