Name:
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 problems 1.
• For problems 2-10, 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. (5 pts) Assume that f′(x0) = f′′(x0) = 0, f′′′(x0) > 0. Answer the following True/False questions (True ⇒ ⃝ ; False ⇒ ×).
• f′(x0) is a local maximum value of f′(x).
• f′(x0) is a local minimum value of f′(x).
• f (x0) is a local maximum value of f (x).
• f (x0) is a local minimum value of f (x).
• the point (x0, f (x0)) is an inflection point of the curve y = f (x).
2. Evaluate the limits.
(a) (5 pts) lim
x→∞
√x sin(1 x).
(b) (5 pts) lim
x→2
(x2 − x − 2)10 (x2 − 10x + 16)10.
3. (5 pts) Given that lim
x→a
f (x)−f(a)
(x−a)2 =−1, does f′(a) exist? If f′(a) exist, compute f′(a). If not, prove it.
4. (5 pts) Use a linear approximation to estimate √ 104.
5. (a) (5 pts) d
dx(x2− 1 x2)10.
(b) (5 pts) d
dx[sin2x + sin (x3)].
6. Given x2+ y2 = 1,
(a) (5 pts) find dydx implicitly;
(b) (5 pts) find ddx2y2 implicitly.
7. (5 pts) Given f (x) = x(x + 1)(x + 2)(x + 3) . . . (x + 100), find f′(0).
8. (a) (5 pts) Given f (x) = 2x2+ 1, find a function F (x) such that F′(x) = f (x).
(b) (5 pts) Given g(x) = sin(2x), find a function G(x) such that G′(x) = g(x).
9. (10 pts) Find the points on the ellipse 4x2+ y2 = 4 that are farthest away from the point (6, 0).
10. (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 (兩車以多快的相對速度接近)?
11. (total 20 points; (a)-(n) no partial credit) Study the function f (x) = x2 x2+ 4 and answer the following questions.
(a) (1 pt) Domain of f : .
(b) (1 pt) Horizontal Asymptote: .
(c) (1 pt) Vertical Asymptote: .
(d) (1 pt) f′(x) = .
(e) (1 pt) Interval(s) of increasing of f : .
(f) (1 pt) Interval(s) of decreasing of f : .
(g) (1 pt) Local maxima of f : .
(h) (1 pt) Local minima of f : .
(i) (1 pt) f′′(x) = .
(j) (1 pt) Interval(s) of concave up: .
(k) (1 pt) Interval(s) 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.
