1. (6 points)For the function whose graph is given, arrange the following numbers in increasing order.
g 0 (1), g 0 (3), g 0 (3.5). Use the blanks provided below.
<
<
1
2. (8 points) Suppose a population of critters is modeled by the function P (t) = 200 + sin(3t) with t
measured in months. How fast is the population changing at 4 months?
Which of the following would correctly describe the answer to this problem? Circle all that apply, if
any.
(a) ∆P on [4, u]
(b)
dP dt t=4
(c)
∆P ∆t t=4
(d) limu→4
(200 + sin(3u)) − (200 + sin(12))
u−4
(e) P 0 (4)
(f) The slope of the tangent line to the graph of P at t = 4.
(g) The slope of the secant line to the graph of P on [4, u].
(h) The change in critters during the first 4 months.
2
3. (8 points) Evaluate the limits below. Round your answer to two decimal places. If an answer does
not exist, write DNE.
tan(x)
x2 + 1
t2 − 4
(b) limt→−2
t+2
(a) limx→0
5
4. (a) (4 points) The function f is given by f (x) = 3ekx + 2 , where k is an unknown constant.
x
Which of the following is the derivative of f ? Circle all that apply, if any.
i.
df
= 3kekx − 10x−3
dx
ii.
df
3
= ekx − 5x−1
dx
k
iii.
df
10
= 3kekx − 3
dx
x
iv.
df
3
= ekx − 5x−1 + 5
dx
k
5
.
x2
which of the following could be the function g(x)? Circle all that apply, if any.
(b) (4 points) The derivative of a function is given by g 0 (x) = 3e3x +
5
i. g(x) = 4 + e3x − x−3
3
ii. g(x) = 9e3x −
5
x
iii. g(x) = 3e3x − 5x−2
iv. g(x) = 4 + e3x −
5
x
3
5. (10 points) Let H(t) be the daily cost, in dollars, to heat an office building when the outside temperature is t degrees Fahrenheit.
(a) What is the meaning of ∆H on the interval [50, 54] degrees in the context of the problem.
(b) What is the meaning of the derivative H 0 (58) in the context of the problem.
(c) Would you expect H 0 (58) to be positive or negative? Explain.
6. (12 points) The table below gives the position of an object (S, in meters) at various times (t, in
seconds).
t (s)
3
5
7.5
9
10
11.5
13
14
15
16
S (m)
24
23
19
17
15
12
8.5
5
3
0
∆S
on [5, 10], Write your answer in a sentence to indicate its meaning in the context of
(a) Find
∆t
the problem.
(b) Estimate S 0 (14), Write your answer in a sentence to indicate its meaning in the context of the
problem.
7. (20 points) The length of an Atlantic herring as a function of age t, in years, can be modeled by the
formula L(t) = 32 − 32e−0.37t for 0 ≤ t ≤ 13, where L is measured in centimeters.
(a) Find L0 (2). Write your answer in a sentence to indicate its meaning in the context of the
problem.
(b) How fast is the length of the Atlantic herring changing when the length is 30 centimeters? Write
your answer in a sentence to indicate its meaning in the context of the problem.
8. (16 points) An object is launched straight upward from a platform. Its height above ground is
h(t) = h0 + v0 t − 16.1t2 where h is in feet and t is in seconds. The graph of h is shown below. The
vertical axis is deliberately without scale. Find h0 and v0 . Be sure to include units in your answers.
(a) Find v0
4
(b) Find h0
9. (12 points) The graph of the electrical potential in a circuit, V measure in volts (V ), as a function
of time, t, measured in milliseconds (ms), is shown below. Fill in the missing values on the table
dV
on the axes below.
below. Then sketch and label the graph of
dt
t (ms)
1
2
1
2
dV
(V/ms)
dt
5
3
4
6