Deck 4: Transcendental Functions

Question 1

Which of the following functions is/are one-to-one?
(a) , (b) 1 - , (c) sgn(x), (d) cos(x) on [0, $\pi$ ], (e) sin(x) on [0, $\pi$ ].

A) only (c) and (d)
B) only (d)
C) only (c) and (e)
D) only (a) and (b)
E) none of the above

Answer 1

only (c) and (d)

Question 2

Find the inverse of the function f(x) = 3 + 7.

A) - 7
B)
C)
D) 3 - 7
E)

Answer 2

Question 3

Assume that the function F(x) = - , x (0 , $\infty$ ) has an inverse. Find a formula for (x).

A) -
B)
C) x - 2
D)
E) 7x - 2

Answer 3

Question 4

Which of the following functions has an inverse?
(a) + x - 1, (b) - x + 1, (c) , (d) 4 + - 15, (e)

A) only (a)
B) only (c) and (e)
C) only (a), (c), and (e)
D) only (a) and (b)
E) none of the above

Question 5

If f is one-to-one, and if f(1) = 3, f(2) = 4, f(3) = 5, f(4) = 8, and f(5) = 20, find .

A) 2
B) 1
C) 3
D) 8
E) 4

Question 6

Find the inverse of the function f(x) = where -5 $\le$ x $\le$ 0.

A) (x) = - , where 0 $\le$ x $\le$ 5
B) (x) = , where 0 $\le$ x $\le$ 5
C) (x) = - , where -5 $\le$ x $\le$ 0
D) (x) = , where -5 $\le$ x $\le$ 0
E) (x) = , where 0 $\le$ x $\le$ 5

Question 7

Find the inverse of the following function: f(x) = , if x $\ge$ 3.

A) (x) = 3 +
B) (x) = 3 -
C) (x) = 2 +
D) (x) = 1 +
E) (x) =

Question 8

Find the inverse of the function f(x) = - 8x + 4, where x $\le$ 4. State its domain.

A) (x) = 4 - . Domain is x $\ge$ -12.
B) (x) = 4 + . Domain is x $\ge$ -12.
C) (x) = 4 + . Domain is x $\ge$ 12.
D) (x) = 4 - . Domain is x $\ge$ 12.
E) f(x) has no inverse.

Question 9

Find the inverse of the following function: f(x) = .

A) (x) =
B) (x) =
C) (x) =
D) (x) =
E) f(x) has no inverse.

Question 10

Let g be the inverse function of f and let y = . Calculate at x = given that f(6) = , , , and .

A) -
B) -
C) -
D)
E)

Question 11

Let f(x) = 2 + 3 +5, x $\ge$ 0, and let y = g(x) be the equation of its inverse function.Find the x and y coordinates of all points on the graph of y = g(x) where the tangent line has slope
.

A) (1, 10)
B) (1, -2)
C) (-2, 1)
D) (10, 1)
E) (1, -2) and (10, 1)

Question 12

Let g be the inverse function of f . Use the table of values below to compute ( ).

A) 2
B)
C) -
D)
E)

Question 13

The function f(x) = is $\textbf{ self-inverse }$
$\textbf{ Note:}$ A function f is self-inverse if ^-1 (x) = f(x) for all x in domain f.

Question 14

An even function defined on (- $\infty$ , $\infty$ ) may have an inverse.

Question 15

Prove that if an odd function f(x) is one-to-one, the inverse function g(y) is also odd.

Question 16

Let f(x) = for all x. Is f invertible? Why? Find .

Question 17

Simplify the expression ( ) - ( ).

A) 3x - 3a
B) 9x - 64a
C)
D) 3x - 4a
E)

Question 18

Given that (5) = 0.02, evaluate (a).

A) 25
B) 5
C) 5a
D) 50
E) 25a

Question 19

Simplify the expression (20) + (5).

A) 2
B) (25)
C) 2.5
D) 4
E) (15)

Question 20

Simplify the expression - - (5)

A) 1
B) 2
C) 5
D) 25
E)

Question 21

If 4 x - ( + 1) + (x - 1) = (y), find y.

A)
B)
C)
D)
E)

Question 22

Solve the exponential equation = 81.

A) x = ±1
B) x = ±
C) x = 0
D) x = ±2
E) x = ±4

Question 23

Solve the exponential equation = 108.

A) x = 2
B) x = 1
C) x = -1
D) x = -2
E) x = 3

Question 24

Let f(x) = .
The domain of f is the set of all real numbers x such that:

A) x (- $\infty$ , 1]
B) x 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 (1 , 3)
C) x 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 [3 , $\infty$ )
D) x 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 (- $\infty$ , 1)
E) x 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 [1 , 3]

Question 25

Solve the exponential equation = 9.

