Calculus I (Dr. McGuffey, MATH 141, USC)
"Daily" interactive graphs to introduce various Calculus concepts in Dr. McGuffey's Math 141 Calculus I course at University of South Carolina. Updated: Fall 2024
Table of Contents
- Limits and Continuity (Pt. 1)
- Instantaneous Rates of Change
- Limits from a Graph
- Continuity
- Limits of Rational Functions
- Limits and Continuity (Pt. 2)
- Evaluating Limits Algebraically (Motivation)
- Formal Definition of Limit
- Derivatives
- The Derivative at a Point
- Differentiability
- The Derivative Function
- Graphs of Derivative Functions
- Differentiation
- Basics of Differentiation
- The Product Rule
- The Chain Rule
- Higher-Order Derivatives
- Implicit Differentiation
- Implicit Differentiation
- Derivatives of Inverse Functions
- Application: Free Fall
- Application: Simple Harmonic Oscillators
- Application: Damped Oscillations
- Applications (Pt. 1): Related Rates & Linearization
- Related Rates (Circle Example)
- Related Rates (Sphere Example)
- Linearization
- Applications (Pt. 2): Analyzing Function Behavior
- Extreme Values & Critical Points
- First Derivatives & Monotonicity
- Second Derivatives & Concavity
- Applications (Pt. 3): Curve Sketching & Optimization
- Integrals & Integration
- Estimating Area with Riemann Sums
- Antiderivatives
- Integration Applications
- The Fundamental Theorem of Calculus (Pt. 2)
Instantaneous Rates of Change
Instantaneous Rates of Change
"Instantaneous" rate of change is a bit of a paradox. For something to change, it must be different at two different times, but an "instant" means one single moment in time. You can't tell how fast something is moving from a single picture; you have to piece together two data points to talk about how fast something is changing. [br][br]However, we also have an intuitive understanding of instantaneous rate of change. The speedometer on your car is giving you an estimate of your current speed at any given moment. If you watched swimming events at the Olympics you could see the instantaneous speeds of the top 3 swimmers at any given time. [br][br]An informal definition: The [b]instantaneous rate of change[/b] of a function [math]f[/math] at input [math]a[/math] is defined to be the [b]limit [/b]of the [b]average rates[/b] of change of f over [i]progressively smaller intervals [/i]containing [math]a[/math]. This means that [i]we can [b]estimate [/b]instantaneous rates of change with average rates of change[/i], as long as we choose a small enough interval for the average rate.
Instructions
The graph below shows the motion of a point along a vertical line over time. On the left is the actual motion of the point; one with an arrow that represents an instantaneous velocity and one with an arrow representing the average velocity. On the right is the position-time graph. [br][list][*]Use the input box to adjust the value of c or click and drag the point P or c on the graph to adjust to a particular time. [/*][*]Use the input box for h to adjust the value of h, which is the horizontal distance (difference in x-values) for the points P and Q on the graph. [/*][*]The check boxes for "Secant" and "Tangent" show the secant line through P and Q and the tangent line at P, respectively. [/*][*]Click on the "Let h approach 0" button to shorten the time interval between P and Q. [/*][/list]
Evaluating Limits Algebraically (Motivation)
Motivation: Limits of Difference Quotients
This preview is about introducing how we will use limits of average rates of change to find instantaneous rates of change. Our goal in the lesson will be to actually evaluate limit expressions exactly without having to rely on estimation techniques with graphs and tables. In other words, we will take a purely [b]algebraic approach [/b]using the difference quotient expression. [br][br]Almost all algebraic techniques for evaluating limits boil down to simplifying an expression to a point of "canceling" a common factor. If you look at the difference quotient function graph, you will notice that this function always has a [b]discontinuity [/b]where h=0. If that discontinuity is [b]removable [/b](i.e., a hole), then canceling the common factor is essentially the algebraic way of filling in this hole so that we obtain a [b]continuous extension [/b]for the difference quotient function. (If the discontinuity is not removable, we will talk later about how this means that the instantaneous rate of change does not exist there.)
