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Marks
[42] 1. Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty.
(a) Assume that z(x, y) is a linear function with slope 2 in the x-direction and slope −3 in the y-direction. If z(1, 1) = 4, find z(− 2 , 1).
Answer:
(b) If f (x, y) = ln(x^2 + y), find lim k→ 0
f (1 + k, 0) − f (1, 0) k
Answer:
(c) Let (x 0 , y 0 ) be a critical point of f (x, y) = −x^2 − y^2 + 6x + 8y − 21. Find (x 0 , y 0 ) and then find the equation of the tangent plane to the surface( f (x, y) at the point x 0 , y 0 , f (x 0 , y 0 )
Answer:
(d) Suppose the marginal revenue in producing x units of a certain product is M R(x) = 300 − 0. 2 x. Find the change in total revenue if production is increased from 10 to 20 units.
Answer:
(e) Find
1 + x x − x^2
dx.
Answer:
(f) If
0
f (x) dx = 2 and f (1) = 3, find
0
5 xf ′(x) dx.
Answer:
(g) You are given the following table of values for f (x):
x 1. 5 2. 0 2. 5 3 f (x) − 0. 6 0. 2 0. 4 0. 8
Estimate
- 5
f (x) dx by using the trapezoidal rule with n = 3.
Answer:
(h) Let p = S(q) = 10(e^0.^02 q^ − 1) be a supply curve, where p denotes the price, and q denotes the quantity supplied. Find the average price over the supply interval [20, 30].
Answer:
(l) Find the constant c such that the function
f (x) = cx^2 (1 − x), 0 ≤ x ≤ 1
is a probability density function.
Answer:
(m) Let X be a continuous random variable having the probability density function
f (x) =
x^4
, x ≥ 1. Find the expected value E(X).
Answer:
(n) Let f (x) be the probability density function of a continuous random variable X, where
1 ≤ x ≤ 5. If the area under the graph of y = f (x) from x = 3 to x = 5 is
, find the probability P (1 ≤ X ≤ 3).
Answer:
Full-Solution Problems. In questions 2–6, justify your answers and show all your work.
[10] 2. Find the total area of all the regions completely enclosed by the graphs of the functions f (x) = x^3 − 3 x + 4 and g(x) = x + 4.
[12] 4. Suppose that $100, 000 is deposited in an account paying 5% interest with continuous com- pounding. Also assume that money is continuously withdrawn from the account at a rate of $10, 000 per year. Find the amount of money in the account at the end of 6 years.
[12] 5. An open rectangular box without a top is to be constructed from material that costs $5 per square foot for the bottom and $2 per square foot for its sides. The bottom of the rectangular box is a square. Use the method of Lagrange multipliers (no credit will be given for any other method) to find the dimensions of the box of greatest volume that can be constructed for $240. You do not need to show that the answer you compute gives the greatest volume.
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