A) -2, -1
B) 2, 1
C) 2, -1
D) 2, -2
E) none of the above

Question 26

If 1296 = 4, what is ?

A) 29
B) 33
C) 28
D) 27
E) 25

Question 27

Solve the equation - = 0 correct to 3 decimal places.

A) -1.414
B) -0.414
C) 0.414
D) 1.414
E) none of the above

Question 28

Solve the logarithmic equation ( - 6x) = 3 + (1 - x).

A) only -4
B) only 2
C) -4 and 2
D) only 1
E) 1 and 2

Question 29

Solve the logarithmic equation ( ) - (6x - 1) = 0.

A)
B)
C)
D)
E)

Question 30

Solve the logarithmic equation log x = 1 - log (x -3).

A) only -5
B) only 5
C) only -2
D) 5 and -2
E) -5 and -2

Question 31

Solve the logarithmic equation 2 ( ) - (2 - ) = 0.

A) x = 1
B) x = 1 and x = -2
C) x = -2
D) x = 0
E) x = -1

Question 32

Evaluate ln .

A) -x
B) x
C) 1
D) -1
E) 0

Question 33

A) 4.5
B) 6.6
C) 3.5
D) 2.2
E) 5.5

Question 34

Evaluate the expression exp(ln3 - ln10).

A)
B)
C) -
D) 30
E)

Question 35

Find ln ( /x).

A) x + ln x
B) 1 - ln x
C) x - ln x
D) - x
E)

Question 36

Solve the exponential equation 5 = 8 correct to 3 decimal places.

A) -1.765
B) 1.755
C) -1.725
D) 1.735
E) 1.693

Question 37

Find the roots of the equation = 2 .

A) 1 - ln(2)
B)
C) 1
D) ln(2)
E) -ln(2)

Question 38

Show that for x > 1, ( ) = .

Question 39

Evaluate the derivative tan( ).

A) ( )
B) 2 ( )
C) sec ( ) tan ( )
D) sec ( ) tan ( )
E) ( )

Question 40

For what value(s) of the constant m is the line y = mx tangent to the curve y = ?

A) e
B) 1
C)
D) ln 2
E)

Question 41

Find a positive number for which the graph of the function f(x) = is tangent to the line y = x.

A) 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11 = e
B) 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11 =
C) 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11= 1
D) 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11 =
E) 11ee7b18_881f_7ad5_ae82_ef6a0704a9e3_TB9661_11 =

Question 42

Find the slope of the curve = 2x + y at the point (1, -1).

A) -2
B) 2
C)
D)
E) undefined

Question 43

Find the slope of the curve - + xy - 1 = 0 at the point (1, 1).

A)
B)
C)
D)
E)

Question 44

Use logarithmic differentiation to find the slope of the line tangent to the curve given by at the point (0, ) on the curve.

Question 45

Evaluate the derivative of .

A) 2cos( )
B) 2cos( )
C) cos( )
D) cos( )
E) -2cos( )

Question 46

A) 24 - 19
B) 6 - 1
C) + C, x $\neq$ -
D) 10 - 5
E) 6 + C

Question 47

Evaluate .

A)
B)
C)
D)
E) 2ln(x)

Question 48

Evaluate the antiderivative dx.

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

Question 49

= .

Question 50

The acceleration of an object moving on the x-axis is 9 . Find the velocity v if the velocity at time is 4 units per second.

A) v = + 3
B) v = 27 - 23
C) v = 3 + 1
D) v = 9 - 5
E) v = + 1

Question 51

Find an equation of the line tangent to the curve - = 2x at the point (0,1).

A) y = 1 -
B) y = 1 +
C) y = -
D) y = -
E) y = -1 +

Question 52

Find if y = .

A) (ln x)
B) 2 (ln x)
C) 2 (ln x)
D) 2 (ln x)
E) (ln x)

Question 53

Find y' if y = .

A) ( ln(x) + )
B) ln(x) +
C)
D) ln(x)
E) x( ln(x) + )

Question 54

Let f(x) = (2x + 1)(3x + 1)(4x + 1)(5x + 1)(6x + 1). Use logarithmic differentiation to calculate (0).

A) 20
B) 1
C) 10
D) 40
E) 5

Question 55

Solve the equation - 2 = 1 for x.

A) -ln 2
B) ln 4
C) ln 2
D) 1
E) 0

Question 56

Find (173) if f(x) = .

A) 0
B) e
C) 175
D) 174
E) 1

Question 57

Evaluate .

A)
B)
C) -
D) -
E) does not exist

Question 58

The function F(x) = ln is an antiderivative of the function f(x) = .

Question 59

Find the derivative of .

