MTH 120 Practice Test #1 Problems 1-21 could be on the no Derive part.
Sections 1.2, 2.2, 2.3, 3.1, 3.3, 3.4, 4.1, 4.2
7 x
13) y = - x 5
Use the properties of limits to help decide whether the limit
exists. If the limit exists, find its value.
x2 - 25
1) lim
x→5 x - 5
2) lim
x→3
4
14) y = x3
x2 + 3x - 18
x2 - 9
Find fʹ(x) at the given value of x.
15) f(x) = x ; Find f′(36).
Give an appropriate answer.
3) Let lim f(x) = 5
x → 10
16) f(x) = -9
; Find f′(-6).
x
Use the product rule to find the derivative.
17) f(x) = (x2 - 3x + 2)(2x3 - x2 + 4)
5x+2
4) Let lim f(x) = x+4
x→ -6
18) f(x) = (4x - 2)(4x3 - x2 + 1)
Find the asymptotes of the function.
3x + 4
5) y = x + 1
6) y = 7) y = Use the quotient rule to find the derivative.
t2
19) g(t) = t - 11
x2 - 36
x - 6
x2 - 4
x
21) y = x2 - 3x + 2
x7 - 2
9
9 - 4x
Use the properties of limits to help decide whether the limit
exists. If the limit exists, find its value.
4x2 + 3x - 1
8) lim
x→∞ -2x2 + 10
9)
20) y = Write a cost function for the problem. Assume that the
relationship is linear.
22) An electrician charges a fee of $55 plus $40 per
hour. Let C(x) be the cost in dollars of using the
electrician for x hours.
5x + 1
lim
x→∞ 13x2 - 7
23) Marginal cost, $150; 30 items cost $5000 to
produce
Find the derivative.
10) f(x) = 2x4 - 6x3 + 1, find fʹ(x)
Solve the problem.
24) Suppose that the demand and price for a certain
model of graphing calculator are related by
p = D(q) = 106 - 2q, where p is the price (in
dollars) and q is the demand (in hundreds).
Find the demand for the calculator if the price is
$41. Round to the nearest whole number if
necessary.
1
1
11) y = x6 - x4
4
2
12) f(x) = 8
7
5
- + x
x
x5
1
32) John owns a hotdog stand. He has found that
his profit is represented by the equation
P(x) = -x2 + 60x + 75, where is x the number of
25) Let the demand and supply functions be
represented by D(p) and S(p), where p is the
price in dollars. Find the equilibrium price and
equilibrium quantity for the given functions.
D(p) = 1840 - 70p
S(p) = 160p - 460
hotdogs. How many hotdogs must he sell to
earn the most profit?
33) Suppose the cost of producing x items is given
by C(x) = 3200 - x3 and the revenue made on the
26) A book publisher found that the cost to produce
1000 calculus textbooks is $25,900, while the cost
to produce 2000 calculus textbooks is $50,600.
Assume that the cost C(x) is a linear function of
x, the number of textbooks produced. What is
the marginal cost of a calculus textbook?
sale of x items is R(x) = 400x - 8x2 . Find the
number of items which serves as a break-even
point.
34) Suppose the cost per ton, y, to build an oil
platform of x thousand tons is approximated by
312,500
. What is the cost for x = 300?
y = x + 625
27) A toilet manufacturer has decided to come out
with a new and improved toilet. The fixed cost
for the production of this new toilet line is
$16,600 and the variable costs are $66 per toilet.
The company expects to sell the toilets for $155.
Formulate a function P(x) for the total profit
from the production and sale of x toilets.
35) If the average cost per unit C(x) to produce x
1500
, what
units of plywood is given by C(x) = x + 50
is the unit cost for 20 units?
28) Regrind, Inc. regrinds used typewriter platens.
The cost per platen is $2.40. The fixed cost to run
the grinding machine is $164 per day. If the
company sells the reground platens for $4.40,
how many must be reground daily to break
even?
Decide whether the limit exists. If it exists, find its value.
36)
lim f(x) and lim f(x)
x→(-1)x→(-1)+
y
4
2
29) Midtown Delivery Service delivers packages
which cost $2.30 per package to deliver. The
fixed cost to run the delivery truck is $325 per
day. If the company charges $7.30 per package,
how many packages must be delivered daily to
make a profit of $75?
-4
-2
2
-2
-4
-6
-8
30) A shoe company will make a new type of shoe.
The fixed cost for the production will be $24,000.
The variable cost will be $31 per pair of shoes.
The shoes will sell for $100 for each pair. How
many pairs of shoes will have to be sold for the
company to break even on this new line of
shoes?
37) lim f(x)
x→1
31) Bob owns a watch repair shop. He has found
that the cost of operating his shop is given by
C(x) = 4x2 - 32x + 235, where x is the number of
watches repaired. What is his minimum cost?
2
4
x
38) lim f(x)
x→0
45) y = 2x - 1 between x = 1 and x = 5
Find the instantaneous rate of change for the function at the
given value.
46) s(t) = t2 + 5t at t = 4
47) f(x) = 5x + 9 at x = 2
Solve the problem.
48) Suppose that the total profit in hundreds of
dollars from selling x items is given by
P(x) = -x2 + 8x - 14. Find the marginal profit
39) lim f(x)
x→ -1/2
at x = 1.
