MATH 140 Exam 1
Name: _____________________________
INSTRUCTIONS:
• The exam is worth 100 points. There are 25 problems (each worth 4 points).
• This quiz is open book and open notes, unlimited time. This means that you may
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as you wish, provided you turn in your exam no later than the due date posted in our
course syllabus.
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you may earn only partial or no credit at the discretion of the professor.
• If you have any questions, please contact me via Instant Message in LEO.
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1
MULTIPLE CHOICE. Choose the one alternative that best completes the statement
or answers the question.
1) Let log 𝑏 𝐴 = 3.420 and log 𝑏 𝐵 = 0.357. Find log 𝑏 𝐴𝐵
(𝐴) 3.777
(𝐵) 9.580
(𝐶) 3.063
(𝐷) 1.221
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
2) Find the average velocity of the function over the given interval:
𝑦 = √2𝑥,
(𝐴) 7
(𝐵) −
3
10
(𝐶)
1
3
(𝐷) 2
[2, 8]
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
2
For the position function, 𝒔(𝒕), make a table of estimated velocities and make a
conjecture about the instantaneous velocity at the indicated time.
3) 𝑠′(𝑡) = 𝑡 2 + 8𝑡 − 2
𝑡
𝑠′(𝑡)
1.9
𝑎𝑡
1.99
𝑡=2
1.999
2.001
A) instantaneous velocity is 18.0
𝑡
1.9
1.99
1.999
𝑠′(𝑡)
16.810 17.880 17.988
2.01
2.001
18.012
2.001
2.01
18.120
B) instantaneous velocity is ∞
𝑡
1.9
1.99 1.999 2.001
𝑠′(𝑡)
19
199
1999 2001
2.01
201
C) instantaneous velocity is 5.40
𝑡
1.9
1.99 1.999 2.001
𝑠′(𝑡)
5.043 5.364 5.396 5.404
2.01
5.436
D) instantaneous velocity is 17.70
𝑡
1.9
1.99
1.999
𝑠′(𝑡)
16.692 17.592 17.689
2.001
17.710
2.001
19.210
2.001
2001
2.001
5.763
2.01
17.808
E) none of the above
3
2.001
18.789
4) Given
lim 𝑓(𝑥) = 𝐿𝑙𝑒𝑓𝑡 ,
lim 𝑓(𝑥) = 𝐿𝑟𝑖𝑔ℎ𝑡 ,
𝑥→0−
𝑎𝑛𝑑 𝐿𝑙𝑒𝑓𝑡 ≠ 𝐿𝑟𝑖𝑔ℎ𝑡
𝑥→0+
Which of the following statements is true? Sketch a graph to illustrate your reasoning.
𝐼. lim 𝑓(𝑥) = 𝐿𝑙𝑒𝑓𝑡
𝑥→ 0
𝐼𝐼. lim 𝑓(𝑥) = 𝐿𝑟𝑖𝑔ℎ𝑡
𝑥→ 0
𝐼𝐼𝐼. lim 𝑓(𝑥) 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
𝑥→ 0
(𝐴) 𝐼𝐼𝐼 𝑜𝑛𝑙𝑦
(𝐵) 𝐼𝐼 𝑜𝑛𝑙𝑦
(𝐶) 𝐼 𝑜𝑛𝑙𝑦
(𝐷) 𝐼 𝑎𝑛𝑑 𝐼𝐼
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Use the table of values of 𝒇(𝒙) to estimate the limit.
5) Let 𝑓(𝑥) =
𝑥−4
, find lim 𝑓(𝑥).
√𝑥−2
𝑥→ 4
A)
; limit = 5.10
B)
; limit = ∞
C)
; limit = 1.20
D)
; limit = 4.0
E) none of the above
4
Give an appropriate answer.
6) Let lim 𝑓(𝑥) = − 4 and lim 𝑔(𝑥) = 2. Find lim [𝑓(𝑥) + 𝑔(𝑥)]2 .
𝑥→ 4
(𝐴) 20
𝑥→ 4
(𝐵) − 6
(𝐶) − 2
𝑥→ 4
(𝐷) 4
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Find the limit.
√1 + 𝑥 − 1
7) lim
𝑥→ 0
𝑥
(𝐴) 𝑙𝑖𝑚𝑖𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
(𝐵) 0
(𝐶)
1
4
1
2
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
(𝐷) 2
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
(𝐷)
Find the limit, if it exists.
𝑥4 − 1
𝑥→ 1 𝑥 − 1
8) lim
(𝐴) 𝑙𝑖𝑚𝑖𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
(𝐵) 4
(𝐶) 0
5
Provide an appropriate response.
9) If 𝑥 3 ≤ 𝑓(𝑥) ≤ 𝑥 for 𝑥 in [−1, 1], find lim 𝑓(𝑥) if it exists.
𝑥→ 0
(𝐴) − 1
(𝐵) 1
(𝐶) 0
(𝐷) 𝑙𝑖𝑚𝑖𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
10-12. Find the limit.
10) lim
𝑥→ 5+ 𝑥 2
(𝐴) 0
11)
(𝐵) − ∞
lim
𝑥→ ∞
(𝐴) 8
1
− 25
(𝐶) 1
(𝐷) ∞
(𝐶) 1
(𝐷)
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
8
8 − (1/𝑥 2 )
(𝐵) − ∞
8
7
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
6
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
12) lim
cos 5𝑥
𝑥
(𝐴) 5
(𝐵) 0
𝑥→ ∞
(𝐶) − ∞
(𝐷) 1
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Find all points where the function is discontinuous.