Instructions
In this lesson we are going to learn algebraic techniques for evaluating limits. The main motivation for being able to do so is to evaluate the limit of a difference quotient expression (i.e., average rate of change) in order to obtain an instantaneous rate of change. [br][list][*]The left part of the graph shows the motion of a moving point, including an arrow to represent the average and instantaneous rates of change. [/*][*]Use the slider tool for h to move the point Q around the point P. [/*][*]Use the respective checkboxes to show/hide the secant between P and Q, the tangent line at P, and the graph of the difference quotient function. [/*][*]When the Secant box is checked, a point will appear. This point takes the slope of the secant line and represents it as a y-coordinate. Adjusting the value of h shows how this slope is a function of h; this is the difference quotient function. Notice it is undefined when h=0, but what value does it seem to approach? [/*][/list]
The Derivative at a Point
The Derivative (at a Point)
As we have seen, the instantaneous rate of change of a quantity can be estimated using the average rate of change over a small interval. And we define the instantaneous rate of change to be the limit of those average rates of change as the width of the interval shrinks to 0. [br][br]The [b]derivative [/b]of a function [math]f[/math] at a point [math]c[/math] in its domain is the [b]instantaneous rate of change[/b] of [math]f[/math] at [math]x=c[/math]. Because we previously introduced instantaneous rates of change as limits of average rates of change, we have a formula for calculating the derivative at [math]c[/math], denoted by [math]f'(c)[/math]. [br][br]Version 1: [math]f'(c)=\lim_{x\to c}\frac{f(x)-f(c)}{x-c}[/math][br][br]Version 2: [math]f'(c)=\lim_{h\to0}\frac{f(c+h)-f(c)}{h}[/math]
Instructions
The graph of a function is shown in the applet. [br][list][*]Use the input box for c or click and drag the point on the graph to change the point where you want to investigate the instantaneous rate of change. [/*][*]Adjust the slider tool for h to move the point Q around P. [/*][*]The "Secant" checkbox will show/hide the secant line between P and Q and the slope of the secant line (i.e., average rate of change). [/*][*]The "Difference Quotient" checkbox will show/hide the graph of the difference quotient function, which has an excluded value when h = 0. [/*][*]The "Tangent" checkbox will show/hide the tangent line at P and its slope. [/*][*]Use the "[math]h\to0[/math]" and observe the relationship between the secant and tangent lines. [/*][/list]
Basics of Differentiation
Basics of Differentiation
As we begin to look at different techniques for differentiation (i.e., finding derivatives), it is important to keep in mind [i]two different types of differentiation rules[/i]. [br][br][b]Derivative Formula:[/b] A rule that simply states the formula for the derivative of a [i]particular function[/i]. For example, [math]\frac{d}{dx}\left[\sin x\right]=\cos x[/math] is a derivative formula. Think of the differentiation operator [math]\frac{d}{dx}[/math] as an operation that takes a function ([math]\sin x[/math]) as an input and produces its derivative function ([math]\cos x[/math]) as an output. [br][br][b]Differentiation Rule: [/b]A rule that states how the differentiation operator "interacts" with function operations (addition, subtraction, multiplication, division, composition). For example, "the derivative of a sum of two functions is the sum of their derivatives," i.e., [math]\frac{d}{dx}\left[f(x)+g(x)\right]=f'(x)+g'(x)[/math]. This rule is true no matter whether f and g are power functions, trig functions, logarithms, etc., as long as they are differentiable. So, the rule is about the differentiation operator and the function operation of addition, not about the specific functions involved.
Instructions
The graph of a function as well as the same function with transformations applied is shown on the left. Compare the derivative of the original function with the derivative of the transformed function. [br][list][*]Use the slider tool for k to adjust the coefficient, which corresponds to a vertical stretch/shrink. [/*][*]Use the slider tool for b to adjust the constant term, which corresponds to a vertical shift. [/*][*]Checkboxes for f(x) and g(x), as well as f'(x) and g'(x), show/hide their graphs. [/*][*]The "tangents" checkbox will show/hide a tangent line segment on each graph. [/*][*]With "tangents" shown, click the "Trace Derivatives" button to trace the graph of the derivative functions on the right. [/*][/list]
Implicit Differentiation
Implicit Differentiation
We will come across some situations where we can describe a relationship between two variables with an equation, but we can't solve the equation to isolate one variable as the output of a function formula. Graphically, this typically means that the graph of the equation does not pass the vertical line test, i.e., there can be multiple y-values paired with a particular x-value. Under certain conditions we can say that one of the variables is an [b]implicit function[/b] of the other. This means that we can't find an explicit function formula, but that if we look at a small enough section of the graph (called a [i]branch[/i]) we would be able to treat the equation more like a function. [br][br]When working with these equations it still makes sense to talk about a tangent line on the graph at each point and therefore we can still think of the derivative as the slope of a tangent line at a point on the graph. We will just have to introduce a new technique, called [b]implicit differentiation[/b], in order to find these derivatives. Essentially this technique will rely on the [b]Chain Rule[/b], treating any occurrence of y as a function of x.
