Math 142
Midterm Exam Review with Answers
Name ________________________
Date __________
The Review has 16 questions. Simplify all answers, include all units when
appropriate.
1. [Sec. 1.3] Solve the following problems.
a. A company’s profit increased linearly from $6 million at the end of year 1 to $14
million at the end of year 3.
i.
Use the two data points (year, profit) coordinates as (1,6) and (3,14), to find
the linear relationship y mx b between x years and y profit.
ii. Find the company’s profit at the end of 5 years.
iii. Predict the year when the profit will be 30 million.
b.
A Telephone company charges a client $24.95 for a service of 200 minutes of
long distance, and charges another client $44.95 for service of 400 minutes of
long distance calls.
i.
Use the data points (minutes, charge) coordinates to find the linear
relationship y=mx+b between x number of minutes and y dollars charged.
ii. Find the company’s charge for 100 minutes of long distance service.
iii. Find the number of minutes of service for a client that was charged $8.15.
2.
[Sec. 1.2] Given the function
2
a. f ( x)
2x 3
b. f ( x)
2
9 x2
i. Write the domain in set notation.
ii. Find the x and y intercepts if any, and write as order pair.
iii. Evaluate f (3)
iv. Graph the function, label three points.
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Math 142
Midterm Exam Review with Answers
3. [Sec. 1.4]
a) A group of students want to raise money to attend an international conference; the
students want to build canoes during the summer in an old warehouse. The rental
of the warehouse is $1500 for the summer, and the materials needed to build one
canoe cost $125. The canoes can be sold for $275 apiece.
i. Write the equation for the Profit.
ii. How many canoes must the students sell to break even?
iii. How many canoes must the students sell to make a profit of at least
$1000?
b) A manufacturer can produce bookcases at a cost of $80 apiece. Sales figures
indicate that if the bookcases are sold for x dollars apiece, approximately150 – x
bookcases will be sold each month.
i. Express the manufacturer’s monthly profit as a function of selling price x.
ii. Find the price where the manufacturer breaks even.
iii. Find the price to maximize the profit.
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Math 142
4.
Midterm Exam Review with Answers
[Sec. 1.1] If, a) f ( x) 3 x 2 and g ( x) 3x
i. Find
iv.
ii. Determine the domain of the function
notation.
and
, write in set
v.
iii. Find the composition f ( g ( x)) , simplify if possible.
iv. Find the composition g ( f ( x)) , simplify if possible.
b. If
and
i.
Find
ii.
Determine the domain of the function
notation.
iii.
Find the composition f ( g ( x)) , simplify if possible.
iv.
Find the composition g ( f ( x)) , simplify if possible.
5. [Sec1.6] a. Graph f ( x)
and
, write in set
2x 3 x  3
Label four points.
8 x x 3
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Math 142
Midterm Exam Review with Answers
Lim f ( x)
x
______________
3
x
Lim f ( x) EXIST
x
Lim f ( x)
/ NOT EXIST
______________
3
(Circle One)
3
x 2 x 4
Label four points.
7 2x x  4
b. Graph f ( x)
Lim f ( x)
______________
Lim f ( x)
______________
x
x
4
4
Lim f ( x) EXIST
x
/ NOT EXIST
(Circle One)
4
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Math 142
Midterm Exam Review with Answers
6. [Sec. 1.5] Find the limit or if it does not exist. If it is infinite indicate whether it is +∞
or -∞
a.
c.
e.
x2
Lim
2
x
2 x
4x 4
3x 2
b.
x( x 2 2)
Lim
x2
x 0
d.
x2 x 5
Lim
1 2 x x3
x
x2
Lim
2
x 7 x
Lim x
x
2
x
8x 7
6x 7
4
16
x2 x 5
Lim1 2 x x3
f. x
7. [Sec 2.1] For the given function use the definition of the derivative to find f ' ( x) ,
Show all the required steps.
a.
f ( x)
4x 2
3x 4
b. f ( x)
7 x2
8. [Sec. 2.2] Find the derivative, f ' ( x)
a.
f ( x)
c. f ( x)
5x 4
2
x4
3x 2
3
x3
6x 3
4x
b.
f ( x)
d.
f ( x)
3
x2
2 x
5
3
x2
9. [Sec 2.3] Find the derivative of the function using the product rule of derivatives,
f’(x), simplify.
a. f ( x)
(x2
x)(3x 1)
b. f ( x)
(4 x 3
3)( x 3)
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Math 142
Midterm Exam Review with Answers
10. [Sec 2.3] Find the derivative, y’, simplify if possible.
a. y
x4
x2 1
x2 1
b. y
x2 x 1
2x 5
11. [Sec 2.4] Find the derivative, f’(x), simplify if possible.
1
a. f ( x)
4x
3
4x 3
b. f ( x)
3
3
x2
3x 4
12. [Sec. 2.4] Evaluate the second derivative of the given function.
a. f ' ' (1) for the function f ( x)
3x 2
5x
b. f ''(1) for the function f ( x)
3x 2
13. [Sec 2.5] Solve the following applied problems, interpret results
a. A steel mill finds that its cost function is C ( x) 8000 x 6003 x dollars,
where x is the daily production of steel (in tons).
i.
Find the Marginal Cost Function.
ii.
Find the marginal cost when 64 tons of steel are produced.
iii.
Interpret the results.
b. The manufacturer determines that the profit P (in dollars) derived from selling x
units is given by P( x) 0.0004 x 3 7 x .
i.
