Deck 3: Differentiation

Question 1

Find the slope of the tangent line to the curve y = 4x - at the point (-1, 0).

A) -1
B) 2
C) 6
D)
E) -2

Answer 1

6

Question 2

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

A) 2x + y - 4 = 0
B) 2x + y + 4 = 0
C) 2x - y - 4 = 0
D) 2x - y + 4 = 0
E) 2x + y = 0

Answer 2

2x + y - 4 = 0

Question 3

Find an equation of the line tangent to the curve y = 2x - at the point where x = 2.

A) 25y = 49x - 1
B) 5y = 49x + 1
C) 25y = 49x + 1
D) 25y = 41x + 1
E) 25x = 49y + 1

Answer 3

25y = 49x + 1

Question 4

Find an equation of the line tangent to the curve y = + 1 at the point where x = 2.

A) y = 12x + 15
B) y = 12x -15
C) y = -12x -15
D) y = -12x + 15
E) y = 15x + 12

Question 5

Find an equation of the line tangent to the curve y = at the point where x = 2.

A) y = -4x + 12
B) y = 4x - 4
C) y = -4x + 4
D) y = 4x + 4
E) y = 4x - 12

Question 6

Find an equation of the line tangent to the curve y = at the point where x = 11.

A) y = x +
B) y = x -
C) y = 4x -
D) y = 4x +
E) y = - x -

Question 7

Find an equation of the line tangent to the curve y = at the point (1, 3).

A) y = - x +
B) y = - x -
C) y = x +
D) y = x -
E) y = 3x - 10

Question 8

Let f(x) be a function such that = - 1. Find the slope of the line tangent to the graph of f at the point (a, f(a)).

A) 3
B) 1 -
C) -3
D) - 1
E) - a + C

Question 9

Find the point(s) on the curve y = such that the tangent lines to the curve at those points pass through (2, -12).

A) (6, 36) and (-2, 4)
B) (6, 36) and (2, 4)
C) (-6, 36) and (-2, 4)
D) (-6, 6) and (-2, 4)
E) (6, -36) and (-2, 4)

Question 10

Find the standard equation of the circle with centre at (1, 3) which is tangent to the line 5x - 12y = 8.

A) - = 1
B) + = 9
C) + = 9
D) + = 8
E) + = 8

Question 11

If the line 4x - 9y = 0 is tangent in the first quadrant to the graph of y = + c, what is the value of c?

A) -
B)
C)
D)
E)

Question 12

Using the definition of the derivative, find the derivative of f(x) = .

A)
B)
C)
D)
E)

Question 13

Find the derivative (x) of the function f(x) = .

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

Question 14

Find the tangent line to the curve y = at the origin.

A) y = - x
B) y = x
C) y = x
D) y = - x
E) y = x

Question 15

Where is the function f(x) = differentiable?

A) at every x (- $\infty$ , $\infty$ )
B) at every x 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 (- $\infty$ , 0) 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 (0, $\infty$ )
C) at every x 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 (- $\infty$ , 3) (3, $\infty$ )
D) at every x 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 (- $\infty$ , 0) 11ee7b09_453f_4329_ae82_2b7669cf7fd7_TB9661_11 (0, 3) 11ee7b09_453f_4329_ae82_2b7669cf7fd7_TB9661_11 (3, $\infty$ )
E) none of the above

Question 16

Find the equation of the straight line that passes through the point P(0,- 3) and is tangent to the curve .

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

Question 17

If f(x) = ( ), calculate f'(5) by using the definition of the derivative.

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

Question 18

Find the slope of the line tangent to the curve y = 1 at the point .

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

Question 19

If f(x) = , calculate f'(-2) directly from the definition of the derivative.

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

Question 20

Let g(x) be a function such that = - . Find (x).

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

Question 21

Calculate the derivative of g(t) = + using the general power rule.

A) 101 - 99
B) 101 - 99
C) -101 - 99
D) 100 - 98
E) 101 + 99

Question 22

If f(x) is an even, differentiable function, then (x)

A) is an odd function.
B) is an even function.
C) is neither odd nor even.
D) may be either even or odd or neither.

Question 23

If the curve y = f(x) has a tangent line at (a, f(a)), then f is differentiable at x = a.

Question 24

If = - $\infty$ , then the graph of f has a tangent line at x = a.

Question 25

If f is continuous at x = a, then f is differentiable at x = a.

Question 26

If exists, then f is continuous at x = a.

Question 27

The domain of the derivative of a function is the same as the domain of the function.

Question 28

Differentiate f(x) = 10 .

A) 10
B) 50
C) 55
D) 50
E) 50x

Question 29

Find if y = 4 + 3 + x - 6.

