Q. Find relative maxima and minima?

When finding relative maxima and minima in the Chapters absolute extrema problem, don't forget to use the first or second derivative test to show that the critical value is a relative maxima or minima. Also, when sketching a graph use EXCEL or graph paper with the scale of the axes labeled. Any other sketch will be marked incorrect.

1. For a certain SUV, M(x) = -.015x² + 1.31x - 7.3, 30 ≤ x ≤ 60, represents the mpg obtained at a speed of x mph. Find the absolute maximum and minimum mpg and the speeds at which they occur. Graph M(X).

2. Differentiate the following implicitly (A) 2xy² - 4xy = x³    (B) x²y² - 3xy = x  C)  x³ + y³ = 3   (D) sqrt(x)+sqrt(y)=2 (sqrt = square root)   E) xy  + y² = x

3. The cost and revenue functions for children's watches are given by C(q) = 250 + 3q + .01q² and  R(q) = 5q + .02q² where q  is monthly production. If production is increasing at a rate of 200 watches per month, when production is 3000 units, find the rate of increase in profits.

4. A 26 foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away from the wall?

5. A farmer wants to construct a rectangular pen next to a barn 60 feet long, using all of the barn as part of one side of the pen. Find the dimensions of the pen with the largest area that the farmer can build if (A) 160 feet of fencing material is available; (B) 250 feet of fencing material is available.