Math 2414 Activity 2 (Due by end of class July 23)

1. a) Use the Intermediate Value Theorem to show that the function xf x e x has a zero on

the interval 1,0 . {Hint: What are the signs of 1f and 0f ?}

b) Use Rolle’s Theorem to show that it has exactly one zero in this interval.

{Hint: What’s f x ?}

2. For what value of k does the equation 2xe k x have exactly one solution?

{Hint: From the graph, you can see cases of no solution, one solution, and two solutions. In

the case of one solution, the function values and derivative values agree. This leads to the

system

2

222

x

x

e k x

ke

x

.}

3. Consider the equation xe x , where is a constant.

a) For which values of does the equation have a unique solution?

b) For which values of does it have no

solution?

c) For which values of does it have two

solutions?

{Hint: Check out the graph.}

xe

2 solutions

x

no solutionx

1 solutionx

2xe

k x

k x

k x {No solution.}

{One solution.}

{Two solutions.}

4. Find the exact absolute maximum value(x and y coordinates)for the function 2 xf x x e .

5. For the function ln 2f x x x x on the

interval

2,

2

1 e

e Find

a) the exact absolute maximum value

(x and y coordinates)

b) the exact absolute minimum value

(x and y coordinates)

{Hint: Check the critical numbers and endpoints.}

6. a) Show that the function 1xf x e x has its minimum value at 0x . What is the

minimum value?

b) Write an inequality between xe and 1 x based on part a).

c) Replace x with 2x in part b) and integrate from 0 to 1 to find upper and lower bounds

for2

1

0

xe dx .

d) Since

0

1

x

x te e dt ,

0

1 1

x

xe t dt , so 2

12

x xe x . Use this new inequality to

find an inequality between xe and a cubic polynomial in x, for 0x .

e) Generalize the previous result to get an inequality between xe and an nth degree

polynomial in x, for 0x .

f) Use the previous results to find limx

kx

e

x, for any positive integer k.

7. By looking at the graph of the function 1

f xx

for 0x , find upper and lower bounds on

ln n , for n an integer larger than 1.

So from the figure, you can see that

1 1 1 12 3 4

1

1n

ndx

x . This implies that

1 1 1 12 3 4

lower bound

lnn

n

Use the following figure to find an upper bound on ln n .

8. Show that ln 11

x

x

xf x e

e

is decreasing for 0x .

{Hint: Show that 0f x for 0x .}

9. Let

2

ln

1 ln

xf x

x

for x in 0, .

1 2 3 n

12

13

14

1n

1

x

1 2 3 n - 1 n

1 12 1

3 11n

1

x

a) By rewriting f as 1

1ln

ln

f x

xx

, find 0

limx

f x

.

b) By rewriting f as 1

1ln

ln

f x

xx

, find limx

f x

.

c) Find the maximum value of f.

(x and y coordinates)

d) Find the minimum value of f.

(x and y coordinates)

10. Calculate 1 1 1

lim1 2n n n n n

by rewriting it as 1 2

1 1 1 1lim

1 1 1 nnn n nn

,

and identifying it as a definite integral.

11. Apply the Mean Value Theorem to the function ln 1f x x for 1x to show that

ln 1 ; 11

xx x x

x

.

{ 0 ln 11

xf x f f c x x

c

for some c between x and 0.}

Case I: If 1 0x , then 1

11 c

and 1

xx

c

. Also,

1 1

1 1c x

, so

1 1

x x

c x

.

Now you take care of Case II. where 0x .

12. Suppose that f x f x for all x and 0 0f . Then 0x xe f x e f x , which

implies from the product rule that 0xe f x , and therefore that xe f x C .

Plugging in 0 for x yields 0C , and we can conclude that f must be the zero function. So

the only solution of f x f x with 0 0f is 0f x . Find as many non-constant

solutions of f x f x with 0 0f as you can.

{Hint: Here’s one: 2; 0

0 ; 0

x xf x

x

.}

13. Show that ln ln

ln2 2

a b a b

for , 0,a b a b .

