### DEPARTMENT OF MATHEMATICS & STATISTICS MATH 1823

 FINAL EXAMINATION APRIL 1995 TIME: 3 Hours

#### ANSWER ALL THE QUESTIONS. ALL YOUR WORK MUST BE SHOWN IN YOUR ANSWER BOOK.

 MARKS (5) 1. A student group produces t-shirts at a cost of \$8.00 each. The fixed cost is \$500.00 per year. The demand is given by x = 1000 - 20p, where p is the price in dollars charged to the consumer, and x is the number bought per year. (a) Find the yearly revenue function. (b) Find the profit function and the marginal profit function. (c) Find the appropriate profit on the sale of the 401st t-shirt. (9) 2. (a) Solve for x (i) e2 lnx - lne2x = 3 (ii) log5(x + 15) = log525 + 1 (b) Simplify (9) 3. Find (a) (b) (c) (5) 4. Find the equation of the tangent to the curve  y = 3x5 + 15x2 + 20x + 5  at the point where x = -1. (8) 5. A water tank is to be constructed with a concrete base, cast iron sides, and open at the top. The base is to be a rectangle whose length is 3 times its width. The volume is to be 12 cubic metres. If the concrete for the base costs \$4.00 per square metre and the cast iron for the sides cost \$6.00 per square metre, what dimensions should the tank have to minimize the cost? What is this minimum cost? (Justify your answer, show all your work.) (15) 6. Find the derivatives of the following functions: (a) (b) g(x) = 3x(5x2 + 9x + 1)4 (DO NOT SIMPLIFY) (c) (SIMPLIFY YOUR ANSWER) (d) (DO NOT SIMPLIFY) (e) (DO NOT SIMPLIFY) (16) 7. For the function f(x) = x3 - 3x2 - 9x + 16 : (a) Find the intervals where f(x) is increasing and where f(x) is decreasing. (b) Find the relative maxima and minima. (c) Find the intervals where f(x) is concave up and where f(x) is concave down. (d) Find the inflection points. (e) Sketch the graph of y = f(x) showing all the above information. (6) 8. The decay of some natural resource is given by Q(t) = Q0e-kt. At time t = 0, the resource measured 10,000 units. At time t = 20, the resource measured 8,000 units. (a) Find the value of Q0. (b) Find an expression for e-k. You may use fractional exponents, a calculator is not needed. (c) What will the resource measure at time t = 40. (A calculator is not needed to solve this.) (d) Give an expression for the value of k. (You do not need to use a calculator; a numerical value is not needed.) (15) 9. Evaluate the following integrals: (a) (b) (c) (d) (e) (8) 10. Find the area between the curves (4) 11. Suppose the marginal profit function for the sale of x units is given (in dollars) by and the profit from the sale of 10 items is \$500.00. Find the profit function.