Instructions
The applet includes 5 examples of graphs of implicit equations. Use the slider tool to change the example. Check the "Normal Line" box to show/hid the normal line (i.e., the line perpendicular to the tangent line).
Related Rates (Circle Example)
Related Rates of Change
Any science or math formula that you can think of typically relates two or more quantities to one another. [br][br]For example, kinetic energy (energy due to motion) is given by [math]K=\frac{1}{2}mv^2[/math]. This relationship allows us to analyze how changes in velocity can lead to changes in energy. Naturally, we could describe this with a derivative (of energy with respect to velocity):[br][br][math]\frac{dK}{dv}=\frac{d}{dv}\left[\frac{1}{2}mv^2\right]=\frac{1}{2}m\cdot2v=mv[/math][br][br]Fun fact: momentum is mass times velocity, so momentum is the rate that kinetic energy changes with respect to changes in velocity. But what happens if the velocity of the moving object is also changing as a function of time? Then, the kinetic energy of the object is a function of velocity, which is a function of time. This creates a chain/composition: [math]t\to v\to E[/math], and we can replace the variables in the equation with functions of time:[br][br][math]E(t)=\frac{1}{2}m\left[v(t)\right]^2[/math][br][br]Now, not only can we ask how the kinetic energy changes with respect to velocity, we can also ask how the kinetic energy changes with respect to time. These are two different rates of change with different units of measurement and different interpretations. How do we differentiate [math][v(t)]^2[/math]in the above equation (with respect to t) without knowing what v(t) is? For example, v(t) could be [math]5-2t[/math], or [math]e^{-0.5t}[/math], or [math]\sqrt{1-t^2}[/math]. [br][br]The premise of related rates problems is that we start with [i]a relationship (equation) between two (or more) quantities[/i] and then [b]use implicit differentiation[/b] (usually with respect to an implicit variable for time, which may not even be present in the equation) to find [i]a relationship (equation) between the rates of change[/i] of these quantities.
Instructions
Start by playing the applet (the play button is in the bottom-left corner of the graphic) and observing the growing circle on the left. Our goal is to examine the rate of change in the circle's area with respect to time. [br][list][*]Use the slider tool or input box for [math]t_0[/math] to set a specific time. [/*][*]Use the slider tool or input box for [math]\Delta t[/math] to set an increment (small change) in time. [/*][*]The variables time, radius, and area create a chain or composition: [math]t\to r\to A[/math]. Observe the three different rates of change involved in relating these three variables. [/*][*]The graph on the right shows the area of the circle as a function of its radius, and the radius is a function of time. Use the input box for r(x) to set how the radius changes over time. (Note that x is being used in place of t because the calculator requires x as the input variable.)[/*][/list]
Extreme Values & Critical Points
Extreme Values & Critical Points
When we talk about extreme values of a function, we mean maximum or minimum values (without specifying which type). [br][br]Global (Absolute) Extreme Values: [br][list][*]f has a global maximum at x=c if [math]f(c)\ge f(x)[/math] for all x in the domain of f. [/*][*]f has a global minimum at x=c if [math]f(c)\le f(x)[/math] for all x in the domain of f. [/*][/list][br]Local (Relative) Extreme Values: [br][list][*]f has a local maximum at x=c if [math]f(c)\ge f(x)[/math] for all x in an interval containing x = c. [/*][*]f has a local minimum at x=c if [math]f(c)\le f(x)[/math] for all x in an interval containing x = c. [/*][/list][br]Critical Points:[br]The critical points of a function f are the points where either [math]f'(c)=0[/math] or [math]f'(c)[/math] does not exist. It turns out that all local extreme values must occur at a critical point. That is, the list of critical points is a list of potential extreme values. However, critical points do not always correspond to extreme values; so, the critical points must be tested to determine if there is an extreme value there.
