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Practice Exam 2 (1479895)
Question 1234567891011121314151617181920212223242526272829303132333435363738394041
1.
Question DetailsLarCalc9 3.1.004.MI. [1373972]
Find the value of the derivative (if it exists) at the indicated extremum. (If an answer does not exist, enter DNE.)
−2 2 3
,
3
3
f(x) = – 3x√x + 1
f ' −2
3
=
0
Tutorial
2.
Question DetailsLarCalc9 3.1.011.MI. [1343927]
Find any critical numbers of the function. (Enter your answers as a comma-separated list.)
f(x) = x8 – 8x7
x=
Tutorial
3.
Question DetailsLarCalc9 3.1.012. [1196826]
Find any critical numbers of the function. (Enter your answers as a comma-separated list.)
g(x) = x5 − 10x3
4.
Question DetailsLarCalc9 3.1.015. [1197752]
Find any critical numbers of the function. (Enter your answers as a comma-separated list.)
(sin(x))2 + cos(x)
0 < x < 2π
x=
5.
Question DetailsLarCalc9 3.1.022. [1122225]
Find the absolute extrema of the function on the closed interval.
f (x) = x3 − 3x,
[0, 4]
minimum
(x, y) = (
1, -2 )
maximum
(x, y) = (
4, 52 )
Question DetailsLarCalc9 3.1.043. [1122213]
6.
Use a graphing utility to graph the function. Locate the absolute extrema of the function on the given interval. (If an answer does not exist, enter DNE.)
minimum
(x, y) = (
6, 9/2 )
maximum
(x, y) = (
DNE )
7.
Question DetailsLarCalc9 3.2.005. [1245553]
Find the two x-intercepts of the function f and show that f '(x) = 0 at some point between the two x-intercepts.
f(x) = x2 − 3x − 10
(x, y) = (
) (smaller x-value)
(x, y) = (
) (larger x-value)
Find a value of x such that f '(x) = 0.
x=
1.5
Question DetailsLarCalc9 3.2.007. [1117244]
8.
Find the two x-intercepts of the function f and show that f '(x) = 0 at some point between the two x-intercepts.
f (x) = x x + 7
(x, y) = (
-7, 0 ) (smaller x-value)
(x, y) = (
0, 0 ) (larger x-value)
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Find a value of x such that f '(x) = 0.
x=
-14/3
9.
Question DetailsLarCalc9 3.2.009. [1117255]
The graph of f is shown. Apply Rolle's Theorem and find all values of c such that f '(c) = 0 at some point between the labeled intercepts. (Enter your answers as a commaseparated list.)
f (x) = x2 + 6x − 27
c =
-3
Question DetailsLarCalc9 3.2.029. [1117242]
10.
The height of a ball t seconds after it is thrown upward from a height of 6 feet and with an initial velocity of 80 feet per second is f (t) = -16t 2 + 80t + 6.
(a) Verify that f(2) = f(3).
f (2) =
102 ft
f (3) =
102 ft
(b) According to Rolle's Theorem, what must be the velocity at some time in the interval (2, 3)?
0 ft/sec
Find that time.
t =
5/2 s
11.
Question DetailsLarCalc9 3.2.040. [1589248]
Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply.)
f (x) = x9, [0,1]
Yes, the Mean Value Theorem can be applied.
No, f is not continuous on [a, b].
No, f is not differentiable on (a, b).
None of the above.
If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f '(c) = f (b) − f (a) . (Enter your answers as a comma-separated list. If
b−a
the Mean Value Theorem cannot be applied, enter NA.)
c =
12.
Question DetailsLarCalc9 3.2.048. [1528431]
Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply.)
f(x) = cos x + tan x, [0, π]
Yes, the Mean Value Theorem can be applied.
No, f is not continuous on [a, b].
No, f is not differentiable on (a, b).
None of the above.
If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f '(c) = f (b) − f (a) . If the Mean Value Theorem cannot be applied,
b−a
explain why not. (Enter your answers as a comma-separated list. If the Mean Value Theorem cannot be applied, enter NA.)
NA
13.
Question DetailsLarCalc9 3.3.005. [1122403]
Use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. (Select all that apply.)
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Increasing:
(-
, -4)
(-4, 4)
(4,
)
none of these
Decreasing:
(-
, -4)
(-4, 4)
(4,
)
none of these
14.
Question DetailsLarCalc9 3.3.010. [1122228]
Identify the open intervals on which the function is increasing or decreasing. (Select all that apply.)
h(x) = 108x − x3
Increasing:
(− , −6)
(−6, 6)
(6,
)
none of these
Decreasing:
(− , −6)
(−6, 6)
(6,
)
none of these
15.
Question DetailsLarCalc9 3.3.021. [1122194]
Consider the following function.
(a) Find the critical numbers of f. (Enter your answers as a comma-separated list.)
x=
-2, 1
(b) Find the open intervals on which the function is increasing or decreasing. (Select all that apply.)
