MIDTERM 1 FOR CALCULUS
Time: 8:10–9:55 AM, Friday, April 20, 2001
No calculator is allowed. No credit will be given for an answer without reasoning.
1. Suppose that a = 3i − 2j + 6k and b = j − 2k. Find (1) [2%] b × −2a,
(2) [2%] ka + 2bk, (3) [2%] projab,
(4) [2%] the unit vector in the direction of a, (5) [2%] cos of the angle between a and b.
2.
(1) [5%] Find an equation of the plane passes through the point (1, 2, 3) and contains the line x = 3t, y = 1 + t, z = 2 − t.
(2) [5%] Find the distance between the two parallel planes: z = x + 2y + 1 and 3x + 6y − 3z = 4.
3.
(1) [5%] Find the acceleration of a particle with the given position function r(t) = t2i + ln tj + tk.
(2) [5%] Find the tangential component of the acceleration vector of r(t) = eti +√
2tj + e−tk.
4. Suppose that r(t) = t33i + t2j + 2tk.
(1) [5%] Find the unit normal vector N(t).
(2) [5%] Find the curvature κ.
5. [10%] Find the area of the shaded region.
6.
(1) [5%] Find the length of the curve x = a(cos θ + θ sin θ), y = a(sin θ − θ cos θ) for 0 ≤ θ ≤ π.
(2) [5%] Find an equation of the tangent line to the curve x = sin t, y = sin(sin t + t) at (0, 0).
7.
(1) [5%] Determine the sequence ©ln(n2) n
ªconverges or diverges. If it converges, find the limit.
(2) [5%] How many terms of the series P∞
n=1(−1)n−1
n2 do we need to add in order to find the sum with error less than 0.01.
8. [10%] Find the radius of convergence and interval of convergence of the series X∞
n=1
(−2)n
√n (x + 3)n.
9. Let f (x) = exx−1.
(1) [5%] Find the power series representation of f in powers of x.
(2) [5%] Differentiate the power series in (1) and show that X∞
n=1
n
(n + 1)! = 1.
10. [10%] If p > 1, evaluate the expression
1 +21p +31p +41p + · · · 1 −21p +31p −41p + · · ·
1
