Exam III Study Guide
Find the derivative.
1) y = 13x-2 + 7x3 + 1x, find f'(x)
1)
A) -26x-3 + 21x2
C) -26x-1 + 21x2 + 1
B) -26x-3 + 21x2 + 1
D) -26x-1 + 21x2
Find the derivative of the given function.
2
2) y = (3x2 + 5x)
A) 18x3 + 45x2 + 25x
2)
B) 36x3 + 90x2 + 50x
C) 36x3 + 45x2 + 50x
D) 18x3 + 45x2 + 50x
Find the slope of the line tangent to the graph of the function at the given value of x.
3) y = x4 + 3x3 - 2x - 2; x = -2
A) -6
B) -4
C) 2
3)
D) 0
Find an equation for the line tangent to given curve at the given value of x.
x3
4) y =
; x=2
2
A) y = 6x - 8
B) y = 8x + 6
C) y = 2x + 8
4)
D) y = 2x - 8
Solve the following.
5) Find all points of the graph of f(x) = 2x2 + 6x whose tangent lines are parallel to the line y - 34x = 0.
A) (7, 140)
B) (9, 216)
C) (10, 260)
D) (8, 176)
Give an appropriate answer.
6) If g′(3) = 4 and h′(3) = -1, find f′(3) for f(x) = 5g(x) - 3h(x) + 2.
A) 25
B) 23
C) 19
5)
6)
D) 17
Use the product rule to find the derivative.
7) f(x) = (3x - 4)(5x3 - x2 + 1)
7)
A) f'(x) = 45x3 + 69x2 - 23x + 3
B) f'(x) = 15x3 + 23x2 - 69x + 3
C) f'(x) = 60x3 - 69x2 + 8x + 3
D) f'(x) = 60x3 - 23x2 + 69x + 3
Use the quotient rule to find the derivative.
x2 - 3x + 2
8) y =
x7 - 2
8)
A)
dy -5x8 + 19x7 - 14x6 - 4x + 6
=
dx
(x7 - 2)2
B)
dy -5x8 + 18x7 - 14x6 - 3x + 6
=
dx
(x7 - 2)2
C)
dy -5x8 + 18x7 - 13x6 - 4x + 6
=
dx
(x7 - 2)2
D)
dy -5x8 + 18x7 - 14x6 - 4x + 6
=
dx
(x7 - 2)2
1
Let f(x) = 8x2 - 5x and g(x) = 7x + 9.
Find the composite.
9) f[g(k)]
A) 392k2 - 973k + 603
9)
B) 392k2 + 973k + 603
D) 56k2 + 35k + 9
C) 56k2 - 35k + 9
10) f[g(-4)]
A) 2983
10)
B) 765
C) 37
D) 148
Find the derivative.
11) y = (x-2 + x)-3
11)
dy 3x4 (2 - x3 )
A)
=
dx
(1 + x3 )3
C)
dy 3x5 (2 - x3 )
B)
=
dx
(1 + x3 )4
dy 3x5 (2 - x3 )
=
dx
(1 + x3 )3
D)
dy 3x4 (2 - x3 )
=
dx
(1 + x3 )4
Solve the problem.
12) The total revenue from the sale of x stereos is given by R(x) = 1000 1 average revenue.
400
A) 0.006 x2
B) 2.5 -
400
x2
C) 0.006 -
x 2
. Find the marginal
400
1000
x2
D) 2.5 -
1000
x2
Find the derivative.
2ex
13) y =
2ex + 1
A)
13)
2ex
(2ex + 1)3
B)
2ex
(2ex + 1)2
C)
2ex
(2ex + 1)
D)
ex
(2ex + 1)2
14) y = 9 x2
14)
A) 9 x2 2x ln 9
B) 9 x2 x ln 9
C) 2x ln 9
D) 9 x2 2x ln x
2x
B)
x2 + 7
2
C)
x
1
D)
2x + 7
Find the derivative of the function.
15) y = ln (7 + x2 )
14
A)
x
15)
Find the derivative.
16) y = ex5 ln x
A)
C)
12)
16)
ex5 + 5ex5 ln x
B)
x
ex5 + 5x5 ex5 ln x
x
D)
2
5x5 ex5 + 1
x
ex5 + 5x4 ex5 ln x
x
Find the derivative of the function.
17) y = log
3x + 8
8
3
A)
2(ln 8)(3x + 8)
17)
B)
3
ln 8
C)
3
ln 8 (3x + 8)
D)
3 ln8
3x + 8
Solve the problem.
