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Mathematics 131, Fall 1999, Final Exam




1. The value of a toy is $60. The toy will depreciate constantly to a value of zero over a period of 20 years.

(a) Express the value of the toy after t years.

(b) When will the value of the toy be $30?


2. Find f'(x) for f(x) = 4 - 6x² using only the definition of the derivative




3. Find dy/dx for each of the following. Algebraic simplification is not necessary.





4. A computer manufacturer estimates that if x computers can be produced each month, the total cost will be C(x) = 90000 + 400 x dollars. With a unit proce U(x) = 1600 - 3 x dollars all computers could be sold.

a) Find the revenue function.

b) Estimate, using your graphing calculator, the "break-even" level of production (cost = revenue).

c) Find the profit function.

d) How many computers should be produced to maximize profit?


5. Suppose f(x) = 4x⁴ - 8x² .

(a) Determine when f(x) is increasing, decreasing, concave up, and concave down.

(b) Find all relative extrema and inflection points.

(c) Sketch the graph of y = f(x).


6. A box is to be constructed with a square base and no top. Material for the sides costs $2 per square foot while material for the bottom costs $8 per square foot. If the box costs $2400, find the dimensions which yield the greatest volume.


7. If xy² + 2x² = 3y defines y implicitly as a function of x,

a. find dy/dx at (1,2) and

b. find an aquation of the tangent line to the graph at (1,2).


8. Using the approximation principle, estimate

without using a calculator.


9. Find the absolute minimum of f(x) = 2x² - 6x + 1 on [ -5/2 , 2 ] .




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