(1)Name: Student ID number: TA/classroom: Guidelines for the test

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.

• For problem-solving (計算與證明題) problems, please show all work, unless instructed other- wise. 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; No Partial Credits) Match the function f (x, y) = (x²+ 3y²)e^−x2−y2 with the graphs

(A) (B) (C) (D)

2. (10 pts) Find the area of the region enclosed by the curve r = sin 2θ, 0≤ θ ≤ π/2.

3. (15 pts)

(a) Given that r(t) =< e^2t, t²− t, cos 2t >, calculate

• (2 pts) lim

t→0r(t) =

• (4 pts)

∫

r(t) dt =

(b) Given the position function r(t) =< sin 2t, cos 2t, t >,

• (2 pts) find the velocity, v(t) = d dtr(t)

• (2 pts) find the unit tangent vector T(t)

• (3 pts) find the principal unit normal vector N(t)

• (2 pts) find the binormal vector B(t) = T(t) × N(t)

Name: Student ID number:

4. (5 pts each) Determine if the series is absolutely convergent, conditionally convergent or divergent.

(a)

∑∞ k=1

(k + 1 k )^k

(b)

∑∞ k=1

2 1 + e^k

(c)

∑∞ k=1

(√^k 2− 1)

(d)

∑∞ k=1

cos kπ k + 1

5. (5 pts) Determine the radius of convergence of the power series.

∑∞ k=1

(3k)!

(k!)³x^k.

6. (5 pts) For f (x) = e^x, find the Taylor polynomial of degree 3 expanded about x = 0.

7. (15 pts) Given that 1 1 + x =

∑∞ k=0

(−1)^kx^k, for − 1 < x < 1,

• (6 pts) find the power series representation of _1+x¹2 and determine the radius and interval of convergence.

• (6 pts) Find the power series representation of tan⁻¹(x) and determine the radius and interval of convergence.

Name: Student ID number:

8. (5 pts) Show that the limit does not exist.

lim

(x,y)→(0,0)

6x³y x⁶+ y²

9. (5 pts)

(x,y)lim→(2,3)

6xy x²+ y² =?

10. (10 pts) Find the indicated partial derivatives.

f (x, y) = x^y− 3xy, x, y > 0; fx, f_y, f_xy, f_xx

11. (5 pts) Find the equation of the tangent plane to the surface at the given point.

z = x²− y²+ 1 at (2, 1, 2)