Download Math 115–Test 1 Sample Problems for Dr. Hukle’s Class

Math 115–Test 1 Sample Problems for Dr. Hukle’s Class
126
pens per month when the selling
p
price is p dollars and each p ≤ 3. A supplier for the bookstore is willing to provide S(p) = 48p2 − 11 pens
per month when the selling price is p dollars each and p ≥ 0.5. Find the equilibrium price and quantity.
1. Demand for a Jayhawk pen at the Union is known to be D(p) =
2. A diver springs from a board over a swimming pool. The height in feet above the water level of the diver
t seconds after leaving the board is given by h(t) = 4 + 12t − 16t2 . With what velocity does the diver hit
the water?
(A) -4 ft/sec
(B) -32 ft/sec
(C) -12 ft/sec
(D)-20 ft/sec
(E) -2.5 ft/sec
3. Market research has shown that the price people are willing to pay for gourmet chocolates is given by the
demand function p = 20 − x. Find the marginal revenue function M R(x).
4. The mathematics office sells sample midterms for 50 cents each. The cost of producing x tests (in cents)
is given by C(x) = 1000 − 60x + x2 .
(a) Find the equations for the revenue function R(x) and profit function P (x).
(b) For what production levels is P (x) = 0? Solve this algebraically.
(c) Find the rate of change of the profit function with respect to the production level x.
(d) Find the rate of change of the profit function when x = 50.
(e) Find the production level which yields maximum profit. (Hint: look at the graph.)
5. If the price of a product per item is given by p(x) = x2 + 2x + 4 and the total cost function is given by
C(x) = 8 + x where x is the number of items produced and sold. Find the profit function P (x). How is the
profit changing when the production and sales are x = 6?
6. Given a number c, find the x-coordinate of the point at which the curve y = x2 + c/x has a horizontal
tangent line.
7. Find the following limits if they exist. If the limit does not exist write DNE.
x2 − 16
=
x→4 x2 − 2x − 8
(a) lim
3t − |t|
=
t→0
t
(b) lim
5x22 − 3x17 + 99
=
x→∞
10x22 − 17
(e) lim 3x2 − 4 =
(d) lim
2x3 + 6
=
x→−∞ x2 + 1
(g) lim
x+3
=
x→∞ x2 + 5x + 6
(j) lim
x→3
x2 + 5x + 6
=
x→ −3
x+3
(h) lim
x2 + 5x + 6
=
x→∞
x2 + 3
(k) lim
1
6x3 + 4x2 − 17x
=
x→∞
x4 + 3x
(c) lim
(4 + h)2 − 16
=
h→0
h
(f) lim
x2 + 5x + 6
=
x→∞
x+3
(i) lim
8. Let f (x) = 2x . Find the following using your graphing calculator, rounding to 3 decimal places.
(a) An equation of the line tangent to f at x = 1.7 (Use zDecimal window)
(b) Find f (.52) and f (3.9)
(c) Find f 0 (2.7).
9. A farmer has 120 meters of fencing with which to make a rectangular pen. The pen is to have one internal
fence running parallel to the end fence that divides the pen into two sections. The length of the larger section
is to be twice the length of the smaller section. Let x represent the length of the smaller section. Find a
function in x that gives the total area of the pens.
10. A tool rental company determines that it can achieve 500 daily rentals of jackhammers per year at a daily
rental fee of $30. For each $1 increase in rental price, 10 fewer jackhammers will be rented. Let x represent
the number of $1 increases. Find the function R(x) giving total revenue from rental of the jackhammers.
11. A homeowner wishes to enclose 800 square feet of garden space to be laid out in the shape of a rectangle.
One side of the space is to have a stone wall costing $50 per running foot. The other three sides are to have
cedar fencing costing $20 per foot. Let x represent the length of the stone wall. Find a function in x giving
total cost of the enclosure.
12. A manufacturer has monthly fixed costs of $40,000 and a production cost of $8 per unit produced. The
product sells for $12 each unit. The profit (or loss) when the production level is 12,000 units is
A) $8000
B) $48,000
C) $40,000
D) -$48,000
E) None of the above.
13. Let f (x) = x − x2 .
(a) Find the slope of the line tangent to the graph of f at the point x = a by using the definition of the
derivative.
To receive credit you must show all your work.
(b) Using your results from part (a), find an equation of the line tangent to the graph at x = 2.
14. The function f is continuous on the closed interval [0, 2] and has values that are given in the table below.
The equation f (x) = 3.5 must have at least two solutions in the interval [0, 2] if k =
x
f (x)
(a) 0
(b)
1
2
(c) 1
(d) 4
0 1 2
1 k 3
(e) 3
15. Carefully graph
2
(√
x + 1 −1 ≤ x < 1
2x − 1 1 ≤ x ≤ 2.
3
f (x) =
16.
The graph of a function f is shown above. Which of the following statements about f is false?
1. f is not continuous at x = x0
2. x = x0 is in the domain of f .
