Review for Second Exam        MAT 131, Fall 1998

The midterm will be held on Tuesday, November 17 at 8:30 pm at locations to be announced. Be sure to bring your Stony Brook ID card and your calculator.

This handout is to help you review and study for the exam. It includes several problems which are similar to those on the exam. These problems are not exactly the same as those on the exam, but they should give you an idea of the kinds of things you need to know. However, you should also do a large number of additional problems from the text. Do not merely limit yourself to the assigned problems or those given here. Solutions to these problems should appear on the web page later this week.

The exam will cover material through section 4.3 in the text, as well as §4.6 and §4.8.

1. Compute the derivative with respect to x for each of the following expressions:

$\pi x^8 - \sqrt{x}$          $x^3 + \tan x$          $x^2 \sin(x^2)$          $\displaystyle{\frac{x^3}{1 + x^4}}$          $\sin(\cos(2x + 1))$
$\ln{\sqrt x}$          $(1 + x^2) \arctan{x}$          $\displaystyle{\frac{\sin(3x)}{3x + 1}}$          $\displaystyle{e^{\cos x}}$          $x\ln x$

2. Consider the elliptic curve C which consists of the set of points for which

x2 -x = y3 - y

(see the graph below).

Write the equation of the line tangent to C at the point (1,0).
Use your answer to part a to estimate the y-coordinate of the point with x-coordinate 3/4 marked A in the figure. Plug your estimate into the equation for C to determine how good it is.

Write the equation of the parabola which best approximates C at the point (1,0).

Use your answer to part c to improve your answer from part b. How close does this new estimate come to being right?

3. A mold culture is growing on the world's largest slice of bread. The culture starts in the center of the bread, and remains approximately circular.

The size of the culture grows at a rate proportional to the square of its diameter. Write a differential equation which expresses this relationship.

Verify that $\displaystyle{y(t) = \frac{1}{C - kt}}$ satisfies the differential equation for any choice of k and C.

If the diameter of the culture was 1 mm at 8 A.M. and 2 mm at noon, what is the size of the culture at 2 P.M.? What about at 3 P.M.? Does anything surprising happen at 4 P.M.?

4. A spotlight is aimed at a building whose base is 20 feet away. If the light is raised so that its angle increases at a constant rate of 5 degrees per second, how fast is the image rising when the light makes a 45 degree angle with the ground?

5. A poster is to be made which requires $150~{\rm in}^2$ for the printed part, and is to have a 3" margin at the top and bottom, and a 2" margin on the sides. What should the dimensions be in order to minimize the total area of the poster?

6. The stiffness of a beam is directly proportional to the product of its width and the cube of its breadth. What are the dimensions of the stiffest beam that can be cut from a cylindrical log with a radius of 2'?

7. At what x value does the maximum of $\ln(x)/x$ occur? What is the maximum value of the function?

8. Compute $\frac{dy}{dx}$ for the curve $\sin(x) + \cos(y) = 1$. What is the slope of the tangent line at the point $(\pi/6, \pi/3)$?


Scott Sutherland