
# 3.4E Exercises

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### Exercise $$\PageIndex{1}$$

1)Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample.

2) Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.

3) When are Rolle’s theorem and the Mean Value Theorem equivalent?

4) If you have a function with a discontinuity, is it still possible to have $$f′(c)(b−a)=f(b)−f(a)?$$ Draw such an example or prove why not.

2. One example is $$f(x)=|x|+3,−2≤x≤2$$,

4. Yes, but the Mean Value Theorem still does not apply

### Exercise $$\PageIndex{2}$$

For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.

1) $$y=sin(πx)$$

2) $$y=\frac{1}{x^3}$$

3) $$y=\sqrt{4−x^2}$$

4) $$y=\sqrt{x^2−4}$$

5) $$y=ln(3x−5)$$

2. $$(−∞,0),(0,∞)$$,

4. $$(−∞,−2),(2,∞)$$

### Exercise $$\PageIndex{3}$$

For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points $$c$$ such that $$f′(c)(b−a)=f(b)−f(a).$$

1) $$y=3x^3+2x+1$$ over $$[−1,1]$$

2) $$y=tan(\frac{π}{4}x)$$ over $$[−\frac{3}{2},\frac{3}{2}]$$

3) $$y=x^2cos(πx)$$ over $$[−2,2]$$

4) $$y=x^6−\frac{3}{4}x^5−\frac{9}{8}x^4+\frac{15}{16}x^3+\frac{3}{32}x^2+\frac{3}{16}x+\frac{1}{32}$$ over $$[−1,1]$$

1. 2 points

3. 5 points

### Exercise $$\PageIndex{4}$$

For the following exercises, use the Mean Value Theorem and find all points $$0<c<2$$ such that $$f(2)−f(0)=f′(c)(2−0)$$.

1) $$f(x)=x^3$$

2) $$f(x)=sin(πx)$$

3) $$f(x)=cos(2πx)$$

4) $$f(x)=1+x+x^2$$

5) $$f(x)=(x−1)^{10}$$

6) $$f(x)=(x−1)^9$$

1. $$c=\frac{2\sqrt{3}}{3}$$

3. $$c=\frac{1}{2},1,\frac{3}{2}$$,

5. Solution: $$c=1$$

### Exercise $$\PageIndex{5}$$

For the following exercises, show there is no $$c$$ such that $$f(1)−f(−1)=f′(c)(2)$$. Explain why the Mean Value Theorem does not apply over the interval $$[−1,1].$$

1) $$f(x)=∣x−\frac{1}{2} ∣$$

2) $$f(x)=\frac{1}{x^2}$$

3) $$f(x)=\sqrt{|x|}$$

4) $$f(x)=[x]$$

Hint:

This is called the floor function and it is defined so that $$f(x)$$ is the largest integer less than or equal to $$x\ Answers 1. Not differentiable 3. Not differentiable ### Exercise \(\PageIndex{6}$$

For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval $$([a,b]\. Justify your answer. 1) \(y=e^x$$ over $$[0,1]$$

2) $$y=ln(2x+3)$$ over $$[−\frac{3}{2},0]$$

3) $$f(x)=tan(2πx)$$ over $$[0,2]$$

4) $$y=\sqrt{9−x^2}$$ over $$[−3,3]$$

5) $$y=\frac{1}{|x+1|}$$ over $$[0,3]$$

6) $$y=x^3+2x+1$$ over $$[0,6]$$

7) $$y=\frac{x^2+3x+2}{x}$$ over $$[−1,1]$$

8) $$y=\frac{x}{sin(πx)+1}$$ over $$[0,1]$$

9) $$y=ln(x+1)$$ over $$[0,e−1]$$

10) $$y=xsin(πx)$$ over $$[0,2]$$

11) $$y=5+|x|$$ over $$[−1,1]$$

1, 3, 5, 9. Yes,

7.The Mean Value Theorem does not apply; discontinuous at $$x=0.$$ ,

11. The Mean Value Theorem does not apply; not differentiable at $$x=0$$.

### Exercise $$\PageIndex{7}$$

For the following exercises, consider the roots of the equation.

1) Show that the equation $$y=x^3+3x^2+16$$ has exactly one real root. What is it?

2) Find the conditions for exactly one root (double root) for the equation $$y=x^2+bx+c$$

$$b=±2\sqrt{c}$$

3) Find the conditions for $$y=e^x−b$$ to have one root. Is it possible to have more than one root?

Under Construction

### Exercise $$\PageIndex{8}$$

For the following exercises, use a calculator to graph the function over the interval $$[a,b]$$ and graph the secant line from $$a$$ to $$b$$. Use the calculator to estimate all values of $$c$$ as guaranteed by the Mean Value Theorem. Then, find the exact value of $$c$$, if possible, or write the final equation and use a calculator to estimate to four digits.

1) $$y=tan(πx)$$ over $$[−\frac{1}{4},\frac{1}{4}]$$

2) $$y=\frac{1}{\sqrt{x+1}}$$ over $$[0,3]$$

3) $$y=∣x^2+2x−4∣$$ over $$[−4,0]$$

4) $$y=x+\frac{1}{x}$$ over $$[\frac{1}{2},4]$$

5) $$y=\sqrt{x+1}+\frac{1}{x^2}$$ over $$[3,8]$$

1. $$c=±\frac{1}{π}cos^{−1}(\frac{\sqrt{π}}{2}), c=±0.1533$$

3. The Mean Value Theorem does not apply.

5. $$\frac{1}{2\sqrt{c+1}}−\frac{2}{c^3}=\frac{521}{2880}; c=3.133,5.867$$

Exercise $$\PageIndex{9}$$

At 10:17 a.m., you pass a police car at 55 mph that is stopped on the freeway. You pass a second police car at 55 mph at 10:53 a.m., which is located 39 mi from the first police car. If the speed limit is 60 mph, can the police cite you for speeding?

Under Construction

### Exercise $$\PageIndex{10}$$

Two cars drive from one spotlight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.

Yes.

### Exercise $$\PageIndex{11}$$

Show that $$y=sec^2x$$ and $$y=tan^2x$$ have the same derivative. What can you say about $$y=sec^2x−tan^2x$$?

Under Construction

### Exercise $$\PageIndex{12}$$

Show that $$y=csc^2x$$ and $$y=cot^2x$$ have the same derivative. What can you say about $$y=csc^2x−cot^2x$$?