MA005: Calculus I

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.

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.

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 Section 1.1: 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.

To review, see Section 1.1: 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.

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 Section 1.2: 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 Section 1.2: 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.

To review, see Section 1.2: Lines in the Plane.

 

1f. 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 1f.1.

Figure 1f.1.

To review, see Section 1.3: Functions and Their Graphs.

 

1g. 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 1g.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.

To review, see Section 1.4: Combinations of Functions.

 

1h. 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 1h.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.

To review, see Section 1.4: Combinations of Functions.

 

1i. 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 Section 1.5: Mathematical Language.

 

1j. 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 Section 1.5: Mathematical Language.

 

1k. 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 Section 1.5: 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
• Tangent line
• Transformations of functions