A) -
B)
C)
D)
E)

Question 60

Find the derivative of y = ln(sin x).

A) ln (cot x)
B) -ln (cot x)
C) tan x
D) cot x
E) ln(cos x)

Question 61

Differentiate the function y = ln(3x).

A) 3 ln(3x) +
B) 3 ln(3x) -
C) 3 ln(3x) +
D) ln(3x) +
E) none of the above

Question 62

Find an antiderivative of cos(2x) .

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

Question 63

Prove the following:

Question 64

The population of a city is growing exponentially, increasing at the rate of 1.6% per year. A new bicycle route must be in place before the population doubles from its current value. How long (rounded up to the nearest year) does the city have before the new bicycle route must be completed?

A) 40 years
B) 55 years
C) 44 years
D) 60 years
E) 50 years

Question 65

A population of bacteria grows continuously at a rate of 15% of its current population. If there are 10,000 bacteria present initially, how many (to the nearest 100) will there be after four hours?

A) 17,500
B) 18,200
C) 16,500
D) 15,300
E) 19,100

Question 66

When a quantity of sugar is placed in a container of water, the sugar dissolves at a rate proportional to the amount of sugar remaining undissolved. Suppose that 30 g of sugar is placed in 1 L of water and that 5 minutes later the concentration of sugar dissolved in the water is 25 g/L.Find a formula for A(t), the amount of sugar that is dissolved t minutes after the sugar was placed in the water.

A) A(t) = 30 g
B) A(t) = 30 g
C) A(t) = 30 g
D) A(t) = 30 g
E) A(t) = 30 g

Question 67

A thermometer reading -7°C is brought into a room kept at 23°C. Half a minute later the thermometer reads 8°C. What is the temperature reading of the thermometer after a further two and a half minutes?

A) 12.5°C
B) 32.5°C
C) 22.5°C
D) 15.5°C
E) 28.5°C

Question 68

The 2000 census recorded a population of 16.1 million in a certain country, while in 2010 the figure was 32.8 million. Assuming that the rate of population growth is proportional to the population, predict the population in the year 2015.

A) 39.8 million
B) 46.8 million
C) 36.8 million
D) 41.8 million
E) 52.8 million

Question 69

A dish with a temperature of 54°C is placed in a fridge, which has a constant temperature of 6°C.If the dish cools to 30°C in a half hour, what is its temperature after another half hour?Assume Newton's Law of Cooling.

Question 70

The rate at which a drug is absorbed into the bloodstream is given by = a - b, where b(t) is the concentration of the drug in the bloodstream at time t minutes after the drug is ingested, and a and b are positive constants. How long does it take for b(t) to rise to half its limiting value as t $\rightarrow$
$\infty$ ?

A) minutes
B) - minutes
C) minutes
D) minutes
E) minutes

Question 71

If a village has a population of 340 people and its population grows continuously at an annual rate of 2.3%, what will the population be in 6 years? (Round your answer to the nearest integer.)

A) 370
B) 381
C) 390
D) 400
E) 412

Question 72

A certain radioactive isotope has a half-life of 37 years. How many years will it take for 100 grams to decay to 64 grams?

A) 18.17 years
B) 33.54 years
C) 23.82 years
D) 28.71 years
E) 24.89 years

Question 73

A certain radioactive isotope decays from 70 grams to 45 grams in 27 years. What is the half-life of the isotope?

A) 40.16 years
B) 42.36 years
C) 48.31 years
D) 50.02 years
E) 36.25 years

Question 74

A certain radioactive isotope has a half-life of 14 years. If 19 grams of the isotope are left after 5 years, how much was present at the beginning?

A) 27.21 grams
B) 24.34 grams
C) 22.10 grams
D) 26.55 grams
E) 34.75 grams

Question 75

An object of temperature 44°C is placed in a freezer which is maintained at a temperature of -16°C. At the end of 10 minutes, the object has cooled to a temperature of 14°C. Assuming Newton's law of cooling, what is the temperature of the object (in degree Celsius) after a long time (that is, as )?

A) -16
B) $\infty$
C) - $\infty$
D) 0
E) -67

Question 76

The temperature x(t) (in degree Celsius) at time t ( in hours) of an object is given by x(t) = -5 +45 . When will the temperature of the object be 0°C?
Express your answer without logarithms.

Question 77

Find (1/ ).

A)
B)
C)
D) -
E)

Question 78

Evaluate .

A)
B)
C) -
D) -
E)

Question 79

Simplify (sin $\pi$ ).

A) 0
B) $\pi$
C)
D) - $\pi$
E) -

Question 80

Simplify cos( (x)).

A)
B) -
C)
D) 0
E) -sin(x)