Increasing:
(− , −2)
(−2, 1)
(1,
(− ,
)
)
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Decreasing:
(− , −2)
(−2, 1)
)
(1,
(− ,
)
(c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.)
relative maximum (x, y) = (
-2, 240 )
relative minimum (x, y) = (
1, -84 )
(d) Use a graphing utility to confirm your results.
16.
Question DetailsLarCalc9 3.3.035. [1267105]
Consider the following function.
x2
f(x) =
x2 − 4
(a) Find the critical numbers of f. (Enter your answers as a comma-separated list.)
x=
(b) Find the open intervals on which the function is increasing or decreasing. (Select all that apply.)
Increasing:
(-
,
-2)
(-2,
0)
(0, 2)
(2,
)
none of
these
Decreasing:
(-
, -2)
(-2, 0)
(0, 2)
(2,
)
none of
these
(c) Apply the First Derivative Test to identify the relative extremum. (If an answer does not exist, enter DNE.)
(x, y) =
relative maximum
(x, y) =
relative minimum
(d) Use a graphing utility to confirm your results.
17.
Question DetailsLarCalc9 3.3.089. [1267037]
The function s(t) describes the motion of a particle along a line.
s(t) = 16t − t 2
(a) Find the velocity function v(t) of the particle at any time t ≥ 0.
v(t) =
(b) Identify the time interval in which the particle is moving in a positive direction.
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(0,
)
(0, 8)
(0, 16)
(16,
)
(8,
)
None
(c) Identify the time interval in which the particle is moving in a negative direction.
(0,
)
(0, 8)
(0, 16)
)
(16,
(8,
)
None
(d) Identify the time at which the particle changes direction.
t =
8
18.
Question DetailsLarCalc9 3.4.006. [1122380]
Determine the open intervals on which the graph is concave upward or concave downward.
y = −x3 − 3x2 − 3
Concave upward:
(− ,
−1)
(−
,
1)
(−1,
)
(1,
)
none of these
Concave downward:
(−
,
−1)
(−
,
1)
(−1,
)
(1,
)
none of these
19.
Question DetailsLarCalc9 3.4.009. [1122475]
Determine the open intervals on which the graph is concave upward or concave downward.
f(x) = −x3 − 9x2 − 5x − 9
Concave upward:
(− ,
−3)
(−
,
3)
(−3,
)
(3,
)
none of these
Concave downward:
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(−
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,
−3)
(−
,
3)
(−3,
)
(3,
)
none of these
20.
Question DetailsLarCalc9 3.4.011. [1122470]
Determine the open intervals on which the graph is concave upward or concave downward.
12
f (x) = 2
x + 12
Concave upward:
(−
, −2)
(−2, 2)
(− , −2)
(−
,
(2,
)
)
none of these
Concave downward:
(−
, −2)
(−2, 2)
(−
, −2)
(−
,
(2,
)
)
none of these
Question DetailsLarCalc9 3.4.017. [1267186]
21.
Determine the open intervals on which the graph is concave upward or concave downward.
− π, π
y = 6x − tan x,
2 2
Concave upward:
− π,
2
π
2
π
,
2
−
0
0,
π
2
none of these
Concave downward:
− π,
2
π
2
− π,
2
0
0,
π
2
none of these
Question DetailsLarCalc9 3.4.021. [1122282]
22.
Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.)
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f (x) = x3 − 6x2 + 17x
(x, y) = (
2, 18 )
Describe the concavity.
Concave upward:
(−
,
2)
(−
,
−2)
(2,
)
(−2,
)
none of these
Concave downward:
(−
,
2)
(−
,
−2)
(2,
)
(−2,
)
none of these
23.
Question DetailsLarCalc9 3.4.041. [1247612]
Find all relative extrema. Use the Second Derivative Test where applicable. (If an answer does not exist, enter DNE.)
f(x) = x3 − 6x2 + 1
relative maximum
(x, y) = (
relative minimum
(x, y) = (
)
)
24.
Question DetailsLarCalc9 3.5.007. [1122271]
Use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate your answer
graphically. (Round your answers to five decimal places. If you need to use
or – , enter INFINITY or –INFINITY, respectively.)
100
x
f (x)
101
-6.50000
104
x
f (x)
105
4.00105
lim f(x) =
102
5.31250
103
4.10714
4.01052
106
4.00011
4.00001
4
x→
25.
Question DetailsLarCalc9 3.5.010. [1126261]
Use a graphing utility to complete the table and estimate the limit as x approaches infinity. Then use a graphing utility to graph the function and estimate your answer
graphically. (Round your answers to five decimal places. If you need to use
or – , enter INFINITY or –INFINITY, respectively.)
100
x
f (x)
101
3.88057
104
x
f (x)
105
3.20000
lim f(x) =
102
3.20513
103
3.20005
3.20000
106
3.20000
3.20000
16/5
x→
26.
Question DetailsLarCalc9 3.5.023. [1267103]
Find the limit. (If you need to use
or – , enter INFINITY or –INFINITY, respectively.)
2x8
lim
x→
x9 − 7
0
Question DetailsLarCalc9 3.5.027. [1122264]
27.
Find the limit. (If you need to use
or – , enter INFINITY or –INFINITY, respectively.)
x
lim
x2 − x
x→−
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-1
28.