A) 16 - 9 + 1
B) 16 + 9 + 1
C) 16 + 9 + 1
D) 16 + 9 - 6
E) 16 + 9 - 5

Question 30

Differentiate the function f(x) = (2 + 5)(3 - x).

A) 30 - 8 + 30x - 5
B) 30 - 8 + 30x + 5
C) 30 + 8 + 30x - 5
D) 30 + 8 - 30x - 5
E) 36 - 6

Question 31

Find the equation of the tangent line to the curve y = (2 - )(1 + + 3x) at the point (1, 5).

A) x - y + 4 = 0
B) x + y - 6 = 0
C) x - y - 4 = 0
D) 6x - y - 1 = 0
E) x + y + 4 = 0

Question 32

Find the points on the curve y = - 6 + 4 where the tangent line is horizontal.

A) ( , -5) and (- , -5)
B) (0, 4), ( , -5), and (- , -5)
C) (0, 4), (- , 5), and ( , -5)
D) (0, 4), ( , -5), and (- , -5)
E) ( , 5) and (- , 5)

Question 33

Given g(x) = which of the following statements is true?

A) g is differentiable at x = -1
B) g is not differentiable at x = -1
C) (-1) = -4
D) g is continuous at x = -1
E) g is continuous from the left at x = -1

Question 34

Lines passing through the point (0, 2) are tangent to the graph of y = - . Find the points of tangency.

A) (1, -1) and (-1, 1)
B) (2, -8) and (-2, -8)
C) (1, -1) and (-2, -8)
D) (2, -8) and (-1, 1)
E) (1, 1) and (-1, -1)

Question 35

Where does the normal line to the curve y = x - at the point (1, 0) intersect the curve a second time?

A) (-2, -6)
B) (- , - )
C) (-1, -2)
D) (0, 0)
E) It does not intersect the curve a second time.

Question 36

Which of the following statements is always true?

A) If f is continuous at c, then it must be differentiable at c.
B) If f is differentiable at c, then it must be continuous at c.
C) If f is not differentiable at c, then it must be discontinuous at c.
D) If f(c + h) = f(c), then f must be differentiable at c.
E) All of the above

Question 37

How many tangent lines to the graph of y = -15 - 10 pass through the point (0, 2)?

A) 0
B) 1
C) 2
D) 3
E) 4

Question 38

Let f(x) = .Find all values of the real number k so that f is differentiable at x = 1.

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

Question 39

There are lines that pass through the point (-1, 3) and are tangent to the curve xy = 1. Find all their slopes.

A) -1 and -9
B) -1 and 9
C) 1 and 9
D) 1 and -9
E) none of the above

Question 40

Find the derivative of .

A)
B)
C)
D)
E)

Question 41

Find the derivative of f(x) = .

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

Question 42

Differentiate the following function: f(x) = .

A)
B)
C)
D)
E) none of the above

Question 43

Differentiate the following function: f(x) = .

A) 14
B) -15
C) -16
D) 17
E) 3

Question 44

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

A) 27x - y + 28 = 0
B) 27x + y + 26 = 0
C) 27y - x - 28 = 0
D) 27y + x - 26 = 0
E) 9x - y + 10 = 0

Question 45

Use the values in the table below to evaluate (-2)

Question 46

Assuming all indicated derivatives exist, ( (c) is equal to

A) (g(c)) (c)
B) (c) g(c) + f(c) (c)
C) (c) (c)
D) (c) (c)
E) ( (c))

Question 47

Let f(x) = (x - 2)( + 4x - 7). Find all the points on this curve where the tangent line is horizontal.

A) and
B) and
C) and
D) and
E)

Question 48

Find . Simplify your answer.

A)
B)
C)
D) -
E)

Question 49

Where does the function f(x) = fail to be differentiable?

A) f(x) is differentiable everywhere.
B) at x = 0
C) at x = 1
D) at x = 0 and x = 1
E) none of the above

Question 50

The function f(x) = is differentiable at x = 0.

Question 51

Differentiate y = sin 4x.

A) 2cos 4x
B) 4cos 2x
C) -4cos 4x
D) 4cos 4x
E) cos 4x

Question 52

Find the derivative of y = tan(cos( )).

A) (-sin(2x))
B) 2x cos( )
C) (-2x sin( ))
D) ( ) cos( ) - tan( ) sin( )
E) -2x (cos( )) sin( )

Question 53

Find the derivative of f(t) = (5t).

A) 15 (5t) sin(5t)
B) -3 (5t) sin(5t)
C) -15 (5t) sin(5t)
D) 15 (5t)
E) 3 (5t)

Question 54

Differentiate sin 2x.

A) sin 2x + cos 2x
B) sin 2x + cos 2x
C) 3 sin 2x - 2 cos 2x
D) sin 2x + 2 cos 2x
E) 3 cos(2x)

Question 55

Question 56

Find the derivative of y = tan( ).

