Student ID number:
TA/classroom:
Guidelines for the test:
• Put your name or student ID number on every page.
• There are 11 problems
• The exam is closed book; calculators are not allowed.
• There is no partial credit for problem 1-3.
• For other problems, please show all work, unless instructed otherwise. Partial credit will be given only for work shown. Print as legibly as possible - correct answers may have points taken off, if they’re illegible.
• Mark the final answer.
1. (2 pts each) f(x) is a continuous function on (−∞,∞) and the graph of its derivative,f′(x), is shown in the figure below.
(Note: lim
x→−∞f′(x) = 0; lim
x→∞f′(x) =∞)
Answer the following True/False questions (True ⇒ ⃝ ; False ⇒ ×).
• (1, f(1)) is an inflection point.
• f has a local maximum at x=−1
• f has a local minimum atx= 1
• f(x) has 3 critical numbers.
2. (2 pts each) Supposef(x) is a continuous function, andF(x) is an antiderivative function of f(x), i.e.,F′(x) =f(x). Answer the following True/False questions (True ⇒ ⃝; False ⇒ ×).
• If f(x) is an odd function, then F(x) is an even function.
• If f(x) is an even function, then F(x) is an odd function.
• If f(x) is a periodic function, then F(x) is a periodic function.
• If f(x) is monotonically increasing, then F(x) is monotonically in- creasing.
Note:
• The graph of an even function is symmetric with respect to the y-axis.
• The graph of an odd function is symmetric with respect to the origin.
• A function f is called monotonic increasing, if for all x and y such that x≤y one hasf(x)≤f(y).
3. (2 pts each) Answer the True/False questions (True ⇒ ⃝ ; False ⇒ ×).
4. Evaluate each of the following limits.
(a) (5 pts) lim
x→0+sinx lnx
(b) (5 pts) lim
x→0+xsinx
5. Find dydx for each of the following.
(a) (5 pts) y=xsinx, x > 0.
(b) (5 pts) y=e2x
√x+ 1
x2+ 2 (2x+ 1)5, x >0.
6. (10 pts) Given that F(x) =
∫ x2 1
et2dt, for x≥0, (a) Find F′(x)
(b) Find (F−1)′(0)
7. Evaluate the given integral (a) (5 pts)
∫
e2xsinx dx.
(b) (5 pts)
∫ √ lnx
x dx,
(c) (5 pts)
∫ 3x
(x+ 1)(x−4)dx,
8. (10 pts) Evaluate the definite integrals
∫ 4 1
e√xdx
9. (a) (5 pts) Evaluate
∫
cos2θ dθ.
(b) (5 pts) Use the trigonometric substitution to evaluate
∫ 1 0
√1−x2dx,
10. (5 pts) Use formulas for indefinite integrals to evaluate
∫ 1
x2−4x+ 5dx.
11. Evaluate the given integral (a) (5 pts)
∫ 1
−1
x−2dx
(b) (5 pts)
∫ ∞
−∞
x dx
