Name:
Student ID number:
TA/classroom:
Guidelines for the test:
• Put your name or student ID number on every page.
• There are 9 problems
• The exam is closed book; calculators are not allowed.
• There is no partial credit for選擇,填充及是非 problems.
• For problem-solving (計算與證明題) problems, please show all work, unless in- structed 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. (5 pts) Given f0(x), find the graph of f (x)?
f0(x) (A) (B) (C)
2. (15 pts)
(a) Find the derivative of f (x) = ln√
e2x(x2+ 1)10/(2x3+ 2), x > 0.
(b) Find the derivative of f (x) = excos (x3+ x).
(c) Find d
dx(x2− 1 x2)10
3. (10 pts) Given the curve x2+ xy + y2 = 3.
(a) find dydx implicitly;
Name: Student ID number:
4. (20 pts) Compute: (Check whether l’Hospital’s rule can be applied before you use it.)
(a) lim
x→9
9− x
√x− 3.
(b) lim
x→0+x ln x
(c) lim
x→0xex
(d) lim
x→0+x2x
5. (5 pts) Given f (x) = x cos x
(x + 1)(x + 2)(x + 3) . . . (x + 100), find f0(0).
6. (5 pts) Given that f (x) =
{x2, x > 0
−x2, x≤ 0 , compute f0(0) by definition (limits).
7. (10 pts) Find the points on the ellipse 4x2+ y2 = 4 that are farthest away from the point (1, 0).
8. (10 pts) Car A is traveling west at 90 km/h and car B is traveling north at 100 km/h. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 60 m and car B is 80 m from the intersection?
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9. (total 20 points; (a)-(n) no partial credit) Study the function f (x) = 1 x2− 9 and answer the following questions.
(a) (1 pt) Domain of f : .
(b) (1 pt) Horizontal Asymptote: .
(c) (1 pt) Vertical Asymptote: .
(d) (1 pt) f0(x) = .
(e) (1 pt) Intervals of increase of f : .
(f) (1 pt) Intervals of decrease of f : .
(g) (1 pt) Local maxima of f : .
(h) (1 pt) Local minima of f : .
(i) (1 pt) f00(x) = .
(j) (1 pt) Intervals of concave up: .
(k) (1 pt) Intervals of concave down: .
(l) (1 pt) Inflection point(s) of f : .
(m) (1 pt) x-intercepts of f : .
(n) (1 pt) y-intercepts of f : .
(o) (6 pts) Sketch the graph of f showing all significant features.
