Deck 7: Logarithmic, Exponential, and Hyperbolic Functions

Question 1

Find the derivative of y with respect to x, t, or ?, as appropriate

-y = ln(ln 8x)

A)
B)
C)
D)

Answer 1

Question 2

Find the derivative of y with respect to x, t, or ?, as appropriate

-y = ln(cos(ln $\theta$ )

A) -tan(ln $\theta$ )
B) tan(ln $\theta$ )
C)
D) -

Answer 2

-

Question 3

Evaluate the integral.

-

A) -7 ln + C
B) ln + C
C) - ln + C
D) - ln + C

Answer 3

- ln + C

Question 4

Evaluate the integral.

-

A) ln
B) ln
C) ln
D) ln

Question 5

Evaluate the integral.

-

A) 6 sin x + C
B) 6 ln + C
C) ln + C
D) ln + C

Question 6

Evaluate the integral.

-

A) ln
B) ln
C)
D) ln

Question 7

Evaluate the integral.

-

A) ln + C
B) ln + C
C) ln + C
D) ln + C

Question 8

Evaluate the integral.

- dx

A)
B)
C)
D)

Question 9

Evaluate the integral.

-

A) -
B) ln 2
C)
D)

Question 10

Evaluate the integral.

-

A) 42 ln 3
B) -42 ln 3
C) 21 ln 3
D) -21 ln 3

Question 11

Evaluate the integral.

-

A) 2 ln 2
B) 2 ln 3
C) 4 ln 2
D) -2 ln 2

Question 12

Evaluate the integral.

- dx

A) 2 ln + C
B) ln + C
C) 2 ln + C
D) -ln + C

Question 13

Evaluate the integral.

-

A) - + C
B) - + C
C) 2 + C
D) - + C

Question 14

Evaluate the integral.

- dx

A) 2 ln + C
B) + C
C) ln + C
D) + C

Question 15

Evaluate the integral.

- dx

A) + C
B) + C
C) + C
D) + C

Question 16

Evaluate the integral.

-

A) - + C
B) - + C
C) -6 + C
D) + C

Question 17

Evaluate the integral.

-

A)

B)

C)

D)

Question 18

Evaluate the integral.

- dx

A) - 5
B) 14
C) 9
D) 5

Question 19

Evaluate the integral.

-

A) + C
B) + C
C) - + C
D) + C

Question 20

Evaluate the integral.

- 4x dx

A) e
B) -
C)
D) 4e

Question 21

Evaluate the integral.

- dx

A) 1 - cos 1
B) 1
C) 1 + cos 1
D) -1

Question 22

Evaluate the integral.

- d $\theta$

A) + C
B) + C
C) 4 ln (1 + ) + C
D) ln (1 + ) + C

Question 23

Evaluate the integral.

- dx

A) 21
B)
C)
D)

Question 24

Evaluate the integral.

- dx

A) 52,464
B)
C) + C
D)

Question 25

Evaluate the integral.

- sin t dt

A)
B)
C)
D) 3

Question 26

Evaluate the integral.

- dx

A)
B)
C)
D)

Question 27

Evaluate the integral.

-

A) + C
B) + C
C) + C
D) + C

Question 28

Evaluate the integral.

-

A) + C
B) 1
C) + C
D) + C

Question 29

Evaluate the integral.

- dx

A)
B) - 1
C) + C
D)

Question 30

Evaluate the integral.

- dt

A) -
B) 2 ln 6
C) 12
D) 6

Question 31

Evaluate the integral.

- dx

A)
B) + C
C)
D) 8

Question 32

Solve the problem.

-A certain radioactive isotope decays at a rate of 1% per 300 years. If t represents time in years and y represents the amount of the isotope left, use the condition that to find the value of k in the equation

A) 0.00897
B) -0.00003
C) 0.00003
D) -0.01005

Question 33

Solve the problem.

-A certain radioactive isotope decays at a rate of 2% per 100 years. If t represents time in years and y represents the amount of the isotope left then the equation for the situation is In how many years will there be 90% of the isotope left?

A) 527 years
B) 1000 years
C) 500 years
D) 246 years

Question 34

Solve the problem.

-A loaf of bread is removed from an oven at and cooled in a room whose temperature is . If the bread cools to in 20 minutes, how much longer will it take the bread to cool to

A) 17 min
B) 4 min
C) 24 min
D) 5 min

Question 35

Solve the problem.