18) Assume the total revenue from the sale of x items is given by R(x) = 31 ln (7x + 1), while the total
cost to produce x items is C(x) = x/4. Find the approximate number of items that should be
manufactured so that profit, R(x) - C(x), is maximum.
A) 54 items
B) 221 items
C) 124 items
D) 173 items
Identify the open intervals where the function is changing as requested.
19) Increasing
18)
19)
f(x)
3
2
1
-2
-1
1
2
3
4
5
6
7
8 x
-1
-2
-3
A) (3, ∞)
B) (3, 6)
C) (-2, 0)
D) (-2, ∞)
Find all the critical numbers of the function.
2
1
20) f(x) = x3 - x2 - 15x + 2
3
2
A) -
5 5
,
2 2
B) -
20)
5
,3
2
C) -3,
5
2
D) 3
Find all values of x (if any) where the tangent line to the graph of the function is horizontal.
21) y = 2 + 8x - x2
A) 4
B) -4
C) 8
Find the open interval(s) where the function is changing as requested.
22) Increasing; f(x) = x2 - 2x + 1
A) (-∞, 0)
23) Decreasing; f(x) = A) (3, ∞)
B) (1, ∞)
C) (0, ∞)
21)
D) -8
22)
D) (-∞, 1)
x+3
23)
B) (-∞, -3)
C) (-3, ∞)
3
D) (-∞, 3)
Solve the problem.
24) Suppose a certain drug is administered to a patient, with the percent of concentration in the
6t
bloodstream t hr later given by K(t) =
. On what time interval is the concentration of the drug
2
t +1
increasing?
A) (0, 6)
B) (1, ∞)
C) (0, 1)
24)
D) (6, ∞)
Find the location and value of all relative extrema for the function.
25)
25)
A) Relative maximum of 3 at -2.
B) Relative minimum of 0 at 2.
C) None
D) Relative maximum of 3 at -2 ; Relative minimum of 0 at 2.
Suppose that the function with the given graph is not f(x), but f′(x). Find the locations of all extrema, and tell whether
each extremum is a relative maximum or minimum.
26)
26)
y
5
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5
x
-1
-2
-3
-4
-5
A) Relative maximum at -2; relative minimum at 2
B) Relative minimum at -4
C) Relative maxima at -2 and 2
D) Relative minimum at -2; relative maximum at 2
Find the x-value of all points where the function has relative extrema. Find the value(s) of any relative extrema.
27) f(x) = x2 + 2x - 3
27)
A) Relative minimum of -4 at -1.
C) Relative maximum of -4 at -1.
B) Relative minimum of 0 at -2.
D) Relative minimum of -2 at 0.
4
Find f"(x) for the function.
28) f(x) = 2x3/2 - 6x1/2
28)
A) 1.5x1/2 + 1.5x-1/2
C) 3x-1/2 + 3x-3/2
29) f(x) =
x
x+1
A) (x + 1)-2
30) f(x) =
A)
B) 1.5x-1/2 + 1.5x-3/2
D) 3x1/2 - 3x-1/2
29)
B) -2(x + 1)-2
C) -2(x + 1)-3
D) (x + 1)-3
ln x
8x
-3 + 2 ln x
8x3
30)
B)
-7 - 2 ln x
9x3
C)
- ln x
8x3
D)
-3 - 2 ln x
8x
Find the requested value of the second derivative of the function.
x
31) f(x) =
; Find f′′ (2).
x+1
A) -
2
27
B) 0
C)
31)
2
3
D)
1
9
Find the open intervals where the function is concave upward or concave downward. Find any inflection points.
32)
32)
A) Concave upward on (0, ∞); concave downward on (-∞, 0); inflection points at (-4, 0), (-1, 0),
7
and , 0
2
B) Concave upward on (-1, ∞); concave downward on (-∞, 2); inflection points at (-1, 0) and (2,
-3)
C) Concave upward on (0, ∞); concave downward on (-∞, 0); inflection point at (0, -1)
D) Concave upward on (-1, ∞); concave downward on (-∞, 2); inflection point at (2, -3)
Find the largest open intervals where the function is concave upward.
33) f(x) = x2 + 2x + 1
A) (-1, ∞)
B) (-∞, -1)
C) None
5
33)
D) (-∞, ∞)
34) f(x) = x3 - 3x2 - 4x + 5
A) (-∞, 1), (1, ∞)
B) (-∞, 1)
C) (1, ∞)
D) None
35) f(x) = 5x - 6e-x
A) (-∞ , 0 )
B) None
C) ( 0 , ∞ )
D) ( -∞, ∞ )
34)
35)
Find any inflection points given the equation.