13)
(𝐴) 𝑥 = −2, 0
(𝐵) 𝑥 = 0, 2
(𝐶) 𝑥 = 2
(𝐷) 𝑥 = −2, 0, 2
Find the number 𝒌, so that 𝒇 is continuous at every point.
14)
7
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
(𝐴) 𝑘 =
1
5
(𝐵) 𝑘 = 5
(𝐶) 𝑘 = 25
(𝐷) 𝑛𝑜𝑡 𝑒𝑛𝑜𝑢𝑔ℎ 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Find an equation for the tangent to the curve at the given point.
15) 𝑓(𝑥) = 6√𝑥 − 𝑥 + 1,
1
(𝐴) 𝑦 = − 𝑥 + 19
2
(36, 1)
1
(𝐵) 𝑦 = − 𝑥 + 1
2
(𝐶) 𝑦 = 1
(𝐷) 𝑦 =
1
𝑥 − 19
2
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Suppose u and v are differentiable functions of x. Use the given values of the
functions and their derivatives to find the value of the indicated derivative.
16) 𝑢(1) = 2,
𝑢′ (1) = −7,
𝑣(1) = 7,
𝑣 ′ (1) = −3
8
𝑑
(3𝑣 − 𝑢) 𝑎𝑡 𝑥 = 1
𝑑𝑥
(𝐴) 23
(𝐵) − 16
(𝐶) − 2
(𝐷) 19
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Solve the problem.
17) The power 𝑃 (in W) generated by a particular windmill is given by 𝑃 = 0.015𝑉 3 where 𝑉
is the velocity of the wind (in mph). Find the instantaneous rate of change of power with
respect to velocity when the velocity is 9.2 mph.
(𝐴) 23.4
𝑊
𝑚𝑝ℎ
(𝐵) 8.5
𝑊
𝑚𝑝ℎ
(𝐶) 0.4
𝑊
𝑚𝑝ℎ
(𝐷) 3.8
Find 𝒚′
1
1
18) 𝑦 = (𝑥 + ) (𝑥 − )
𝑥
𝑥
9
𝑊
𝑚𝑝ℎ
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
(𝐴) 2𝑥 −
1
𝑥2
(𝐵) 2𝑥 +
1
𝑥3
(𝐶) 2𝑥 +
1
𝑥2
(𝐷) 2𝑥 +
2
𝑥3
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Suppose that the functions f and g and their derivatives with respect to x have the
following values at the given values of x. Find the derivative with respect to x of
the given combination at the given value of x.
19)
(𝐴) 8
𝑓(𝑔(𝑥)),
(𝐵) − 32
(𝐶) − 20
𝑥=4
(𝐷) 24
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Find dy/dt.
20) 𝑦 = 3𝑡(5𝑡 + 4)5
10
(𝐴) 3(5𝑡 + 4)4 (30𝑡 + 4)
(𝐵) 15𝑡(5𝑡 + 4)4
(𝐶) 3(5𝑡 + 4)5 (10𝑡
4
(𝐷) 3(30𝑡 + 4)
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
+ 4)
Solve the problem.
21) The position of a particle moving along a coordinate line is 𝑠 = √2 + 2𝑡, with 𝑠 in meters
and 𝑡 in seconds. Find the particle's velocity at 𝑡 = 1 seconds.
(𝐴) 1 𝑚/𝑠
(𝐵)
1
𝑚/𝑠
2
(𝐶)
1
𝑚/𝑠
4
(𝐷) −
1
𝑚/𝑠
2
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Find 𝒅𝒚/𝒅𝒙 by implicit differentiation.
22) 𝑥 1/3 − 𝑦1/3 = 1
𝑥 2/3
(𝐴) − ( )
𝑦
𝑦 2/3
(𝐵) ( )
𝑥
𝑥 2/3
(𝐶) ( )
𝑦
𝑦 2/3
(𝐷) − ( )
𝑥
11
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Find the derivative of 𝒚 with respect to 𝒙.
23) 𝑦 = ln(ln 6𝑥)
(𝐴)
1
𝑥
(𝐵)
1
ln 6𝑥
(𝐶)
1
6𝑥
(𝐷)
1
𝑥 ln 6𝑥
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Solve the problem.
24) A piece of land is shaped like a right triangle. Two people start at the right angle of the
triangle at the same time, and walk at the same speed along different legs of the triangle. If
the area formed by the positions of the two people and their starting point (the right angle)
is changing at 5 𝑚2 /𝑠, then how fast are the people moving when they are 3 𝑚 from the
right angle? (Round your answer to two decimal places.)
(𝐴) 1.67 𝑚/𝑠
(𝐵) 0.83 𝑚/𝑠
(𝐶) 3.33 𝑚/𝑠
12
(𝐷) 1.80 𝑚/𝑠
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒
Solve the problem. Round your answer, if appropriate.
25) Water is being drained from a container which has the shape of an inverted right
circular cone. The container has a radius of 4.00 inches at the top and a height of 8.00
inches. At the instant when the water in the container is 6.00 inches deep, the surface level
is falling at a rate of 0.9 𝑖𝑛/𝑠. Find the rate at which water is being drained from the
container.
(𝐴) 36.8 𝑖𝑛3 /𝑠
(𝐵) 42.4 𝑖𝑛3 /𝑠
(𝐶) 25.4 𝑖𝑛3 /𝑠
13
(𝐷) 24.3 𝑖𝑛3 /𝑠
(𝐸) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑜𝑣𝑒