(ii) Show that 4a · b = ||a + b||2− ||a − b||2

MIDTERM 1 FOR CALCULUS

Date: 2000, June 1, 8:10–10:00AM Each problem is worth 10 points.

1. Suppose that a = 2i − 3j + k, b = −2i + 3k. Find:

(i) a · b, (ii) b × 2a, (iii) (b × a) × b, (iv) proj_ba,

(v) ||a||.

2.

(i) Let l1, l2 be lines that pass through the origin and have direction vectors d1 = i + 2j + 4k, d2= −i − j + 3k respectively. Find an equation for the plane that contains l1 and l2.

(ii) Show that 4a · b = ||a + b||²− ||a − b||². 3. Find the limits:

(i) limn→∞sin n n , (ii) limx→eln(ln x) ln x−1, (iii) limx→0(cosh x)^4/x (iv) lim_x→0+sin x ln x.

4.

(i Determine whetherP_∞

k=1

¡_{ln k}

k

¢_k

converges or diverges.

(ii) Determine whetherP_∞

k=1(−1)^{k 2}_k2^√+1^k converges absolutely, converges conditionally or diverges.

5. A nonnegative function f defined on (−∞, ∞) is a probability density function if Z _∞

−∞

f (x) dx = 1.

(i) Let k > 0. Show that the function f defined by

f (x) =

( ke^−kx, if x ≥ 0;

0, if x < 0 is a probability density function.

(ii) Let f be defined as in (i). Compute the mean µ given by µ =

Z _∞

−∞

xf (x) dx.

6. Find the interval of convergence for

X∞

k=1

k²+ k x^k . 7. Evaluate

(i) R₃

0 x

(x²−1)^3/2dx, (ii) R₁

−∞e^(x−ex)dx.

1

2 MIDTERM 1 FOR CALCULUS

8. Let f (x) = ^ex_x⁻¹.

(i) Find the power series representation of f in powers of x.

(ii) Differentiate the power series in (i) and show that X∞

n=1

n

(n + 1)! = 1.

9. Suppose that the function f (x) is infinitely differentiable on an interval containing 0, and suppose that f⁰⁰(x) + f (x) = 1 and f (0) = 1, f⁰(0) = 1. Find the power series representation of f in powers of x. Find the radius of convergence.

10. Let a and b be positive numbers with b > a. Define two sequences {an} and {bn} as follows:

a1 = ^a+b₂ b1 = √

ab an = ^an−1+b₂ ⁿ⁻¹

bn = p

an−1bn−1, for n = 2, 3, 4, . . .

(i) Use mathematical induction to show that an−1> an> bn > bn−1for n = 2, 3, 4, . . ..

(ii) Prove that {an} and {bn} are convergent sequences and that limn→∞an= limn→∞bn.