Exam 4: Applications of Differentiation

A lakefront runs east-west. A man in a rowboat is 5 miles due north of point A on the shore. He wishes to get to B, 5 miles due east of A, in the least time. He is able to row 3 miles per hour and walk 5 miles per hour. What is the minimum time, in minutes?

The length of a rectangle is increasing at the rate of 2 feet per second, while the width is increasing at the rate of 1 foot per second. When the length is 5 feet and the width is 3 feet, how fast is the area increasing?

Find the most general antiderivative of the function.(a) $f ( x ) = \pi ^ { 3 }$ (b) $f ( t ) = \sec ^ { 2 } t + t ^ { 2 }$ (c) $f ( \theta ) = \left( e ^ { \frac { \theta } { 2 } } + e ^ { - \frac { \theta } { 2 } } \right) ^ { 2 }$ (d) $f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } + 1 } { x ^ { 4 } }$

Find the most general antiderivative of $\sec ^ { 2 } x + \sin x$ .

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

Consider the function $f ( x ) = x ^ { 2 }$ on the interval $\left[ 0 , \frac { 1 } { 2 } \right]$ . According to the Mean Value Theorem, there must be a number c in $\left( 0 , \frac { 1 } { 2 } \right)$ such that $f ^ { \prime } ( c )$ is equal to a particular value d. What is d?

A revenue function is given by $R ( x ) = 6000 x - 0.2 x ^ { 3 }$ . where x is the number of units sold. Find the number of units x that produces maximum revenue.

Use Newton's method to approximate the root of $f ( x ) = x ^ { 3 } + 4 x + 2$ that lies between $x = - 1$ and $x = 0$ .

Suppose that the cost function (in dollars) for producing Fuzzy Widgets® is given by $C ( x ) = 0.004 x ^ { 3 } - 0.02 x ^ { 2 } + 6 x + 1000$ . Determine the minimal marginal cost.

Draw a direction field for the function $f ( x ) = \frac { 1 - \cos x } { x ^ { 2 } } , 0 < x < 2 \pi$ . Use this direction field to sketch several members of the family of antiderivatives.

The graph of the first derivative $f ^ { \prime }$ of a function f is given below. (a) On what intervals is f increasing? Explain.(b) At what values of x does f have a local maximum and local minimum? Explain.(c) On what intervals is f concave up? concave down? Explain (d) Where are the coordinates of the inflection points of f? Explain

Find all value(s) of c (if any) that satisfy the conclusion of the Mean Value Theorem for the function $f ( x ) = \frac { 1 } { 1 + x }$ on the interval $[ 0,1 ]$ .

A revenue function is given by $R ( x ) = 6000 x - 0.4 x ^ { 3 }$ . where x is the number of units sold. Find the marginal revenue when $x = 50$ .

A ladder 10 feet long is leaning against a wall. If the top of the ladder is sliding down the wall at 4 feet per second, how fast is the angle between the top of the ladder and the wall changing when the foot of the ladder is 8 feet away from the wall?

If $f ^ { \prime \prime } ( x ) = - 4 \sin 2 x , f ( 0 ) = 0 , f ^ { \prime } ( 0 ) = 2$ , find the value of $f \left( \frac { \pi } { 4 } \right)$ .

Find all value(s) (if any) that satisfy the condition of the Mean Value Theorem for the function $f ( x ) = 10 - \frac { 16 } { x }$ on the interval $[ 2,8 ]$ .

A plastic right cylinder with closed ends is to hold V cubic feet. If there is no waste in construction, find the ratio between the height and diameter that results in the minimum use of materials.

On what interval is the function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ increasing?

A gas pipeline is to be constructed from a storage tank, which is right on a road, to a house which is 600 feet down the road and 300 feet back from the road. Pipe laid along the road costs $8.00 per foot, while pipe laid o the road costs $10.00 per foot. What is the minimum cost for which this pipeline can be built?

Find the value of the limit $\lim _ { x \rightarrow 0 } ( 1 - x ) ^ { \frac { 1 } { x } }$ . Use L'Hospital's Rule, if applicable.