A) x ( )
B) 4x )
C) 2x )
D) 2x sec( ) tan( )
E) ( )

Question 57

Find the derivative of the following function: y = (cos x).

A) -2sin x
B) 2sin x
C) -2sin x
D) -2sin x
E) -2tan(x)cos(x)sin(x)

Question 58

Let y = . A simplified expression for is given by

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

Question 59

Find the slope of the curve y = cos at the point where x = .

A) -
B) -
C) -
D) -
E) The slope is not defined at x = .

Question 60

Find all points in the interval [0, $\pi$
] where the curve y = 2 x - sin(2x) has a horizontal tangent line.

A) and
B) and
C) and
D) and
E) The tangent line is never horizontal.

Question 61

Find if y = .

A) 12
B) 5
C) 15
D) 20
E) 10

Question 62

Find the second derivative of g(x) = .

A) (t) = 2
B) (t) = - 3
C) (t) = 3
D) (t) = -2
E) (t) = 4

Question 63

Question 64

Find (2) given that = . - 7

A) 12
B) 0
C) 6
D) 5
E) 10

Question 65

let y = , x > 0. Show that = .

Question 66

Calculate the third derivative of f(x) = x.

A) -2sin(2x)
B) -4sin(2x)
C) -2cos(2x)
D) -4sin x
E) -2(x)

Question 67

Find a formula for the nth derivative of the function y = .

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

Question 68

Find the second derivative of the function f(x) = .

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

Question 69

Find the second derivative of the function f(x) = sin(2x).

A) - 6x sin(2x) + 8 cos(2x) - 4 sin(2x)
B) 6x sin(2x) + 8 cos(2x) - 4 sin(2x)
C) 6x sin(2x) + 12 cos(2x) - 4 sin(2x)
D) - 6x sin(2x) + 12 cos(2x) + 4 sin(2x)
E) 6x sin(2x) + 12 cos(2x) + 4 sin(2x)

Question 70

Find the second derivative of the function f(x) = .

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

Question 71

A spherical balloon is being inflated. Find the rate of change of volume with respect to the radius when the radius is 5 cm.

A) 200 $\pi$ /cm
B) 100 $\pi$ /cm
C) 300 $\pi$ /cm
D) 400 $\pi$ /cm
E) 500 $\pi$ /cm

Question 72

Find the rate of change of the volume of a cube with respect to its edge length x when x = 4 m.

A) 40 /m
B) 42 /m
C) 48 /m
D) 50 /m
E) 8 /m

Question 73

A spherical balloon is being inflated. Find the rate of increase of the surface area (S = 4 $\pi$ ) with respect to the radius when r = 2 m.

A) 16 $\pi$ /m
B) 8 $\pi$ /m
C) 12 /m
D) 24 /m
E) 4 $\pi$ /m

Question 74

Find the rate of change of the area of a circle with respect to its circumference C.

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

Question 75

The electrical resistance R of a wire of unit length and cross-sectional radius x is given by R = , where K is a non-zero constant real number. By approximately what percentage does the resistance R change if the diameter of the wire is decreased by 6%?

A) -6%
B) -9%
C) 12%
D) 6%
E) -12%

Question 76

The cost in dollars for a company to produce x pairs of shoes is . Find the marginal cost function.

A) (x) = 1 + 0.02x + 0.0006
B) (x) = 1 + 0.01x + 0.0002
C) (x) = 3 + 0.02x + 0.0003
D) (x) = 3 + 0.02x + 0.0006
E) (x) = 3 + 0.01x + 0.0006

Question 77

The population (in thousands) of the city of Abbotsford is given by , with t in years and with corresponding to 1980. What was the rate of change of P in 1986?

A) 9.6 thousand per year
B) 8.6 thousand per year
C) 7.6 thousand per year
D) 8.9 thousand per year
E) 4.4 thousand per year

Question 78

The daily cost of production of x widgets in a widget factory is C dollars, where . What is the cost per widget, and what value of x will make the cost per widget as small as possible?

A) dollars, x = 2,000
B) dollars, x = 4,000
C) dollars, x = 8,000
D) dollars, x = 400
E) dollars, x = 4,000

Question 79

If the cost of mining x kg of gold is C(x) = A + Bx + C dollars where A, B, and C are positive constants, which of the following statements is true for a given positive value of x?

A) The marginal cost C'(x) is greater than the cost C(x + 1) - C(x) of mining 1 more kg.
B) The marginal cost C'(x) is less than the cost C(x + 1) - C(x) of mining 1 more kg.
C) The marginal cost C'(x) is equal to the cost C(x + 1) - C(x) of mining 1 more kg.
D) There is not enough information to make any conclusion.
E) None of the above

Question 80

By approximately what percentage does the volume of a cube change if the edge length changes by 1%?

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