Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Curve Sketching

Calculus

5. Graphical Applications of Derivatives

Curve Sketching

Previous problemNext problem

2:53 minutes

Problem 4.2.40

Textbook Question

Mean Value Theorem and graphs Find all points on the interval (1,3) at which the slope of the tangent line equals the average rate of change of f on [1,3]. Reconcile your results with the Mean Value Theorem. <IMAGE>

Verified step by step guidance

First, understand the Mean Value Theorem (MVT): It states that for a continuous function f on the closed interval [a, b] that is differentiable on the open interval (a, b), there exists at least one point c in (a, b) such that the derivative at c, f'(c), is equal to the average rate of change of the function over [a, b].

Calculate the average rate of change of the function f on the interval [1, 3]. This is given by the formula: (f(3) - f(1)) / (3 - 1).

Find the derivative of the function, f'(x), which represents the slope of the tangent line at any point x.

Set the derivative f'(x) equal to the average rate of change calculated in step 2. Solve this equation to find the value(s) of x in the interval (1, 3) where the slope of the tangent line equals the average rate of change.

Verify that the solution(s) found in step 4 satisfy the conditions of the Mean Value Theorem: the function must be continuous on [1, 3] and differentiable on (1, 3). If these conditions are met, the solution(s) are valid according to the theorem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mean Value Theorem

The Mean Value Theorem (MVT) states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the derivative of the function at c equals the average rate of change of the function over [a, b]. This theorem is fundamental in connecting the behavior of a function to its derivative.

Average Rate of Change

The average rate of change of a function f over an interval [a, b] is calculated as (f(b) - f(a)) / (b - a). This value represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)). Understanding this concept is crucial for applying the Mean Value Theorem, as it provides the value that the derivative must equal at some point within the interval.

Tangent Line and Derivative

The slope of the tangent line to a function at a given point is represented by the derivative of the function at that point. This slope indicates the instantaneous rate of change of the function. In the context of the Mean Value Theorem, finding points where the tangent line's slope equals the average rate of change is essential for identifying the specific points c where the theorem holds true.

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Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

82. y' = sin t, for 0 ≤ t ≤ 2π

Textbook Question

85. y' = x^(-2/3) (x - 1)

Textbook Question

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

46. y = cos(x) + √3 * sin(x), 0 ≤ x ≤ 2π

Textbook Question

Let ƒ(x) = (x - 3) (x + 3)²

f. State the x- and y-intercepts of the graph of ƒ.

Textbook Question

{Use of Tech} Tree growth Let b represent the base diameter of a conifer tree and let h represent the height of the tree, where b is measured in centimeters and h is measured in meters. Assume the height is related to the base diameter by the function h = 5.67+0.70b+0.0067b².

a. Graph the height function.

Textbook Question

{Use of Tech} Fuel economy Suppose you own a fuel-efficient hybrid automobile with a monitor on the dashboard that displays the mileage and gas consumption. The number of miles you can drive with g gallons of gas remaining in the tank on a particular stretch of highway is given by m(g) = 50g−25.8g²+12.5g³−1.6g⁴, for 0≤g≤4.

a. Graph and interpret the mileage function.

Textbook Question

b. Graph and interpret the gas mileage m(g)/g.

Textbook Question

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x² - 6x

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