Exam A
1.(5pts) What is the natural domain of f (x) =
x−3
?
3x2 − 3
(a) {x ∈ R : x 6= 1}.
(b) {x ∈ R : x 6= 1, x 6= −1}. (c) {x ∈ R : x 6= 3}.
(d) {x ∈ R : x 6= −3}.
(e) {x ∈ R : x 6= −1}.
2.(5pts) A mattress store has fixed expenses of $800 per month. Each mattress costs $100 to
make and sells for $550. Assuming all mattresses made are sold, which of the expressions
below best describes the profit of this store as a function of the number x of mattresses
made?
(a) f (x) = 550x − 800
(b) f (x) = 550x − 700
(d) f (x) = 800x − 450
(e) f (x) = 550x − 900
(c) f (x) = 450x − 800
x2 − 25
is equal to
x→5 x − 5
3.(5pts) The limit lim
(a) −5
(b) 15
(d) ∞
(e) 0
(c) 10
4.(5pts) Given the piecewise function f (x) below, choose a value c which makes the function
continuous at x = −2.
 3
 x + 5x2 + 6x
,
x 6= −2
f (x) =
x+2
c,
x = −2.
(a) −∞
(b) ∞
(d) 0
(e) -2
(c) -3
5.(5pts) Elizabeth puts her savings of $200, 000 in an account that pays 4% interest compounded quarterly. Which of the following represents how much money will be in the account
after 5 years?
(a) $200, 000e0.04·5
(d) $200, 000(1 +
0.04 5·4
)
4
(b) $200, 000(1 +
0.04 4
)
4
(e) $200, 000(1 + 0.04)5·4
(c) $200, 000(1 + 0.04)4
Exam A
6.(5pts)
i.
ii.
iii.
iv.
Which of the following statements are true for the function f (x) = ex ?
lim f (x) = ∞.
x→∞
f has a horizontal asymptote at y = 1.
For some real numbers x and y, f (x + y) 6= f (x)f (y)
The natural domain of f is all real numbers.
(a) none are true
(b) all are true
(d) i and iv
(e) i, ii, and iv
(c) ii and iii
7.(5pts) If log2 (A) = 3, what is log4 (A)?
9
4
(b)
1
2
(d) 9
(e)
3
2
(a)
(c)
√
3
8.(5pts) Suppose that the demand curve for a certain product is modeled by
3456
p=
(q + 29)
where p is the price and q is the quantity. Recalling that the revenue
R(q) = pq
compute
lim R(q)
q→∞
(a) 29
(b) 3456
(d) 0
(e) ∞
(c) 119.17
9.(5pts) Use the approximation log2 3 ≈ 1.585 and log2 5 ≈ 2.322 to estimate log2 45.
(a) 5.492
(b) 6.229
(d) 3.68037
(e) 5.83338645
(c) −0.848
Exam A
10.(5pts) Which answer below is the following limit. lim
n→∞
(a) 0
3
1+
n
n
?
(c) e3
(b) ∞
(d) The limit does not exist. (e) e1/3
11.(12pts) Find the natural domain and equations of all vertical and horizontal asymptotes of
the function
x+1
f (x) =
(x − 2)(x + 3)(x − 4)
Compute all one-sided limits at all vertical asymptotes.
12.(12pts) At the upcoming board meeting, a company selling smartphones will be deciding its
marketing budget. If the company spends x millions of dollars on advertising, the resulting
revenue R is given by
x − 500
R(x) = 50 −
(x − 5)2 + 25
Part 1. Compute the following limits:
a) lim R(x) b) lim R(x)
x→∞
x→0
Part 2. The advertising budget is a fixed cost. Suppose the company has additional fixed
cost of 30 million dollars. Write down a function for the total cost, C, as a function of money
spent on advertising, x. Write down the function that describes the company’s profit, P , as
a function of money spent on advertising x.
Part 3. If the company is currently set to spend 2 million on advertising, would your
recommend to the board for them to double down and increase the spending on marketing
to 4 million? (Assume that the board wishes to maximize profit.)
Part 4. Would you recommend increasing the advertising budget to 7 million? Suppose
the board wished to maximize revenue. Would you recommend going from 2 million to 7
million?
13.(12pts) You have $10,000 dollars to invest and in 10 years you need to have $20,000.
(a) Assuming all investments compound continuously, what interest rate do you need to
get?
(b) How long will it take to get $40,000?
Exam A
1. Solution. Since f (x) is a quotient of polynomials, we just need to avoid those x such that
the denominator is equal to zero. These are the solutions of 3x2 − 3 = 0, which are x = 1
and x = −1.
