Exam 4: Applications of Differentiation

Suppose that C(x) is the number of dollars in the total cost of producing x tables (x > 6) and $C ( x ) = 25 + 4 x + 18 x ^ { - 1 }$ . Find the marginal cost when $x = 15$ .

(a) Draw a direction field for the function $f ( x ) = \sin x$ . Use this direction field to sketch several members of the family of antiderivatives.(b) Compute the general antiderivative for $f ( x ) = \sin x$ explicitly and sketch several specific antiderivatives. Compare these results with your sketches from part (a).

A particle moves along a path described by $y = 4 - x ^ { 2 }$ . At what point along the curve are x and y changing at the same rate?

Let $f ( x ) = 2 x ^ { 4 } - 3 x ^ { 3 } - 10 x ^ { 2 } + 20$ . Use the graphs of f, $f ^ { \prime }$ , and $f ^ { \prime \prime }$ to estimate, to two decimal places, each of the following: (a) The critical numbers of f.(b) The intervals where f is increasing.(c) The x values where f has inflection points.(d) The intervals where f is concave downward.

Find the absolute minimum value for the function $f ( x ) = \frac { x - 1 } { x + 1 } \text { for } 0 \leq x \leq 1$

Find the minimum value of the function $f ( x ) = x \ln x$ .

Find the minimum value of the function $f ( x ) = x ^ { 2 } - 2 x + 2$ .

Given $f ( x ) = x ^ { 2 } - 3$ , use Newton's method to find the iterative formula for $x _ { n + 1 }$ .

(a) Explain why Newton's Method is unable to find a root for $x ^ { 3 } - 3 x + 4 = 0$ if $x _ { 1 } = 1$ or $x _ { 1 } = - 1$ .(b) Using $x _ { 1 } = - 2$ ,use Newton's Method to approximate a root of this equation to four decimal places.

Find all critical numbers for the function $f ( x ) = \frac { x - 1 } { x + 1 }$ .

On what interval is the graph of $f ( x ) = \left( 1 - \frac { 1 } { x } \right) ^ { 2 }$ concave downward?

A rectangular swimming pool with a horizontal bottom is being drained. If its length and width are 25 ft and 20 ft, and the water level is falling at the rate of $\frac { 1 } { 2 }$ ft/min, how fast is the water draining?

Find the interval on which the graph of $f ( x ) = \ln \left( x ^ { 2 } + 1 \right)$ concave upward?

Consider $f ( x ) = e ^ { - x ^ { 3 } }$ .(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f.(c) Find the intervals of concavity and the inflection points.(d) Using the information in (a), (b), and (c) to sketch a graph of f.

Find the shortest distance from the point $( 4,0 )$ to a point on the parabola $y ^ { 2 } = 2 x$ .

Find the x-coordinate of the point on the graph of $y = \frac { 1 } { x ^ { 2 } + 3 }$ that has the maximum slope.

Find f(x) if $f ^ { \prime \prime } ( x ) = \frac { 2 } { \sqrt { x } } , f ( 4 ) = 1$ .

Find f(x) if $f ^ { \prime \prime } ( x ) = \cos x , f ^ { \prime } \left( \frac { \pi } { 2 } \right) = 1 , f \left( \frac { \pi } { 2 } \right) = 3$ .

Mr. Wilson is standing near the top of a ladder 24 feet long which is leaning against a vertical wall of his house. Dennis, the little boy next door, ties a rope from his tricycle to the bottom of the ladder and starts to pull the foot of the ladder away from the house wall. The bottom end of the ladder begins to slide away from the wall at the rate of 1 foot per second.(a) How fast is the top of the ladder sliding down the wall when the foot of the ladder is 8 feet from the wall? (b) How fast is the top of the ladder sliding down the wall when the top of the ladder is 16 feet from the ground? (c) How fast is the angle between the top of the ladder and the wall changing when the foot of the ladder is 8 feet from the wall?

A rancher wishes to fence in a rectangular corral enclosing 1300 square yards and must divide it in half with a fence down the middle. If the perimeter fence costs $5 per yard and the fence down the middle costs $3 per yard, determine the dimensions of the corral so that the fencing cost will be as small as possible.