49) The total cost to produce x handcrafted wagons
is C(x) = 100 + 8x - x2 + 2x3 . Find the rate of
change of cost with respect to the number of
wagons produced (the marginal cost) when
x = 8.
Graph to find the limit.
6x2 + 7x - 4x4
40) lim x→∞ 6x2 - 8x + 2
A) -∞
C) Does not exist
Find the equation of the tangent line to the curve when x
has the given value.
x3
50) f(x) = ; x = 4
4
B) ∞
D) 1
Solve the problem.
51) The graph shows the total sales in thousands of
dollars from the distribution of x thousand
catalogs. Find the average rate of change of sales
with respect to the number of catalogs
distributed from 10 to 50.
Sales
(in thousands)
4x7 - x + 7
41) lim x→∞ 8x2 - x - 8
Find all points where the function is discontinuous.
42)
70
70
60
40
60
30
40
50
50
43)
60
40
30
20
10
10
20
Number (in thousands)
Find the average rate of change for the function over the
given interval.
44) y = 4x3 + 3x2 + 5 between x = -4 and x = 1
3
50
Solve the problem.
61) If the price of a product is given by
1024
P(x) = + 1000, where x represents the
x
Find the x-values where the function does not have a
derivative.
52)
demand for the product, find the rate of change
of price when the demand is 8.
62) The profit in dollars from the sale of x thousand
compact disc players is P(x) = x3 - 5x2 + 10x + 5.
Find the marginal profit when the value of
x is 12.
53)
Write an equation of the tangent line to the graph of y = f(x)
at the point on the graph where x has the indicated value.
63) f(x) = (-2x2 + 5x - 2)(-2x + 5), x = 0
64) f(x) = Solve the problem.
54) Suppose the demand for a certain item is given
by D(p) = -3p2 + 7p + 7, where p represents the
-4x2 - 6
, x = 0
2x - 3
Solve the problem.
65) The total cost to produce x units of perfume is
C(x) = (8x + 2)(6x + 7). Find the marginal
average cost function.
price of the item. Find Dʹ(5).
55) The revenue generated by the sale of x bicycles
is given by R(x) = 20.00x - x2 /200. Find the
66) The demand function for a certain product is
given by:
marginal revenue when x = 900 units.
Find the slope of the line tangent to the graph of the
function at the given value of x.
56) y = x4 + 8x3 + 2x + 2; x = 1
D(p) = 6p + 250
.
8p + 19
Find the marginal demand Dʹ(p).
7
57) y = - x; x = 4
x
67) The total revenue for the sale of x items is given
by:
Find an equation for the line tangent to given curve at the
given value of x.
58) y = x3 - 25x + 5; x = 5
R(x) = Find all values of x (if any) where the tangent line to the
graph of the function is horizontal.
59) y = x3 - 12x + 2
100 x
.
7 + x3/2
Find the marginal revenue Rʹ(x).
Solve the following.
60) Find all points of the graph of f(x) = 3x2 + 9x
whose tangent lines are parallel to the line
y - 39x = 0.
4
Answer Key
Testname: 120PRACTICETEST1
1) 10
3
2)
2
3)
4)
5)
6)
5
14
Vertical asymptote at x = -1; horizontal asymptote at y = 3
No asymptotes; hole at x = 6
9
7) Vertical asymptote at x = horizontal asymptote at y = 0
4
8) - 2
9) 0
10) 8x3 - 18x2
11)
dy
= 3x5 - x3
dx
12) fʹ(x) = - 7
25
4
+ - 2
3/2
x
x6
x
13)
dy
7
1
= - - dx
5
x2
14)
dy
3
= dx
4
4 x
15)
1
12
16)
1
4
17) fʹ(x) = 10x4 - 28x3 + 21x2 + 4x - 12
18) fʹ(x) = 64x3 - 36x2 + 4x + 4
19) gʹ(t) = t2 - 22t
(t - 11) 2
20)
dy x2 +4
= dx
x2
21)
dy -5x8 + 18x7 - 14x6 - 4x + 6
= dx
(x7 - 2)2
22) C(x) = 40x + 55
23) C(x) = 150x + 500
24) 3250 calculators
25) $10; 1140
26) $24.70
27) P(x) = 89x - 16600
28) 82 platens
29) 80 packages
30) 348 pairs
31) $171
32) 30 hotdogs
33) 8 items
34) $101,351.35
5
Answer Key
Testname: 120PRACTICETEST1
35) $21.43
36) -2; -7
37) Does not exist
38) -2
39) -1
40) A
41) ∞
42) x = 1
43) x = -2, x = 0, x = 2
44) 43
1
45)
2
46) 13
47) 5
48) $600 per item
49) $376 per wagon
50) y = 12x - 32
3
51)
4
52) x = 0
53) x = 0, x = 3
54) -23
55) $11.00
56) 30
11
57) - 16
58) y = 50x - 245
59) 2, -2
60) (5, 120)
61) -16
62) $322
63) y = 29x - 10
4
64) y = x + 2
3
65) 48 - 14
x2
66) Dʹ(p) = 67) Rʹ(x) = -1886
(8p + 19)2
50(7x-1/2 - 2x)
(7 + x3/2)2
6