Give the speed of the particle at t = π

FINAL FOR CALCULUS

Date: 2000, January 17, 1:10–3:00PM Each problem is worth 10 points.

1.

(i) At time t, a particle has position

x(t) = 1 − cos t, y(t) = t − sin t.

Find the total distance traveled from t = 0 to t = 2π. Give the speed of the particle at t = π.

(ii) Find the area of the surface generated by revolving the curve y = cosh x, x ∈ [0, ln 2] about the x-axis.

2.

(i) Find the tangent(s) to the curve

x(t) = −t + 2 cos1

4πt, Y (t) = t⁴− 4t² at the point (2, 0).

(ii) Find the area of the region common to the circle r = 2 sin θ and the lima¸con r = ³₂− sin θ.

3.

(i) Verify that sinh 2t = 2 sinh t cosh t.

(ii) Compute f⁰(x) where f (x) = x^2x. 4.

(i) Two years ago, there were 4 grams of a radioactive substance . Now there are 3 grams. How much was there 10 years ago?

(ii) Determine the exact values of sec⁻¹(−√

2) and sin⁻¹(sin 7π/4).

5.

(i) Let f be continuous and define F by F (x) =

Z _x

0

· t²

Z _t

1

f (u) du

¸ dt.

Find F⁰(x) and F⁰⁰(x).

(ii) Compute the limit

x→0lim

1 − cos 4x 9x² .

6. Sketch the graph of the continuous function f that satisfies the conditions:

f⁰⁰(x) > 0 if |x| > 2, f⁰⁰(x) < 0 if |x| < 2;

f⁰(0) = 0, f⁰(x) > 0 if x < 0, f⁰(x) < 0 if x > 0;

f (0) = 1, f (2) = ¹₂, f (x) > 0 for all x, and f is an even function.

7. Evaluate the given integral

(i) Z

x(x + 1)⁹dx,

1

2 FINAL FOR CALCULUS

(ii) Z

cos θ

sin²θ − 2 sin θ − 8dθ.

8.

(i) Evaluate the integral Z

dx e^x√

4 + e^2x.

(ii) Show that the polynomial p(x) = x³+ ax²+ bx + c has no extreme values if and only if a²< 3b.

9. Show that, if u and v are differentiable functions of x and f is continuous, then d

dx

·Z _v

u

f (t) dt

¸

= f (v)dv

dx− f (u)du dx. Then compute

d dx

·Z _x²

x

dt t

¸ . 10.

(i) Use mean value theorem to show that, if f is continuous on [x, x + h] and differentiable on (x, x + h), then

f (x + h) − f (x) = f⁰(x + θh)h for some number θ between 0 and 1.

(ii) Let h > 0. Suppose that f is continuous on [a−h, a+h] and differentiable on (a−h, a)∪(a, a+h).

Use (i) to show that if

x→alim⁻f⁰(x) = lim

x→a⁺f⁰(x) = L, then f is differentiable at a and f⁰(a) = L.