-In a chemical reaction, the rate at which the amount of a reactant changes with time is proportional to the amount present, such that , when t is measured in hours. If there are of reactant present when t = 0, how many grams will be left after ? Give your answer to the nearest tenth of a gram.

A) 3.7 g
B) 11 g
C) 7.3 g
D) 0.1 g

Question 36

Solve the problem.

-The charcoal from a tree killed in a volcanic eruption contained 64.7% of the found in living matter. How old is the tree, to the nearest year? Use for the half-life of

A) 1720 years
B) 5700 years
C) 3581 years
D) 2482 years

Question 37

Solve the problem.

-The barometric pressure p at an altitude of h miles above sea level satisfies the differential equation If the pressure at sea level is 29.92 inches of mercury, find the barometric pressure at 22,000 ft.

A) 6.5 in.
B) 68.85 in.
C) 13 in.
D) 0.37 in.

Question 38

Solve the problem.

-The amount of alcohol in the bloodstream, A, declines at a rate proportional to the amount, that is, . If for a particular person, how long will it take for his alcohol concentration to decrease from 0.10% to 0.05%? Give your answer to the nearest tenth of an hour.

A) 2.1 hr
B) 1.4 hr
C) 0.3 hr
D) 2.8 hr

Question 39

Write the word or phrase that best completes each statement or answers the question.

-Verify the identity using the definitions of hyperbolic functions. coth x =

Question 40

Write the word or phrase that best completes each statement or answers the question.

-Verify the identity using the definitions of hyperbolic functions. coth (-x) = -coth x

Question 41

Write the word or phrase that best completes each statement or answers the question.

-Use the fundamental identity x - x = 1 to verify the identity. 1 - x = x

Question 42

Write the word or phrase that best completes each statement or answers the question.

-Derive the formula given that d/dx (cosh x) = sinh x and d/dx (sinh x) = cosh x. d/dx (coth x) = - x

Question 43

Write the word or phrase that best completes each statement or answers the question.

-Verify the identity. cosh( x) = , for all x

Question 44

Compute dy/dx for the function.

-y = sinh 7x

A) 7 cosh 6x
B) 7 cosh 7x
C)
D) 7 sinh 7x

Question 45

Compute dy/dx for the function.

-y = 6x

A) 2 cosh 6x
B) 12 sinh 6x cosh 6x
C) 2 sinh 6x cosh 6x
D) 12 cosh 6x

Question 46

Compute dy/dx for the function.

-y =

A) -
B)
C)
D)

Question 47

Compute dy/dx for the function.

-y = ln sinh 5x

A)
B) coth 5x
C) 5 coth 5x
D) 5 csch 5x

Question 48

Determine the indefinite integral.

-

A) cosh 8x + C
B) 8x + C
C) - cosh 8x + C
D) cosh 8x + C

Question 49

Determine the indefinite integral.

-

A) x + C
B) sech x tanh x + C
C) + C
D) x + C

Question 50

Determine the indefinite integral.

-

A) + C
B) x + C
C) ln(1 + cosh x) + C
D) -ln(1 + cosh x) + C

Question 51

Determine the indefinite integral.

-

A) x + C
B) x x + C
C) - + C
D) + C

Question 52

Determine the indefinite integral.

- (Simplify the answer using the definitions of hyperbolic functions.)

A) + C
B) x + + C
C) + C
D) cosh x + C

Question 53

Determine the indefinite integral.

- , x > 4

A) + C
B) + C
C) + C
D) + C

Question 54

Determine the indefinite integral.

-

A) + C
B) + C
C) - + C
D) - + C

Question 55

Determine the indefinite integral.

- , x < ln 4

A) + C
B) - + C
C) - + C
D) + C

Question 56

Determine the indefinite integral.

-

A) + C
B) + C
C) - + C
D) - + C

Question 57

Evaluate the definite integral.

-

A) -
B)
C)
D)

Question 58

Evaluate the definite integral.

-

A) ln
B)
C) ln 2
D) ln

Question 59

Evaluate the definite integral.

-

A) -14
B)
C) 14
D) 28

Question 60

Evaluate the definite integral.

- (Give an approximate answer rounded to two decimal places if necessary.)

