Solutions for Second Exam Review Questions MAT 131, Fall 1998

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

**Solution:**

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

(see the graph below).

**a.**- Write the equation of the line tangent to
*C*at the point (1,0).

**Solution:**

We use implicit differentiation to obtain 2*x*- 1 = 3*y*^{2}*y*' -*y*'. Solving for*y*' gives . Plugging in at the point (1,0) says the slope of the relevant tangent line is . Thus, the line tangent to*C*at (1,0) is*y*= 0 -1(*x*-1)-- that is,*y*=1-*x*.

**b.**- 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.

**Solution:**

Plugging into the equation for the tangent line gives . Trying the point in the equation for*C*, we obtain

which is off from being true by 3/64, or about 0.0469.

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

**Solution:**

We need to determine*y*'', so we take the derivative of*y*' from part**a**. Again, we use implicit differentiation, this time together with the quotient rule.

Thus, . This means our desired parabola is 1 -*x*- (*x*-1)^{2}.

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

**Solution:**

Here we obtain the estimate Plugging into the equation for*C*gives

off by , or about .00659, a dramatic improvement.

**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.

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

**Solution:**

Let's let*y*(*t*) be the diameter of the culture. This changes at a rate proportional to (that is, a constant times) its square, so we have*y*' =*ky*^{2}

**b.**- Verify that
satisfies the differential
equation for any choice of
*k*and*C*.

**Solution:**

We just need to check that this particular*y*(*t*) satisfies the equation in**a**. Taking the derivative, we have

This is exactly*k*(*y*(*t*))^{2}, so the equation holds.

**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.?

**Solution:**

Let's let*t*=0 correspond to 8 A.M., so we have

That is,*C*=1.Since the diameter is 2 at noon, four hours later, we have

Solving for*k*gives 1 - 4*k*= 1/2, or*k*=1/8, so our particular solution is

At 2 P.M., the size of the culture is given by

*y*(6) = 1/(1 - 6/8) = 4, so it is 4 mm across.At 3 P.M., the culture has a diameter of 1/(1-7/8), doubling to 8

*mm*in one hour.At 4 P.M., the universe comes to an end, because the size of the mold is now infinite.

**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?

**Solution:**

It is helpful to draw a figure. Let's call the angle the spotlight makes with the ground

Let's also call the distance from the ground to where the spotlight hits the building

It is probably safe to assume that the building is at least
approximately perpendicular to the ground, so

Differentiating this with respect to

Plugging in what we know gives the result:

or about .

**5.**
A poster is to be made which requires

**Solution:**

If we let

Thus, , and

**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'?

**Solution:**

We want to maximize the product

So,
, and

**7.**
At what *x* value does the maximum of occur? What is the
maximum value of the function?

**Solution:**

Write
. Then
.
Thus, *f*'(*x*)=0 if , and *f*'(*x*) does not exist if *x*=0.
Thus *x*=*e* is a relative maximum, and it is a global maximum because
of the shape of the graph. The maximum value is 1/*e*.

**8.**
Compute

**Solution:**

Using implicit differentiation,

At , the slope is 1.