This is an actual M403K midterm given on September 22, 1992. On that test I allowed a ``crib sheet'' and a non-graphing calculator.

The syllabus has changed some in 8 years, so this exam doesn't cover
exactly the same material as our first midterm will. In particular, problem
4 is Chapter 2 material, which will be tested in the *second* midterm.

**Problem 1. Tangent lines**

Consider the curve

a) Find *dy*/*dx* when *x*=1.

b) Find the slope of the line that is tangent to this curve at the point (1,4).

c) Find the equation of this line.

**Problem 2. Derivatives and limits**

A race car driver is circling the track at Indianapolis. The table below
gives his position at various times (*x* denotes time, in hours. *y*
denotes distance, in miles).

How fast is he going at time *x*=1?

**Problem 3. Limits and continuity**

Consider the function

a) Find the limit of f(x) as x approaches -1 from below, as x approaches -1 from above, and the overall limit , if they exist.

b) Is *f*(*x*) continuous at *x*=-1? Why or why not?

**Problem 4. Graphs and optimization**

Consider the function .

a) Find all the partition points of *f*. For what values of *x*
is *f*(*x*) positive?

b) Find all the critical points of *f*. For what values of *x*
is *f*(*x*) increasing?

c) Find all the local minima of *f*(*x*).

**Problem 5. Taking derivatives**

Take the derivatives, with respect to *x*, of the following functions.
You do NOT need to simplify your results:

a)

b)

c)

d)

**Problem 6. Taking limits**

Evaluate the following limits:

a)

b)

c)

d)