A) 1
B) 0.71
C) 0.73
D) 0.75

Question 61

Find the value of the x-coordinate of the points at which the functions intersect, and approximate the area of the region described. Round to two decimal places when necessary.

-f(x) = sinh x, g(x) = tanh x; the region bounded by the graphs of f, g, and ln 5

A) x = 0; 0.54
B) x = 0; 0.22
C) x = 0; 1
D) x = 0; 0.64

Question 62

Evaluate the derivative.

-f(x) =

A)
B)
C)
D)

Question 63

Evaluate the derivative.

-f(t) = 5

A)
B)
C)
D)

Question 64

Evaluate the derivative.

-f(v) = 3

A)
B)
C)
D)

Question 65

Evaluate the derivative.

-f(x) = ( 6/x)

A)
B) -
C)
D)

Question 66

Evaluate the derivative.

-f(x) = 9 (cos x)

A) -
B) ln sin x
C) -
D) -

Question 67

Evaluate the definite integral. Express the answer in terms of natural logarithms.

-

A) ln( + 3)
B) ln( + )
C) ln( + 2)
D) ln

Question 68

Evaluate the definite integral. Express the answer in terms of natural logarithms.

-

A) ln
B) ln 2
C) ln
D) ln

Question 69

Evaluate the definite integral. Express the answer in terms of natural logarithms.

-

A) ln - 1
B) ln
C) ln 6
D) ln

Question 70

Evaluate the definite integral. Express the answer in terms of natural logarithms.

-

A) - ln
B) ln
C) -ln
D) ln

Question 71

Evaluate the definite integral. Express the answer in terms of natural logarithms.

-

A) ln( 8 + 3 )
B) ln( )
C) ln 9
D) ln( 8 + )

Question 72

Solve the problem.

-Find the volume interior to the inverted catenary kiln (an oven used to fire pottery) shown in the figure. Give an exact answer in terms of logarithms.

A) 22(3 ln(3 + ) + )
B) 22(3 ln(3 + ) - )
C) 11(3 ln(3 + ) - )
D) 66 ln(3 + ) -

Question 73

Solve the problem.

-When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after t seconds is given by where m is the mass of the object in kilograms, g = 9.8 m/s² is the acceleration due to gravity, and k is a physical constant. A BASE jumper (m = 76kg) leaps from a tall cliff and performs a ten-second delay (she free falls for 10 s and then opens her chute). How far does she fall in 10 s? Assume k = 0.2 Round the answer to one decimal place if necessary.

A) 14.5 m
B) 331.2 m
C) 361.9 m
D) 72.4 m

Question 74

Solve the problem.

-A BASE jumper's velocity (in meters per second) after t seconds is given by where m is the mass of the object in kilograms, g = 9.8 m/ s² is the acceleration due to gravity, and k is a physical constant. How fast is the BASE jumper falling at the end of a 12 second delay? Use m = 75kg and k = 0.2 Round the answer to one decimal place if necessary.

A) 19.2 m/s
B) 58.2 m/s
C) 63.2 m/s
D) 698 m/s

Question 75

Use a calculator to evaluate the expression, or state that the value does not exist. Round the answer to four decimal places when necessary.

-coth 2

A) 3.7622
B) 0.964
C) 0.2658
D) 1.0373

Question 76

Use a calculator to evaluate the expression, or state that the value does not exist. Round the answer to four decimal places when necessary.

- 5

A) 1.0001
B) 0.9999
C) 0.2027
D) does not exist

Question 77

Use a calculator to evaluate the expression, or state that the value does not exist. Round the answer to four decimal places when necessary.

- 9

A) 4.0515
B) 0.0998
C) 0.0002
D) 0.1109

Question 78

Use a calculator to evaluate the expression, or state that the value does not exist. Round the answer to four decimal places when necessary.

-csch x

A) -2.9085
B) -1.571
C) 0.0371
D) -0.9103

Question 79

Use a calculator to evaluate the expression, or state that the value does not exist. Round the answer to four decimal places when necessary.

-

A) -0.1105
B) -0.0552
C) -0.2209
D) -0.0131

Question 80

Evaluate the expression without using a calculator, or state that the value does not exist. Simplify the answer to the extent possible.

-sech 0

A) 0
B) 1
C) 2
D)