Math 161

Sample Midterm 2

Dr. Wilson

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In problems 1 - 6, find any x and y intercepts, asymptotes, horizontal, vertical, or otherwise, find where the function assumes positive values and where it assumes negative values, the places where the first derivative is either 0 or does not exists, where the function is increasing and decreasing, find the places where the second derivative is 0 or doesn't exist, and where the function is concave up and concave down. Find the local maxima and minima and points of inflection, and sketch the graph.

1.

y = x3 - 3x

2.

y = 3x5 - 5x 3

3.

4.

5.

6.

7. A person is going to make a box by taking a square piece of cardboard, which is 12 inches on a side, like the one in the picture to the right, cutting squares out of the corners, and folding the edges like the other figure in the picture to the right. How big of a square should they cut out of the corners in order to maximize the volume of the resulting box?

8. A rectangular sheet of metal 20 feet long and 2 feet wide is bent down the middle to form a "V". Two triangular pieces of metal are soldered onto the ends to form a prismatic trough as in the picutre to the right. What does the height of the triangle have to be to maximize the volume of the trough?

9. What are the dimensions of a right circular cylindrical can with a given surface area which will result in the maximum volume?