3. lim f (x) = lim f (x)
x→x−
0
x→x+
0
4. lim f (x) 6= f (x0 ).
x→x0
5. f is not differentiable at x = x0 .
17. Show there is a real number that is equal to its cube plus one.
18. Let f (x) = 1/x. When simplified, the difference quotient
(A)
1
x(x+h)
(B)
1
x(h−x)
(C) −
1
x2
(D)
1
x(x−h)
(E)
f (x + h) − f (x)
becomes
h
−1
x(x+h)
1
by using the limit definition of the
x + 17
1
slope. Using this answer, find the equation of the line tangent to the graph of f (x) =
at x=17.
x + 17
19. Find the slope of the line tangent to the graph of f (x) =
20. Consider the graph of f(x) shown to the right.
3
y
(a)
lim
x→(−1)−
f (x) =
3
(b) lim f (x) =
x→−1
(c) lim f (x) =
(d) f (2) =
(e) f (1) =
(f) lim f (x) =
x→2
2
f(x)
1
!4 !3 !2
x→2+
!1
1
2
3
!1
(f) f (x) is NOT continuous at x =
(g) f (x) is NOT differentiable at x =
√
x+1
21. Let f be the function defined by
. The domain of f is
x−2
A) (−∞, 2) ∪ (2, ∞) B) (−1, 2] ∪ [2, ∞)
C) [−1, 2) ∪ (2, ∞)
D) [1, ∞)
E) None of the above
22. The following data is known about the function f and g.
t
0
1
3
5
f (t) f 0 (t) g(t) g 0 (t)
0
6
0
-1
1
0
5
-2
3
2
1
-3
2
-1
3
7
Find the value of:
(a)
d
[f (t)g(t)] at t = 3.
dt
(b)
d g(t)
(
) at t = 5.
dt t2
(c)
d
((f (t) + g(t)) at t = 1.
dt
23. If we correctly calculate the derivative of f (x) = x2 + 17 directly from the definition of derivative ,
which of the following will appear in our calculations?
(A) lim x2 + h
h→0
(B) lim x2 + h
x→0
(C) lim 2x + h
24. Let y = x17 . What is the definition of
x17
h→0 h
x17
(F) lim
x→0 h
(A) lim
(x + h)17
h→0
h
(B) lim
(D) lim 2x + h
x→0
h→0
(E) lim x + h
h→0
(F) lim x + h
x→0
dy
dx ?
(x + h)17
x→0
h
(C) lim
(x + h)17 − x17
h→0
h
(D) lim
√
25. Find a function f (x) and a real number a such that
4
f 0 (a)
= lim
h→0
h+9−3
.
h
(x + h)17 − x17
x→0
h
(E) lim
x
26.What is the profit in terms of the cost C(x) to produce x units and the unit price p(x) at which x units
will sell ?
(A) x(p(x) − C(x)) (B) p(x) − xC(x) (C) p(x) + xC(x) (D) xp(x) − C(x) (E) None of the above
27. A dishwasher company can produce up to 100 dishwashers per week. Sales experience indicates that the
manufacturer can sell x dishwashers per week at price p where p and x are related by the demand function
p = 600 − 3x. The company’s weekly fixed costs are 4000 dollars and production records show that it costs
150x + 0.5x2 dollars to make x dishwashers.
1. Find the Cost function C(x) and compute the total cost of manufacturing 75 dishwashers in a week.
2. Find the Average Cost function C(x) and compute the average cost at a weekly production level of
75 dishwashers.
3. Find the Marginal Revenue function M R(x) and use it to estimate the revenue received with the sale
of the 75th dishwasher in a week.
4. Find the Profit function P (x) and compute the profit earned from the sale of 75 dishwashers.
True or False Questions:
1. If f (x) is not defined at x = a, then lim f (x) does not exist.
T
F
2. If both lim f (x) and lim f (x) exist, then lim f (x) exists.
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F
f (x)
does not exist.
g(x)
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F
f (x)
does not exist.
g(x)
T
x→a
x→a−
x→a
x→a+
3. If lim f (x) and lim g(x) exist, then lim
x→a
x→a
x→a
f (x)
exists.
g(x)
4. If lim f (x) = 0 and lim g(x) = 0, then lim
x→a
x→a
x→a
5. If lim f (x) = L 6= 0 and lim g(x) = 0, then lim
x→a
x→a
x→a
6. If lim f (x) exists, then f is continuous at x = a.
x→a
T
7. If f (x) is continuous at x = a, then lim f (x) exists.
x→a
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F
F
F
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F
8. If f (x) is continuous at x = x0 , then f (x) has a derivative at x = x0 .
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F
9. If f (x) is differentiable at x = x0 , then f (x) is continuous at x = x0 .
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F
10. If f (x) and g(x) are continuous, then
f (x)
is continuous.
g(x)
11. If f is differentiable at x = a, then lim f (x) exists.
x→a
12. If f and g are differentiable, then
T
T
F
F
d
d
d
(f (x)g(x)) =
f (x) g(x).
dx
dx
dx
5
T
F