Let 2

ln2 2

a b a bL x x

a b

be the tangent line to the graph of ln x at

2

a bx

. Since 2

1ln 0x

x , we have that lnL x x for

2

a bx

. This means

that ln ln

2 2

L a L b a b . If you work out the left side of the inequality, you’ll get the

result.

14. Let ln 1

; 0x

f x xx e

.

a) Find the maximum value of f x (x and y coordinates).

b) Find all positive numbers x so that e xx e .

{Hint: If e xx e , then ln 1 ln 1

ln 0 0x x

e x x f xx e x e

.}

15. For 1t , 1 1

t t , so

1 1

1 10 ln 2 1

x x

x dt dt xt t

.

a b 2

a b

ln ln

2

a b

ln2

a b

L x

a) Use the previous result to find ln

limx

x

x. {Hint: Give it a squeeze.}

b) Use the previous part to find 0

lim lnx

x x

. {Hint: Let 1

xt

in lnx x , and let t .}

16. a) Show that if b a and f is a continuous function with 0f x f a b x for

a x b , then

b b

a a

f x f a b xdx dx

f x f a b x f x f a b x

.

{Hint: Use the substitution u a b x in the integral on the left.}

b) Find the actual value of the previous integral.

{Hint: If

b

a

f xI dx

f x f a b x

, then from part a),

b

a

f a b xI dx

f x f a b x

.

So add these two equations and solve for I.}

c) Use the previous part to evaluate

4

2

ln 9

ln 9 ln 3

xdx

x x

.

17. Consider the function 1

1

1 x

f xe

. Determine the following limits:

a) limx

f x

b) limx

f x

c) 0

limx

f x

d) 0

limx

f x

18. Consider the function 1

1

1

x

x

ef x

e

. Determine the following limits:

a) limx

f x

b) limx

f x

c) 0

limx

f x

d) 0

limx

f x

19. Find the following limits:

a) 0

lim log 2aa

b) 1

lim log 2aa

c) 1

lim log 2aa

d) limlog 2aa

{Hint: ln 2

log 2ln

aa

.}

20. Find the point on the curve 2ln 1y x x at which both the first and second derivatives

are equal to zero. Is this an inflection point for the curve?

21. Suppose that f is a function with the property that for all positive real numbers x and y,

f xy f x f y .

a) Find 1f .

b) Show that if 1f exists, then f x exists for all positive x.

{Difference Quotient.}

22. Consider the function n xf x x e , where n is an even integer greater than zero. What’s the

connection between the x-coordinate of the local maximum of f and the x-coordinates of the

two inflection points of f ?

23. If log 32

b

b

b, then what’s the value of b?

24. Consider the function 1 1

1 1

x xf x

e e.

a) Find 0

limx

f x . b) Find 0

limx

f x . c) Find 0

limx

f x

.

{Hint: Combine the two fractions in the formula for f into one.}

25. Differentiate the following functions using logarithmic differentiation:

a) sinx xf x x

b)

cossin

xf x x c)

2xf x x

26. Find the intervals of increase and decrease for the following functions:

a) log 2xf x b) 12

logxf x

27. In the function xf x x both the base and the exponent are variable. Its derivative can be

calculated using logarithmic differentiation. What if the number of x’s in the exponent is

also variable? Specifically, define the continued exponentiation function, xxxcont x x ,

where there is x number of x’s in the exponent. For example,

33 273 3 7,625,597,484,9873 3 3 3cont , which has over 3.6 billion digits. So far the definition

of cont x makes sense only for positive integer values. How could we define 3

2cont

?

Perhaps

13 2

23

23 3

2 2cont

,

131

31 1

3 3cont

, and

2 12

2

2 2cont

. If you

see the pattern, then expand the following:

a) 5

2cont

b) 10

3cont

c) 5cont

28. Express the following in terms of ln2 and/or ln5 .

a) ln10 b) 12

ln c) 15

ln d) ln 25 e) ln 2 f) 3ln 5 g) 120

ln h) 12ln2

29. Find the maximum value of 1xf x x on the interval 0, .

30. a) Show that as x decreases towards 1, ln

1

xf x

x

increases. In other words, show that

0f x for 1x .

