This study guide will help you get ready for the final exam. It discusses the key topics in each unit, walks through the learning outcomes, and lists important vocabulary terms. It is not meant to replace the course materials!
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.