Find the Marginal Profit Function.
ii.
Find the marginal profit for a production level of 52 units.
iii.
Interpret the results.
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Math 142
Midterm Exam Review with Answers
14. [Sec. 2.4] A company’s profit function is P( x) 13x 1700
a. Find the Relative Rate of Change of the Profit Function.
b. Find the Marginal Profit Function.
c. Evaluate the Relative Rate of Change of the Profit Function at x = 250.
15. [Sec. 2.3] Solve the following applied problems, interpret results.
d. The number of weeds (in hundreds) n in lawn x weeks after a weed killer was
applied is given by n( x)
x 2 2 x 3 . Find the rate of change in the
number of weeds in the lawn at the end of the second week.
1
e. The velocity of an object in meters per second is v(t ) 1 9t 2 , 0 t
.
3
Find the acceleration of the subject when t = .15 sec.
16. [Sec 3.1 & 3.2] For the following functions:
f. The function, f ( x)
x3
6x 2
9 x 1,
g. The function, f ( x)
x4
4x 3
20 x 2 12
i. Make a sign diagram of f ' ( x )
ii. Identify the relative extreme points and inflection points
iii. Sketch the graph
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Math 142
Midterm Exam Review with Answers
Midterm Review Math 142 Answers:
#1
a.
i. y = 4x + 2
iii. @ the end of the 7th year
b.
i. y = .1x + 4.95 or y
1
x 4.95
10
ii. y = $22 million
ii. y = $ 14.95
iii. 32 minutes
#2
a.
i.
xx
3
2
iii. f (3) =
ii. x-intercept: None & y-intercept:
2
3
iv. Graph curve with label on any of three points (2, 2); (1, -2);
#3
#4
)
b.
i. { x| -3 x 3 }
ii. x-intercept:
iii. f (3) = 2
iv. Graph curve with label on any of three points: (0, -1); (3, 2); (-3,2)
a.
i. P( x) 150 x 1500
iii. x 17 canoes
b.
i. P( x)
x 2 230 x 12000
iii. price of $115 per unit
a.
i. f(-2) = -1; f((3) = 3 ; g(3) = 9
ii. Domain f(x) {R}; g(x) {R}
iv. g(f(x)) = 9 – 3x2
b.
ii. 10 canoes
ii. @ price of $80 and $150 per unit
iii. f(g(x)) = 3 – 9x2
i. f(-2) = Undefined; f(0) = Undefined; g(3) = 5
1
ii. Domain f(x) x; x
; g(x){R}
iii. f ( g ( x))
3
iv. g ( f ( x))
6x 4
3x 1 1
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Math 142
#5
Midterm Exam Review with Answers
a. Piecewise lines with points (0, 8), (1, 7), (2, 6), (3, 5); stop at the point (3, 5)
And line with points (3, 3), (4, 5), (5, 7), (6, 9); (3, 5) do not exist.
lim f ( x) 5
x
3
lim f ( x) 3
lim f ( x) do not exist.
x
3
x
3
b. Piecewise lines with points (0, 7), (1, 5), (2, 3), (3, 1), (4, -1); stop at the point
(4, -1) and line with points (4, 2), (5, 3), (6, 4); (4, -1) do not exist.
lim f ( x) 1
x
4
lim f ( x) 2
lim f ( x) do not exist.
#6
x
4
x
4
a. 0
3
4
e. 0
b.
d. Does not exist
c.
#7
a. f’(x) = 8x – 3
b. f’(x) = -2x
#8
a. f’(x) = 20x3 – 6x + 6
b. f’(x) =
c. f’(x) =
8
x
5
9
-4
x4
#9
a. f’(x) = 9x2 + 8x + 1
#10
a. y’ =
2x5 4x3
( x 2 1) 2
d. f’(x) =
2
3
3 x
1
x
10
33 x 5
b. f’(x) = 16x3 – 36x2 – 3
b. y’ =
2 x 2 10 x 7
(2 x 5) 2
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Math 142
#11
Midterm Exam Review with Answers
3(12 x 2 4)
a. f’(x) =
(4 x 3 4 x 3) 4
#12
a. f”(1) =
#13
a.
b. f”(1) =
4000
x
2
3( x 2
4
5
i. MC(x) =
2x 3
b. f’(x) =
200
3
x2
3x 4) 3
9
4
ii. MC(64) = 487.5
iii. The cost is increasing by $487.5 for every ton of steel when the mill is
producing 64 tons of steel.
b.
i. MP(x) = .0012x2 + 7
ii. MP(52) = 10.24
iii. The profit is increasing at $10.24 per unit when manufactures
produces 52 units.
13
13x 1700
#14
a. RRCP(x) =
b. MP(x) 13
c. RRCP(250) = 0.008
#15.
a. n’(2)=-2, the rate of change is decreasing 2 hundred units per week.
b. a(.15) = -2.7, the object is decreasing in acceleration of 2.7 m/sec.2.
#16. a.
i)
+ - +
ii)
Extreme Points (1,3) Relative Max
(3,-1) Relative Min.
Inflection Point (2, 1)
b.
i)
ii)
- + - +
Extreme Points (-5,-387) Rel Minimum
(0,-12) Rel. Maximum
(2, -44) Rel. Minimum.
Inflection points (1.0817, -28.97), (-3.082, -228.8)
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Math 142
Midterm Exam Review with Answers
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