**
MATH 1823**

FINAL EXAMINATION APRIL 1995 | TIME: 3 Hours |

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.
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(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 401^{st} t-shirt.
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(9) | 2. | (a) | Solve for x
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(i) e^{2 lnx} - lne^{2x} = 3
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(ii) log_{5}(x + 15) = log_{5}25 + 1
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(b) | Simplify | ||||||

(9) | 3. | Find | |||||

(a) | (b) | (c) | |||||

(5) | Find the equation of the tangent to the curve y = 3x^{5} + 15x^{2} + 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.)
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(15) | 6. | Find the derivatives of the following functions: | |||||

(a) | |||||||

(b) | g(x) = 3^{x}(5x^{2} + 9x + 1)^{4} (DO NOT
SIMPLIFY)
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(c) | (SIMPLIFY YOUR ANSWER) | ||||||

(d) | (DO NOT SIMPLIFY) | ||||||

(e) | (DO NOT SIMPLIFY) | ||||||

For the function f(x) = x^{3} - 3x^{2} - 9x + 16 :
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(a) | Find the intervals where f(x) is increasing and where f(x) is
decreasing.
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(b) | Find the relative maxima and minima. | ||||||

(c) | Find the intervals where f(x) is concave up and where f(x)
is concave down.
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(d) | Find the inflection points. | ||||||

(e) | Sketch the graph of y = f(x) showing all the above
information.
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The decay of some natural resource is given by Q(t) = Q_{0}e^{-kt}.
At time t = 0, the resource measured 10,000 units.At time t = 20, the resource measured 8,000 units.
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(a) | Find the value of Q_{0}.
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(b) | Find an expression for e^{-k}. You may use fractional
exponents, a calculator is not needed.
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(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
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and the profit from the sale of 10 items is $500.00. Find the profit function. |