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Mathematics LibreTexts

4.5E: Exercises for Section 4.4

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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.

Answer
One example is f(x)=|x|+3,2x2

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)(ba)=f(b)f(a)? Draw such an example or prove why not.

Answer
Yes, but the Mean Value Theorem still does not apply

In exercises 5 - 9, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer.

5) y=sin(πx)

6) y=1x3

Answer
(,0),(0,)

7) y=4x2

8) y=x24

Answer
(,2),(2,)

9) y=ln(3x5)

In exercises 10 - 13, 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)(ba)=f(b)f(a).

10) [T] y=3x3+2x+1 over [1,1]

Answer
2 points

11) [T] y=tan(π4x) over [32,32]

12) [T] y=x2cos(πx) over [2,2]

Answer
5 points

13) [T] y=x634x598x4+1516x3+332x2+316x+132 over [1,1]

In exercises 14 - 19, use the Mean Value Theorem and find all points 0<c<2 such that f(2)f(0)=f(c)(20).

14) f(x)=x3

Answer
c=233

15) f(x)=sin(πx)

16) f(x)=cos(2πx)

Answer
c=12,1,32

17) f(x)=1+x+x2

18) f(x)=(x1)10

Answer
c=1

19) f(x)=(x1)9

In exercises 20 - 23, 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].

20) f(x)=|x12|

Answer
Not differentiable

21) f(x)=1x2

22) f(x)=|x|

Answer
Not differentiable

23) 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.)

In exercises 24 - 34, determine whether the Mean Value Theorem applies for the functions over the given interval [a,b]. Justify your answer.

24) y=ex over [0,1]

Answer
Yes

25) y=ln(2x+3) over [32,0]

26) f(x)=tan(2πx) over [0,2]

Answer
The Mean Value Theorem does not apply since the function is discontinuous at x=14,34,54,74.

27) y=9x2 over [3,3]

28) y=1|x+1| over [0,3]

Answer
Yes

29) y=x3+2x+1 over [0,6]

30) y=x2+3x+2x over [1,1]

Answer
The Mean Value Theorem does not apply; discontinuous at x=0.

31) y=xsin(πx)+1 over [0,1]

32) y=ln(x+1) over [0,e1]

Answer
Yes

33) y=xsin(πx) over [0,2]

34) y=5+|x| over [1,1]

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

For exercises 35 - 37, consider the roots of each equation.

35) Show that the equation y=x3+3x2+16 has exactly one real root. What is it?

36) Find the conditions for exactly one root (double root) for the equation y=x2+bx+c

Answer
b=±2c

37) Find the conditions for y=exb to have one root. Is it possible to have more than one root?

In exercises 38 - 42, 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.

38) [T] y=tan(πx) over [14,14]

Answer
c±0.1533
c=±1πcos1(π2)

39) [T] y=1x+1 over [0,3]

40) [T] y=|x2+2x4| over [4,0]

Answer
The Mean Value Theorem does not apply.

41) [T] y=x+1x over [12,4]

42) [T] y=x+1+1x2 over [3,8]

Answer
12c+12c3=5212880
c3.133,5.867

43) 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?

44) Two cars drive from one stoplight 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.

Answer
Yes

45) Show that y=sec2x and y=tan2x have the same derivative. What can you say about y=sec2xtan2x?

46) Show that y=csc2x and y=cot2x have the same derivative. What can you say about y=csc2xcot2x?

Answer
It is constant.

4.5E: Exercises for Section 4.4 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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