Instructions
Use the input boxes below to define a function f(x) on an interval [a,b]. Use the checkboxes for a and b to exclude the endpoints from the domain of the function. Use the checkboxes to show/hide the critical points and local extrema in the interior of the interval (i.e., not at the endpoints). Use the "Trace y" button to trace points on the y-axis as you move the point c across the domain.
Estimating Area with Riemann Sums
Estimating Integrals with Riemann Sums
To estimate the area under the graph of a function over an interval [a, b] using rectangles:[br][list][*]Partition the interval [a, b] into n sub-intervals of length [math]\Delta x=(b-a)/n[/math]. This fixes the width of the rectangles and creates sub-intervals [math][x_0,x_1],[x_1,x_2],\ldots,[x_{n-1},x_n][/math]. [/*][*]Choose a method for determining the height of the rectangles using the value of the function at some point in the sub-interval. For the midpoint rule, for example, choose the midpoint [math]x_i^*[/math] of the sub-interval [math][x_{i-1},x_i][/math] and evaluate [math]f\left(x_i^*\right)[/math] to be the height of rectangle [math]i[/math]. [/*][/list]Calculate the area of each rectangle by multiplying the height [math]f\left(x_i^*\right)[/math] by the width [math]\Delta x[/math], and add up all the areas:[br][list][*][math]f\left(x_1^*\right)\Delta x+f\left(x_2^*\right)\Delta x+\cdots+f\left(x_n^*\right)\Delta x[/math] [br][br][math]=\sum_{i=1}^nf\left(x_i^*\right)\Delta x[/math][/*][/list]
Instructions
Use the input box to define the function f(x). [br][list][*]Click "Trace Area" and use the slider tool for c to trace out the area under the curve. When you drag c all the way to the right endpoint it will reveal the actual area under the graph expressed in integral notation. [/*][*]Check the Rectangles and Partition boxes to reveal rectangles that can be used to estimate this area. [/*][*]The "Rectangles Height" slider determines how the heights of the rectangles are determined. The default is the Midpoint Rule (i.e., the rectangle height is determined by the value of the function at the midpoint of each subinterval), but you can slide to see the Left Sum or Right Sum. [/*][*]The partition slider determines how many rectangles n, which simultaneously determines the widths of the rectangles. [/*][/list]
The Fundamental Theorem of Calculus (Pt. 2)
The Definite Integral
This lesson is about the definite integral [math]\int_a^bf'(x)dx[/math], which is essentially the net area "under" the graph of a derivative function f'(x). [br][br]If f'(x) represents a velocity, then [math]f'(x)\Delta x[/math] is essentially a very small change in position based on moving at a velocity of [math]f'(x)[/math] over a very small time [math]\Delta x[/math].[br][list][*]If velocity is positive, then these small changes in position are positive (i.e., position is increasing, moving upward). The area "under" the derivative is counted positively. [/*][*]If velocity is negative, then these small changes in position are negative (i.e., position is decreasing, moving downward). The area "under" (between the curve and the x-axis) is counted negatively. [/*][/list]If we add up all these little areas, each of which represents a small change in position, we get a total change in position over the time from x=a to x=b. So, the definite integral of a velocity function f'(x) over the interval [a, b] can be interpreted as a change in position (i.e., change in f(x)) from x = a to x = b. That is:[br][br][math]\int_a^bf'(x)dx=\Delta f=f(b)-f(a)[/math][br][br]Where we are going with this: This is called the Fundamental Theorem of Calculus (Part 2), and this gives us a way of evaluating definite integrals. But, to apply this theorem, we need to be able to start with a derivative f'(x) and find the function f(x) from which it is derived. We refer to this process as antidifferentiation and refer to f(x) as an antiderivative of f'(x). We will study this in more detail over the next two lessons.
Instructions
The applet shows the motion of a point P on the left, with a graph of its position f and velocity f' on the right. [br][list][*]Adjust the slider tool for n to create more data points along the graph of f. This simultaneously shrinks the width of the equal-sized sub-intervals of width [math]\Delta x[/math]. [/*][*]Use the checkboxes for f and f' to show the position and velocity graphs, respectively. [/*][*]Use the Slopes checkbox to show average velocity slopes on the graph of f. [/*][*]Use the AVG velocity checkbox to plot a new graph (a "step" function) whose value over an interval is the (constant) average velocity of the point over that interval. [/*][*]Use the Area checkbox to show the area "under" the curve over each interval. [/*][/list]