|TIME: 3 Hours|
|(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) e||(ii) log||(b)||Simplify ||(9)||3.|| Find
||(a)||(b)||(c)||(5)||Find the equation of the tangent to the curve y = 3||(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
||(a)||(b)||g(x) = 3||(c)|| (SIMPLIFY YOUR ANSWER)
||(d)|| (DO NOT
||(e)|| (DO NOT
||For the function f(x) = x3 - 3||(a)||Find the intervals where f(x) is increasing and where f(x) is
||(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
||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