DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 1823

INSTRUCTIONS:

(a)Answer each question in your answer booklets. All your work must be shown.
(b)For most questions calculators are not required; on such questions the calculator may only be used to check accuracy.
MARKS
(15)1.Find the derivative of each of these functions:
(a)
(b)
(c)
(d)
(e)
(12)2.Determine these limits:
(a)
(b)
(c)
(d)
(13)3.(a)Find the equation of the tangent line to the graph of when x = 1.
(b)Simplify:
(c)Recall that an investment of $P compounded continuously for x years at (100r)% p.a. is worth . If the annual interest rate is 5%, how long does it take for $1000 to double in value?
(d)Set up (but do not evaluate) the limit required when computing the derivative of from the definition.
(8)4.A concert promoter finds from a consumer survey that she can sell 500-5x tickets, if the price is set at x dollars each. On the other hand, concert costs run at $10,000 plus an additional $20 for each ticket.
(a)Give the revenue function (as a function of x).
(b)Give the cost function.
(c)Give the profit function.
(d)What ticket price should the promoter charge in order to maximize her profit? Be sure to justify your answer using the methods of calculus.
(14)5.For the function :
(a)Determine the intervals where is increasing and where is decreasing.
(b)Find the relative maxima and minima.
(c)Determine the intervals where is concave up and where is concave down.
(d)Sketch the graph of .
(5)6.If the marginal cost function is given by and the fixed cost (i.e. at the 0 production level) is 500, determine the cost function.
(14)7.Evaluate the following integrals:
(a)
(b)
(c)
(d)
(6)8.Find the area enclosed by the curves
and y = 2 - x