Question DetailsLarCalc9 3.5.088. [1245583]
A business has a cost of C = 0.5x + 250 for producing x units. The average cost per unit is given by the following.
C= C
x
Find the limit of the average cost as x approaches infinity.
0.5
29.
Question DetailsLarCalc9 3.6.017. [1197119]
Analyze and sketch a graph of the function. Find any intercepts, relative extrema, points of inflection, and asymptotes. (If an answer does not exist, enter DNE.)
intercepts
(x, y) = (
)
(smaller x-value)
(x, y) = (
)
(larger x-value)
relative minimum
(x, y) = (
relative maximum
(x, y) = (
point of inflection
(x, y) = (
)
)
)
Find the equation of the asymptote.
Use a graphing utility to verify your results.
30.
Question DetailsLarCalc9 3.6.048. [1122334]
Suppose f(0) = 9 and 2 ≤ f '(x) ≤ 6 for all x in the interval [-5, 5]. Determine the greatest and least possible values of f(5).
least
19
greatest
39
Question DetailsLarCalc9 3.6.070. [1197380]
31.
Create a function whose graph has the given characteristics. (There is more than one correct answer.)
Vertical asymptote: x = 3
Slant asymptote: y = -x
y=
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32.
Question DetailsLarCalc9 3.7.008. [1267195]
Find two positive numbers that satisfy the given requirements.
The sum of the first number squared and the second number is 51 and the product is a maximum.
(first number)
(second number)
33.
Question DetailsLarCalc9 3.7.013. [1122337]
Find the point on the graph of the function that is closest to the given point.
Function
Point
16, 1
f(x) = x2
2
(x, y) = (
2, 4 )
34.
Question DetailsLarCalc9 3.7.020.MI. [1356893]
On a given day, the flow rate F (in cars per hour) on a congested roadway is given by the equation below, where v is the speed of traffic in miles per hour.
v
F=
24 + 0.01v2
What speed will maximize the flow rate on the road? (Round your answer to one decimal place.)
49 mi/h Tutorial
35.
Question DetailsLarCalc9 3.7.021. [1122187]
A farmer plans to enclose a rectangular pasture adjacent to a river. (see figure). The pasture must contain 405,000 square meters in order to provide enough grass for the
herd. What dimensions will require the least amount of fencing if no fencing is needed along the river?
x=
900 m
y=
450 m
Question DetailsLarCalc9 3.8.001. [1122270]
36.
Complete two iterations of Newton's Method for the function using the given initial guess. (Round your answers to four decimal places.)
f(x) = x2 − 3, x1 = 1.9;
n
f(xn)
xn
f(xn)
f '(xn)
f '(xn)
xn −
f(xn)
f '(xn)
1
1.9000
0.6100
3.8000
0.1605
1.7395
2
1.7395
0.0258
3.4789
0.0074
1.7321
37.
Question DetailsLarCalc9 3.8.004. [1267075]
Complete two iterations of Newton's Method for the function using the given initial guess. (Round your answers to four decimal places.)
f(x) = tan x, x1 = 0.06
f(xn)
n
xn
1
0.06
2
f(xn)
f '(xn)
f '(xn)
0.0001
xn −
f(xn)
f '(xn)
0.0601
1.0036
0.0599
0.0001
0.0001
1.0000
0.0001
0.0000
38.
Question DetailsLarCalc9 3.9.001. [1197162]
Find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. (Round your answers to four decimal
places.)
f(x) = x2, (4, 16)
T(x) =
3.9
x
3.99
4
4.01
4.1
f(x)
15.2100
15.9201
16.0000
16.0801
16.8100
T(x)
15.2000
15.9200
16.0000
16.0800
16.8000
39.
Question DetailsLarCalc9 3.9.004. [1267719]
Find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. (Round your answer to four decimal
places.)
x , (3,
3)
f(x) =
x
2.9
2.99
3
3.01
3.1
f(x)
1.7029
1.7292
1.7321
1.7349
1.7607
T(x)
1.7032
1.7292
1.7321
1.7349
1.7609
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40.
Question DetailsLarCalc9 3.9.009. [1245592]
Use the information to evaluate and compare ∆y and dy. (Round your answers to four decimal places.)
y = x4 + 5
x = −2
∆x = dx = 0.01
∆y =
-0.3176
dy =
-0.3200
41.
Question DetailsLarCalc9 3.9.030. [1122433]
The measurement of the edge of a cube is measured to be 11 inches, with a possible error of 0.02 inch. Use differentials to approximate the possible error and the relative
error in computing the following values. (Round your answers to three decimal places.)
(a) the volume of the cube
possible error ±
7.260 in3
relative error ±
0.005
(b) the surface area of the cube
possible error ±
2.640 in2
relative error ±
0.004
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Name (AID): Practice Exam 2 (1479895)
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Author: Downs, Jeff ( downsj4@wnc.edu )
Last Saved: Oct 13, 2010 05:26 PM PDT
Permission: Protected
Randomization: Person
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10/13/2010 6:00 PM