[10%] Find the area of the region bounded by the graphs of f (x

CALCULUS FINAL EXAM— PAPER C JANUARY 8, 2004

~ʐ»,ÀU™pæñ»5é (C) 1Ÿ-â±

No credit will be given for an answer without reasoning.

1. Evaluate (a) [8%]

Z ₁

0

(x + 3)√ 2 − x dx

(b) [8%]

Z ₁

0

e^2x e^2x+ 1dx (c) [8%]

Z ₄

0

|2x − 1| dx (d) [8%]

Z ₁

0

x (x + 1)¹⁰dx

2. [8%] Find the derivative of y = (x²+ 1)^2x+5.

3. The marginal revenue (iÒГç) for the sale of a product can be modelled by dR

dx = 50 − 0.02x + 100 x + 1, where x is the quantity demanded.

(1) [6%] Find the revenue function R.

(2) [4%] Find the revenue when 1500 units are sold.

4. [8%] A company purchases a new machine (˛7ø«hœÂ) for which the rate of depreciation (~H§0) can be modelled by

dV

dt = 10, 000 (t − 6), 0 ≤ t ≤ 5

where V is the value of the machine after t years. Set up and evaluate the definite integral that yields the total loss of value of the machine over the first 3 years (#|‡úœÂgMí,¸Ü).

5. [10%] Use the Midpoint Rule with n = 4 to approximate (V¡) Z ₂

0

5 x³+ 1dx.

6. [10%] Find the area of the region bounded by the graphs of f (x) = (x − 1)³, and g(x) = x − 1 from x = 0 to x = 2.

7. Find the volume of the solid obtained by revolving the curve y =x²

3 from (0, 0) to (1,1 3) (a) [6%] about x-axis.

(b) [6%] about the line y = −3.

8. [10%] The demand (Û°) and supply (X@) functions for a product are

Demand: p = −0.3x + 10 and Supply: p = 0.1x + 2

where x is the number of units (in millions). Find the consumer (¾‘6) and producer (`¨6) surpluses (Œì) for this product.

Hint: Let (x₀, p₀) be the point at which a demand function and a supply function intersect. Economists (%Èç 6) call the area of the region bounded by the graph of the demand function, the horizontal line (®() p = p0, and the vertical line (ò() x = 0 the consumer surplus. Similarly, the area of the region bounded by the graph of the supply function, the horizontal line p = p₀, and the vertical line x = 0 is called the producer surplus.

1