Download Math Problems: Limits, Derivatives, and Integrals and more Exams Mathematics in PDF only on Docsity!
Part 1 – Multiple Choice (52 points)
Read each question carefully; each problem is worth 4 points.
You can use the following information if needed.
Special angle formulæ
θ cos θ sin θ tan θ
π
6
π
4
π
3
- Compute
d
dx
(x ln(4x)).
A: ln(4x)
B: 1 + ln(4x)
C: 14 + ln(4x)
D:
1 4 x
E: (^) x^4
- If f ′(x) = x(1 − x^2 ) for − 3 ≤ x ≤ 2 , in which of the following intervals is f (x) always increas-
ing?
A: (− 1 , 0)
B: (1, 2)
C: (− 3 , 0)
D: (− 3 , −2)
E: (− 1 , 1)
- Below is the graph of the second derivative f ′′(x) of the function f (x). Which of the following must be
true?
A: f (2) = 0.
B: f (x) has a local maximum at x = 2.
C: f (x) has a local minimum at x = 2.
D: f ′(2) = 0.
E: f (x) has an inflection point at x = 2. x
y
1
2
- sin(tan−^1 x) =
A:
√x 1+x^2 B: √x 1 −x^2
C: √^1 1+x^2
D: √^1 1 −x^2
E:
1 + x^2
- Compute lim x→ π 3
cos x −
1 2 π 3 −^ x^
A: −
B:
C: − 1 / 2
D: 1 / 2
E: 1
- Find the domain of f (x) = 3 sin
− 1 (2x − 5).
A: [− 1 , 1]
B: [−
π 2 ,^
π 2 ]
C: (−∞, ∞)
D: [0, 1]
E: [2, 3]
- The number N (t) of bacteria in a culture triples every 10 hours. N (t) satisfies the differential equation:
dN
dt
= kN. Compute k.
A: ln 10 3
B: ln 3 10
C: 10 ln 3
D: 103
E:
ln 3 ln 10
- An antiderivative of xex^ is:
A:
x^2 2 e
x
B: xex^ + ex
C: xex^ − ex
D: x^2 ex
E:
1 2 e
x^2
- Which of the definite integrals below is equal to lim n→∞
∑^ n
i=
1 n e
1+ (^) ni
A:
0
ex^ dx
B:
0
e
x (^2) dx
C:
1
ex^ dx
D:
1
e1+x^ dx
E:
0
e1+2x^ dx
PART 2 (52 points)
Refer to the front for instructions.
- a) (6pts) Compute lim x→+∞
x
1 x
b) (6pts) Compute lim x→ 0 +^
x
1 x
- The atmospheric pressure is often modeled by assuming that the rate of change of the pressure p with
respect to the altitude x (height above sea level) is given by
dp
dx
= kp, where k is a constant.
The atmospheric pressure at sea level is 1000 millibars and the pressure at 10 km is 250 millibars.
a) (4pts) Express the pressure p in terms of k and x.
b) (4pts) Compute k. (You can give your answer in terms of logarithms.)
c) (4pts) At what altitude will the pressure be 500 millibars? (You can give your answer in terms of logarithms.)
- (10pts) A rectangular box with open top has height h, length l and width w. The length of the box is twice its
width and the volume of the box is 9 ft^3. The material for the base costs $10 per ft^2 and the material for the sides costs $5 per ft^2. Find the dimensions of the box that will minimize the cost of the material. What is the
minimal cost? Show that your answer gives a minimum.