2. Solution. Each mattress made and sold gives a profit of $550 − $100 = $450. Thus, selling
x mattresses gives 450x as a profit. Finally, we subtract the fixed expenses, and we get
f (x) = 450x − 800.
3. Solution. Simplifying the expression in the limit, we get
x2 − 25
(x − 5)(x + 5)
=
=x+5
x−5
x−5
Then, plugging in x = 5 gives 10.
Exam A
4. Solution. We see that for x 6= −2 we have
x(x + 2)(x + 3)
x3 + 5x2 + 6x
=
= x(x + 3).
f (x) =
x+2
x+2
As we accept the fact that polynomials are continuous functions, we can therefore evaluate
the limit as x goes to −2 by plugging in −2 for x. Thus
lim f (x) = lim x(x + 3) = (−2)((−2) + 3) = −2.
x→−2
x→−2
Setting c = −2 therefore makes the function f continuous.
5. Solution. Recall that when compounding interest, we multiply the principal, which in this
case is $200, 000 by the number obtained in the following procedure. We take the annual
interest rate, which in this case is 4%, divide it by how many times it is compounded per
unit of time (note that our units of time are years), in this case 4 as there are 4 quarters per
year, add 1, and finally exponentiate this value to number of times the interest is calculated.
As there are 4 quarters per year, and Elizabeth keeps the money in the account for 5 years,
the interest is compounded 5 · 4 = 20 times. Thus her account has $200, 000(1 + 0.04
)5·4 after
4
5 years.
6. Solution.
i. As we saw in class, as x gets arbitrarily large, so does ex , therefore lim ex = ∞. Therefore
x→∞
i is true.
ii. The limit as x goes to −∞ is 0, so y = 0 is a horizontal asymptote. The limit as x goes
to ∞ is discussed in i and does not generate an asymptote. Therefore ii is false.
iii. By the laws of exponents, for any two real numbers x and y, f (x + y) = ex+y = ex ey =
f (x)f (y). Therefore iii is false.
iv. ex is defined for all real numbers x, hence its natural domain is all real numbers. Therefore
iv is true.
Exam A
7. Solution. We are told A = 2log2 (A) = 23 = 8. Hence we want log4 (8). But 23 = 8 =
4log4 (8) = 22 log4 (8) so 2 log4 (8) = 3.
8. Solution. lim
q→∞
3456q
3456
3456
=
= lim = 3456
q→∞
(q + 29)
1+0
1 + 29
q
9. Solution. 45 = 9 · 5 = 32 · 5 so log2 (45) = log2 (32 · 5) = 2 · log2 3 + log2 5 = 2 · 1.585 + 2.322.
10. Solution. From the book, lim
n→∞
n
r n
3
r
1+
= e so lim 1 +
= e3 .
n→∞
n
n
Exam A
11. Solution. The natural domain will be the set of those real numbers x for which the denominator of f (x) does not vanish. This is all real numbers except x = 2, x = −3 and x = 4.
These are also the vertical asymptotes of f (x). As for horizontal asymptotes, we compute
both limits
lim f (x), lim f (x)
x→∞
x→−∞
These are both zero, since the power in the denominator (three) is higher than the power in
the numerator (one). Thus, the only horizontal asymptote is y = 0.
12. Solution.
Part 1.
a) lim R(x) = 50
x→∞
b) lim R(x) = 50 −
x→0
0 − 500
500
= 50 +
= 60
2
(0 − 5) + 25
50
Part 2.
C(x) = 30 + x
P (x) = R(x) − C(x) = 50 −
x − 500
x − 500
− (30 + x) = 20 − x −
2
(x − 5) + 25
(x − 5)2 + 25
Part 3. We see that P (2) ≈ 32.6471 and P (4) ≈ 35.0769. Therefore, the profit has increased
when the spending on marketing was doubled from 2 million to 4 million dollars. Thus, yes,
we would recommend that the company increase its spending on marketing in this case.
Part 4. P (7) = 30 which is less than P (2) so we would not recommend going up to 7
million. Hence, if the board wishes to maximize revenue we should recommend that we go
to 7 million.
13. Solution. (a) P (t) = P0 ert where P0 = 10000, so P (1) = 10000er . But P (10) = 20000 so
20000
e10r =
= 2 so 10r = ln(2) and hence r = 0.6931471806
= 0.0693147806.
10
10000
(b) By part (a), your investment takes 10 years to double. SO it will take 10 more years to
double again so to get to $40,000 will take 10 + 10 = 20 years.
OR
P (t) = 10000 · 2t/10 so solve 10000 · 210t = 40000 or 2t/10 = 4 = 42 or 10t = 2 and t = 20.