USC Math 122 Calculus for Business and Social Sciences
Table of Contents
- Function Basics
- 1A. Linear Functions
- 1B. Functions and Function Notation
- 1C. Average Rate of Change
- 1D. Function Behavior
- Common Nonlinear Functions
- 2A-1. Exponential Functions
- 2A-2. Logarithmic Functions
- 2B. Power Functions
- 2C-1. Combining Functions with Operations
- 2C-2. Function Composition
- The Derivative Concept
- 3A-1. The Derivative at a Point
- 3A-2. The Difference Quotient
- 3A-3. Instantaneous Rate of Change
- 3B-1. Linearization & Differentials
- 3B-2. Linear Approximation
- The Derivative Function
- 4-A. The Derivative Function
- 4-B. Graphs of Derivatives
- 4-C. Basic Differentiation Rules
- Differentiation Rules
- Derivatives & Function Behavior
- 6A. Critical Points & First Derivatives
- 6B. Second Derivatives
- 6C. Second Derivatives & Function Behavior
- 7. Optimization & Applications
- 7A. Extreme Value Theorem
- Integration Basics
1A. Linear Functions
Instructions:
[list][*]Use the input boxes and slider tools for [math]m[/math] and [math]b[/math] to adjust the slope and vertical intercept of the line, respectively. [/*][*]Use the input box for [math]x_0[/math] to move the point on the line. Use the slider tool for [math]h[/math] to adjust how far away the second point is from the first point. [/*][*]Use the slider tool for [math]\Delta x[/math] and then for [math]\Delta y[/math] to see a visual representation of the slope of the line. [/*][/list]
Linear Functions
[b]Slope-Intercept Form: [/b]Linear functions have the form [math]y=mx+b[/math], where [math]m[/math] is the slope of the line and [math]b[/math] is the vertical intercept. [br][br][b]Slope: [/b]The slope between any two points [math](x_1,y_1)[/math] and [math](x_2,y_2)[/math] can be calculated by:[br][br][math]m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}[/math][br][br][b]Point-Slope Form: [/b]If a point [math](x_0,y_0)[/math] on the line and the slope [math]m[/math] of the line are both known, then you can write an equation of the line using the slope formula by leaving the coordinates of a second point as variables [math](x,y)[/math]. By multiplying both sides of the slope formula by the denominator to get rid of the fraction, we obtain the point-slope form of a linear equation: [br][br][math]y-y_0=m(x-x_0)[/math]
2A-1. Exponential Functions
Instructions:
[list][*]Use the input boxes on the left to set the parameters [math]y_0[/math] and [math]a[/math] for the exponential function of the form [math]f(x)=y_0\cdot a^x[/math]. Use the slider tools to adjust these values and see how they effect the graph/behavior of the function. [/*][*]Use the input box for [math]x_0[/math] to set the location of the point [math]P[/math]. Use the slider tool for [math]h[/math] to set the location for the point [math]Q[/math]. [/*][*]The text displays information to show how the [math]y[/math]-values of [math]P[/math] and [math]Q[/math] are related through the growth/decay factor. [/*][/list]
Exponential Functions
An [b]exponential function [/b]has the form [math]f(x)=y_0\cdot a^x[/math], where [math]y_0[/math] and [math]a[/math] are constants and [math]x[/math] is the independent variable. Notice that the [b]coefficient [/b][math]y_0[/math] determines the vertical intercept of the graph and that the [b]growth factor[/b] [math]a[/math] determines the "steepness" of the graph. [br][br]The fundamental characteristic of an exponential function is that changes in the input correspond to [i]repeated multiplication[/i] in the output. If you move 2 units in the [math]x[/math] direction (i.e., [math]h=2[/math]), then you have to multiply the [math]y[/math]-coordinate by the growth/decay factor 2 times. [br][br]Based on the values of [math]y_0[/math] and [math]a[/math], you should be able to predict whether the graph will be increasing or decreasing. Based on the shape of the graph, what can you say about the concavity?
