FINAL FOR CALCULUS
Date: Monday, Jun 17, 2002 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1. [10%]
(i) Find the exact value of sin7π6. (ii) Find the derivative of f (t) = sin√
t.
2. [10%]
(i) The nth term of a sequence is given by an =1+1−√√nn. Determine whether the sequence converges or diverges. If the sequence converges, find its limit.
(ii) Evaluate dy/dx at (π2,π2) when sin 2x + cos y = 0.
3. [10%] Evaluate the double integral R R
A2xy dA where A is the area enclosed by the curves y =√ x and y = x2.
4. [10%]
(i) Suppose that the average wage earner in a certain country saves 8% of his or her take-home pay and spends the other 92%. Estimate the impact that a proposed $10 billion tax cut will have on the economy over the long run in terms of the additional spending generated.
(ii) Find the area under the curve y = cot x and above the x-axis from x = π4 to x = π2.
5. [10%] Suppose X is a normal random variable with µ = 35 and σ = 3. Find the following values by using the table appended in the exam.
(i) P (X ≤ 44) (ii) P (30 ≤ X ≤ 40)
6. [10%] The life of a light bulb is the random variable x (in hours) associated with the probability density function f (x) = ke−kx over the interval [0, ∞) where k is a constant. Suppose we know that the mean is 1000 hours.
(i) Find the value k.
(ii) Find the variance of the random variable.
7. [10%] Find the solution of the initial value problem: y0 = y2ex+ y2 and y(0) = 1.
8. [10%] Estimate the value √3
7.8 by the following two methods:
(i) Use two iterations (i.e., find x2) of the Newton-Raphson method with the initial guess x0 = 2 on the function f (x) = x3− 7.8.
(ii) Use the second Taylor polynomial of g(x) =√3
x at x = 8.
9. [10%] Find the Taylor series of the function f (x) = sin 2x at x = π6. And also find the interval of convergence.
10. [10%] A company has a monthly advertising budget of $60, 000. Their marketing department esti- mates that if they spend x dollars on newspaper advertising and y dollars on television advertising, then the monthly sales will be given by
z = f (x, y) = 90x1/4y3/4
dollars. Determine how much money the company should spend on newspaper advertising and on television advertising per month to maximize its monthly sales.
1
