CALCULUS FINAL EXAM— PAPER A JANUARY 8, 2004
~Ê»,ÀU™pæñ»5é (A) 1Ÿ-â±
No credit will be given for an answer without reasoning.
1. Evaluate (a) [8%]
Z 4
0
|2x − 1| dx (b) [8%]
Z 1
0
x (x + 1)10dx
(c) [8%]
Z 1
0
(x + 3)√ 2 − x dx
(d) [8%]
Z 1
0
e2x e2x+ 1dx
2. [8%] Find the derivative of y = (x2+ 1)5x+2.
3. [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).
4. 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.
5. 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.
6. [10%] Use the Midpoint Rule with n = 4 to approximate (V¡) Z 2
0
5 x3+ 1dx.
7. [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.
8. [10%] The demand (Û°) and supply (X@) functions for a product are
Demand: p = −0.2x + 12 and Supply: p = 0.3x + 2
where x is the number of units (in millions). Find the consumer (¾‘6) and producer (`¨6) surpluses (ì) 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.
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