3A-1. The Derivative at a Point
Instructions:
[list][*]Use the input box for f(x) to change the function formula. Use the input box for c or click and drag the point on the graph to move it along the graph. [/*][*]Adjust the slider tool for h to adjust how far apart (horizontally) the points P and Q are on the graph. Click the "[math]h\to0[/math]" button to bring Q closer to P. [/*][*]Use the Secant check box to show the secant line between P and Q and to display the slope of this line as a y-coordinate (in dark blue). Moving the slider tool for h or clicking the "h \to 0" button will leave a trace of this point on the graph (generating a graph of the difference quotient function). [/*][*]Use the Difference Quotient check box to show/hide the graph of the difference quotient function around the point P. Note that this function is undefined at x = c but that there is just a hole in the graph. What is the y-value of this point? [/*][*]Use the Tangent check box to show/hide the tangent line to the graph of f(x) at the point P. Observe the relationship between the hole in the difference quotient function and the slope of the tangent line. [/*][/list]
The Derivative at a Point
The derivative of a function f at a point x=c is the instantaneous rate of change of the function at that point. We define this instantaneous rate to be the [b]limit [/b]of the average rates of change over progressively smaller intervals around the point x=c. [br][br]The [b]limit [/b]concept is represented in the graph above by clicking the button [math]h\to0[/math] repeatedly. Theoretically, the point Q can be brought closer and closer to P without ever reaching it. As we bring the point Q closer to the point P we recalculate the slope of the [b]secant [/b]line (i.e., [b]average rate of change[/b]). As we do this we notice that the slopes of the secant lines seem to "[b]settle down[/b]" at (or [b]converge [/b]to) a particular value. This value is the slope of the [b]tangent [/b]line, which we also define to be the [b]instantaneous rate of change[/b], aka the [b]derivative[/b].
4-A. The Derivative Function
Instructions:
[list][*]Use the input box for f(x) to define a function. Use the input box and slider tool for c to move the point [math]P=(c,f(c))[/math] along the graph of the function. [/*][*]Use the "Estimate Slope" checkbox to show/hid a movable line segment. Click and drag the "move" point to estimate the tangent line at this point. Observe the point marked (x) on the right graph. Click the "Mark Point" button to leave a trace of this point. [/*][*]Repeat the above process at multiple points in the interval between a and b. This will generate several (estimated) points on the graph of the derivative function. [/*][*]Use the "Tangent" checkbox to show/hide the actual tangent line segment and trace the actual graph of the derivative function. [/*][/list]
4-A. The Derivative Function
The derivative function y = f'(x) is a [b]function [/b]whose outputs are the instantaneous rates of change of f(x) at each x value in its domain. This function has its own representations as a table, graph, or even a formula.
6A. Critical Points & First Derivatives
Instructions:
[list][*]Use the input box to define the function y = f(x). Use the input boxes for a and b to define the endpoints of the domain of f. Use the checkboxes to include/exclude the endpoints. [/*][*]Use the checkbox for Critical Points or Interior Extrema to show/hide critical points or interior extreme values, respectively. [/*][*]Use the checkbox for Monotonicity to highlight the section of the graph corresponding to the location of the point c. [/*][*]Use the Derivative checkbox to show the graph of the derivative function. [/*][/list]
Critical Points & Monotonicity
Our goal is to be able to describe [b]transitions [/b]in the [b]behavior [/b]of a function (e.g., from increasing to decreasing, from concave up to concave down). These changes in a function correspond to easier-to-find changes in its derivative(s). Because the [b]monotonicity [/b](increasing/decreasing behavior) of a function is tied to the [b]sign [/b](positive/negative) of its derivative, describing the monotonicity of f requires finding where the derivative f' is positive, negative, or zero. [br][br]Assuming certain nice properties of the derivative function (i.e., that it is continuous), the derivative could not change from positive to negative (or vice versa) without passing through a value of 0. A[b] critical point[/b] is a point where the derivative is equal to zero (or does not exist). Hence, a critical point is a location where the function [i]has the potential to change monotonicity[/i]. [br][br]The illustration above demonstrates that the critical points of a function give [i]potential candidates[/i] for [b]extreme values[/b] (maximum/minimum values). [i]While all extreme values occur at critical points, not every critical point results in an extreme value. So, we have to [b]test [/b]critical points to determine whether there is a maximum or minimum value there. [/i]
7A. Extreme Value Theorem
Instructions:
Use the input box for f(x) to define the function.
Extreme Value Theorem
Extreme values (maximum/minimum values) can be classified as [b]local [/b]or [b]global[/b]. [br][list][*]A maximum/minimum is [b]local [/b]if it is only the largest/smallest value of f in a small viewing window around it. In other words, if you move far enough away from a local maximum/minimum you can find other values of the function that are larger/smaller. [/*][*]A maximum/minimum is [b]global [/b]if it is the largest/smallest value across the entire domain of f. In other words, a global maximum is the largest local maximum value and a global minimum is the smallest local minimum value. [/*][/list][br]The [b]Extreme Value Theorem [/b]states that a function defined on a closed interval [a,b] is guaranteed to have both a global maximum and a global minimum. To find global extreme values:[br][list][*]Find all the critical points of f between the endpoints of the domain, a and b. [/*][*]Evaluate f(x) at the critical points and endpoints. These are the only locations where f can have maximum or minimum values. [/*][*]Because a global maximum/minimum is guaranteed to exist (by the theorem), the largest f(x) value gives the global maximum and the smallest f(x) value gives the global minimum. [/*][/list]