36) f(x) = 3x2 + 12x
36)
A) No inflection points
C) Inflection point at (4,-12)
B) Inflection point at (-2,-12)
D) Inflection point at (-4,-12)
37) f(x) = 2x3 - 15x2 + 24x
37)
A) Inflection points at (4, -16), (1, 11)
B) Inflection point: (0, 0)
C) No inflection points
D) Inflection point at
5
5
,2
2
Suppose that the function with the given graph is not f(x), but f′(x). Find the open intervals where the function is concave
upward or concave downward, and find the location of any inflection points.
38)
38)
y
120
80
40
-5 -4 -3 -2 -1
-40
1
2
3
4
5
x
-80
-120
A) Concave upward on (-3, 3); concave downward on (-∞, -3) and (3, ∞); inflection points at -3
and 3
B) Concave upward on (-∞, -3) and (3, ∞); concave downward on (-3, 3); inflection points at -3
and 3
C) Concave upward on (-∞, -3) and (3, ∞); concave downward on (-3, 3); inflection points at -20
and
20
D) Concave upward on (-∞, 0); concave downward on (0, ∞); inflection point at 0
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain.
39) f(x) = x2 - 4; [-1, 2]
Maximum
A) -3 at x = -1
B) 0 at x = 2
C) -3 at x = 1
6
D) 0 at x = -2
39)
40) f(x) =
1
; [-4, 1]
x+2
40)
Minimum
1
A) at x = 0
2
B)
C) No absolute minimum
1
at x = 1
3
D) -
1
at x = -4
2
Find the absolute extrema if they exist as well as where they occur.
41) f(x) = -3x4 + 16x3 - 18x2 + 4
41)
A) Absolute maximum of -1 at x = 1; no absolute minima
B) No absolute extrema
C) Absolute maximum of 31 at x = 3; no absolute minima
D) Absolute maximum of 12 at x = 2; no absolute minima
Solve the problem.
42) P(x) = -x3 +
27 2
x - 60x + 100, x ≥ 5 is an approximation to the total profit (in thousands of dollars)
2
42)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must
be sold to maximize profit.
A) 5 hundred thousand
B) 4 hundred thousand
C) 4.5 hundred thousand
D) 5.5 hundred thousand
Find dy/dx by implicit differentiation.
43) x3 + y3 = 5
y2
A)
x2
44) x1/3 - y1/3 = 1
y 2/3
A) x
B)
43)
x2
C) -
y2
y2
D) x2
y2
44)
x 2/3
B) y
x 2/3
C)
y
45) x ln y + y = x4 y6
dy 4x3 y7 - y ln y
A)
=
dx
x - 6x4 y6 + y
C)
x2
y 2/3
D)
x
45)
dy 4x3 y7 - y ln y - 1
=
dx
x - 6x4 y6 + y
4x3 y6 - ln y
B)
dy
=
dx x - 6x4 y6 + y
D)
dy
4x3 y7 - ln y
=
dx x - 6x4 y6 + 1
Find the equation of the tangent line at the given point on the curve.
46) x2 + y2 + 2y = 0; (0, -2)
A) y = -x
B) y = -2
C) x = 0
7
46)
D) y = -x - 2
Solve the problem.
47) S(x) = -x3 - 9x2 + 165x + 1300, 5 ≤ x ≤ 20 is an approximation to the number of salmon swimming
47)
upstream to spawn, where x represents the water temperature in degrees Celsius. Find the
temperature that produces the maximum number of salmon.
A) 5°C
B) 19°C
C) 20°C
D) 6°C
48) The velocity of a particle (in
ft
) is given by v = t2 - 8t + 7, where t is the time (in seconds) for which
s
it has traveled. Find the time at which the velocity is at a minimum.
A) 8 sec
B) 7 sec
C) 3.5 sec
8
D) 4 sec
48)
Answer Key
Testname: EXAMIII STUDYGUIDE_2015
1) B
2) B
3) C
4) A
5) A
6) B
7) C
8) D
9) B
10) A
11) B
12) C
13) B
14) A
15) B
16) C
17) A
18) C
19) A
20) B
21) A
22) B
23) C
24) C
25) D
26) A
27) A
28) B
29) C
30) A
31) A
32) C
33) D
34) C
35) B
36) A
37) D
38) B
39) B
40) C
41) C
42) A
43) C
44) D
45) A
46) B
47) A
48) D
9