Exam II Sample Questions (from OLD exams)
4 + 6x − 4x3
=
x→∞ 3 − 12x + 7x3
1. lim
(a)
4
3
4
7
1
(c) −
2
(d) 0
(b) −
(e) ∞
2. Which of the following is an equation of the line tangent to y = ln x at x = 2?
1
2
1
(x − 2)
2
1
(x − 2) + ln 2
2
1
1
(x − 2) +
x
2
1
(x − 2) + ln 2
x
(a) y = x
(b) y =
(c) y =
(d) y =
(e) y =
f (x + h) − f (x)
=
h→0
h
3. If f (x) = x4 − 10, then lim
(a) 1
f (h)
h
(c) 4x3 − 10
(b)
4x3 h + 6x2 h2 + 4x3 h + h4 − 20
h
(e) 4x3
(d)
4. Find the derivative of y = x3 (x4 + 6)
(a)
(b)
(c)
(d)
(e)
7x5 + 18x3
7x5
7x6 + 18x2
7x6 + 6x3
7x6 + 6x3 + 6x2
1
5. If f (x) = 3x , then f 0 (x) =
(a) x3x−1
(b) 3(2x )
(c)
3x
ln(3)
(d) 3x ln(3)
(e) 3x
6. You are told that a function has the following derivatives:
f 0 (x) = (x − 5)2 (x − 2) and f 00 (x) = 3(x − 3)(x − 5)
Which of the following gives the largest interval on which f is decreasing?
(a)
(b)
(c)
(d)
(e)
(−∞, 2)
(3, 5)
(−∞, 2) ∪ (3, 5)
(−∞, 2) ∪ (5, ∞)
(2, 5)
7. Using the function described in #6, which of the following gives the largest interval on which f
is concave up?
(a)
(b)
(c)
(d)
(e)
(−∞, 3) ∪ (5, ∞)
(4, ∞)
(3, 5)
(2, 3) ∪ (5, ∞)
(−∞, ∞)
8. An account with an initial deposit of $1000 at 6% annual interest compounded monthly will
grow to A = 1000(1.005)12t , where A is in dollars and t is the number of years after starting the
account. How fast is the account growing after 2 years? (include appropriate units)
2
9. Find the derivatives of the following functions (you do not have to simplify).
(a) f (x) =
x3 + 1
x2 − ex
(b) f (x) = 3x2 ln(x)
(c) g(x) = x−1/2 · 4x
10. Find the rst and second derivatives of f (x) = e−x (you do not have to simplify).
2
3
100
11. The demand for a 16 GB USB ash drive is given by p =
, where x is the number of thousand
1+x
ash drives and p is the price in dollars. The cost (in thousands of dollars) for producing them
1
is given by C(x) = 6 + x. Find the marginal prot when 3 thousand ash drives are produced.
6
What does your answer tell you about producing more?
7
6
12. Find all relative maxima, relative minima, and inection points of f (x) = x7 − x6 + 1.
13. Find the equation of the line tangent to f (x) = (x2 + x)100 at x = 1.
4
14. Given the graph below is the graph of the DERIVATIVE of f (x), answer each of the following:
Graph of FIRST DERIVATIVE of f
M
N
P
Q
R
S
T
(a) List the intervals where f is increasing.
(b) List the x-coordinates where f has a local minimum
(c) List the intervals where f is concave down
15. A rectangular eld has a barn on one side (see gure below). If 240 feet of fencing are to enclose
the eld with none needed along the barn, nd the dimensions of the eld which maximize the
area.
5
16. Use the four-step graphing method to analyze each of the following for f (x) = xe−x :
(a)
(b)
(c)
(d)
Intercepts and Asymptotes
Intervals of Increasing and Decreasing
Intervals of Concave Up and Concave Down
Graph
6