{Hint:

2

1 ln

1

x x xf x

x x

. If you can show that the numerator is negative for 1x ,

you’ll have the result. For 1 lng x x x x , 1 0g , now see if you can show

that 0g x for 1x .}

b) Replace x with 11x

to show that 11x

xh x is increasing for 0x .

{Hint:

1

1

1

ln 1ln 1

1 1

xx

x

x

, and f x is increasing if and only if f x

e is increasing.}

31. Suppose that f is a continuously differentiable function such that 1f x f x for all x,

and 0 0f . What is the largest possible value of 1f ?

{Hint: Multiply the inequality by xe . Recognize the left side as xe f x, and integrate

from 0 to 1.}

32. Let xf x e kx with 0k .

a) Show that f has a local minimum at lnx k .

b) Find the value of k for which this local minimum value of f is as large as possible.

33. Show that the equation

2 cos 0x

f x

e x x has exactly one solution.

{Hint: Check out 0f and f . Also examine f x , and consider Rolle’s Theorem.}

34. Find the exact extrema (x and y coordinates)for the function arcsin 2f x x x .

35. You are sitting in a classroom next to the wall looking at the whiteboard at the front of the

room. The whiteboard is 12 ft. long and starts 3 ft. from the wall you are sitting next to.

Your viewing angle is given by 1 1

15 3cot cotx xx . How far from the front of the

classroom should you sit to maximize your viewing angle?

There is a unique circle that passes through the two points on the left and right edges of the

white board at your eye level and your eye. What’s special about the point on the circle at

your eye?

Evaluate the following integrals:

36. 21 4

dx

x 37. 217

dx

x {Hint: 21717 }

38.

2

2

24 3

dt

t

{Hint: Factor the 3.} 39.

ln 3

2

01

x

x

e dx

e {Hint: Let xu e .}

Top View

3 ft.

12 ft.

x

40.

0

2

1

6

3 2

dt

t t

{Complete the square.} 41.2 6 10

dy

y y {Complete the square.}

42.

1cos

21

xe dx

x

{Hint: Let 1cosu x .}

43. For 0 1x , 2 2 3 20 4 2 4 4x x x x .

So by taking square roots, you get that 2 2 3 20 4 2 4 4x x x x .

And by taking reciprocals, you get 2 2 3 2

1 1 1

4 4 4 2x x x x

.

Integrate this inequality from 0 to 1 and find upper and lower bounds on the value of 1

2 3

0

1

4dx

x x .

44. a) Show that

144

2

0

1 22

1 7

x xdx

x

, using 44 8 7 6 5 41 4 6 4x x x x x x x and

completing the long division

6

2 8 7 6 5 4

8 6

7 6 5 4

1 4 6 4

4 5 4

x

x x x x x x

x x

x x x x

.

b) For 0 1x ,

4 4 44 4 4

2

1 1 1

2 1 1

x x x x x x

x

. This means that

1 1 144

4 44 4

2

0 0 0

111 1

2 1

x xx x dx dx x x dx

x

. Evaluate

1

44

0

1x x dx to get an

inequality of the form 22

7m M , and see how close

22

7 is to .

45. Find the exact inflection point of 11 tanf x x x (x and y coordinates).

{Hint: Check out f x .}

46. Two towns A and B are 8 miles apart. A third town C is located 5 miles from both A and B.

If the point P, equidistant from A and B is such that the sum of the distances PA, PB, and

PC is the least possible, how far is the point P from C?

The sum of the lengths in terms of the angle is given by

1 33 8sec 4tan ; 0 tan

4L

.

A B

C

P

4,0

4,0

0,3

8

47. A 27 ft. ladder is placed vertically against an 8 ft. high fence. The lower end of the ladder is

then pulled directly away from the fence. If the ladder is kept in contact with the top of the

fence, what is the greatest horizontal distance the ladder ever projects beyond the fence?

The horizontal projection as a

function of is given by

1 827cos 8cot ; sin

27 2P

.

