Math 152
Sample Exam 2

1.

Use the formula

$$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) -f(x)}{h}$$

to compute the derivative of f(x) = x².

2.

Find y' for each of the following functions:

(a)

$y = \sqrt[3]{x}$

(b)

$y = x^2 \sin(x)$

(c)

$y = \frac{x^3-x^2}{3x+1}$

(d)

$y = \sin (x^2+3)$

(e)

y = e^-3x

(f)

y = x²(4+x²)⁴

3.

Let f(x) = x³ - 3x -1

(a)

Find and classify (as a relative maximum, a relative minimum, or neither) all critical points, if any, of f.

(b)

On what intervals of x, if any, is f increasing? On what intervals of x, if any, is f decreasing?

(c)

Find all inflection points, if any, of f.

(d)

On what intervals of x if any, is f concave upwards? On what intervals of x if any, is f concave downwards?

(e)

Sketch the graph of f

4.

Let S(x) be the number of woody species in several Malaysian rainforests for a phosphorus and potassium concentration index of x. See the figure below.

(a)

Explain in words what S'(x) represents.

(b)

Using the graph of S(x) sketch the graph of S'(x) when the phosphorus and potassium concentration index is between 0 and 3.

5.

Suppose a function $B(t) = \frac{30 + 10t}{2+t}$ describes the biomass (in grams) of a bacteria population t hours after the start of an experiment. Find B'(t) and give its units. Is the biomass always increasing, always decreasing, or sometimes increasing and sometimes decreasing for $t \ge 0$?

6.

Some birds tend to avoid flights over large bodies of water during daylight hours. (It is speculated that more energy is required to fly over water because are generally rises over land and falls over water during the day.) Suppose an adult bird with this tendency is taken from its nesting area on the edge of a large lake to an island 10 miles upstream and 5 miles offshore and is the released. If the resistance over water is 1.4 times as much over land, how far up shore (x, in miles) should the bird head in order to minimize the total energy expended in returning to its nesting area?

(NOTE: Energy expended = distance times resistance. You may assume distance over land has resistance 1.)

7.

A forest company plans to log a certain area of fir trees after a given number of years. The average number of board feet obtained per tree over the given period is known to be (50-.5x), where x is the number of trees per acre, and x lies between 35 and 80. What density of trees should be maintained in order to maximize the amount of timber.