48. Apply the Mean Value Theorem to the function 1tanf x x for 0x to show that

1

2tan ; 0

1

xx x x

x

.

{ 1

20 tan

1

xf x f f c x x

c

for some c between x and 0.}

49. Use differentiation to show that 1 1

; 01 2

tan tan

; 02

x

xx

x

.

50. Find the angle in the following figure:

{Hint: 6

6

3 3 3tantan

3

.}

6

3

3 3

51. Find the value of 1 1tan tana b

x x

when x is the x-coordinate of the maximum value

of 1 1tan tana b

f xx x

, for 0a b .

52. a) Use the fact that for 1

02

x , 4 2

1 11

1 1x x

, to find upper and lower bounds for

the integral

12

4

0

1

1dx

x .

b) Use the fact that for 0 1x , 2 4 2

1 1 1

2 4 4x x x

, to find upper and lower

bounds for the integral

1

2 4

0

1

4dx

x x .

53. Evaluate 10 5

1

1dx

x x x . Factor 10x from the radicand to get

2

6 5 5

1

1

dx

x x x ,

let 5u x , and then complete the square.

54. Use differentiation to show that 1 1

4

1tan tan

1

xx

x

is an identity.

55. Show that 1 12tan tan 1

xf x x

x

is a constant function.

56. Show that 1 14 2

tan lnx x for 1x .

{Hint: For 1 14 2

tan lnf x x x , 1 0f . See if you can show that 0f x for

1x to get the result.}

57. Find all functions f such that f is continuous and

2 2 2

0

100

x

f x f t f t dt for all real x.

{Hint: Differentiate both sides and use 2 2 22a b a ab b .}

58. Find all continuous functions f so that 0

1

x

f x f t dt .

{Hint: Differentiate the integral equation and then use the integral equation to determine an

initial condition.}

59. Suppose that f is a continuous, non-constant function on 0,1 with

1

0

0f x dx . Consider

the function

1

0

af xg a e dx .

a) Find the value of 0g .

b) Find the value of 0g .

{Hint: According to a theorem of Leibniz,

1 1

0 0

af x af xdg a e dx f x e dx

da .}

c) Show that 0g a for all a.

{Hint:

1 1

2

0 0

af x af xdg a f x e dx f x e dx

da .}

d) What’s the minimum value of g?

{Hint: The graph of g is concave-up.}

60. a) Use the fact that 2

21

1

x

x

to show that

12

2

0

11

xdx

x

.

b) Use the fact that

1 12 2

2 2 2

0 0

1 1

1 1 1

x xdx dx

x x x

to find the exact value of

12

2

01

xdx

x .

61. If

1

ln

1

x

tf x dt

t

, then find the value of 1

ef e f .

{Hint:

1

1

1 1

ln ln

1 1

ee

e

t tf e f dt dt

t t

. Let 1t

u in the second integral to get

1

2

1 1

ln ln

1

e e

t udt du

t u u

. Now rewrite the sum of the integrals as 2

1

ln lne

t t tdt

t t

, and

simplify the integrand.}

62. Find the value of

12

2

2

ln

1

xI dx

x

.

{Hint: Let 1x

u .}

63. Show that ln x is not a rational function restricted to 0, .

{Hint: Suppose that

ln N x

xD x

, where N and D are polynomial functions with no

common factors. Differentiation leads to

2

2

1

D x N x D x N xD x x D x N x D x N x

x D x. This

means that x must be a factor of D x , and therefore, 1 kD x x D x with 1k

and 1 0 0D . Substituting this into the previous equation leads to

2 2 1 1

1 1 1 1

k k k kx D x x D x N x x D x N x kx D x N x , and dividing by kx

leads to 2

1 1 1 1 kx D x xD x N x xD x N x kD x N x . Plugging in zero

for x leads to 1 0 0 0 0 0 kD N N . What does this imply is a factor of

N x , and what can you conclude?}

64. Use differentiation to verify the identity 1 1

2tan sin

1

xx

x

for 1 1x .

65. Use differentiation to verify the identity 1 11

22

1tan sin

41

xx

x

for 1 1x .