Exam 4: Applications of Differentiation

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

While driving through a certain state, a tourist passed through a turnpike toll booth and got a ticket punched for 2:10 p.m. Down the road 74 miles, the tourist exited the turnpike and handed in his ticket which was punched at 3:05 p.m. In addition to the toll amount, he was handed a speeding ticket for driving in excess of 80 miles per hour in a 65 mph zone. Is the speeding ticket justifiable? Explain.

Is it true that $\int \cos ^ { 2 } x d x = \sin ^ { 2 } x + C$ ? Explain why or why not.

A large, closed shipping container with a square base is to be made from 1000 square feet of fiberboard. Find the dimensions of the container with the greatest volume.

Let $f ( x ) = \frac { \ln x } { x } .$ Find the maximum value of f.

A custard pie is thrown vertically up from the ground with velocity 48 ft/s. Find the greatest height that it attains.

A hot air balloon is rising vertically. An observer on the ground is 2000 meters from the point where the balloon took off. The angle between the ground and the observer's line of sight increases at a rate of $\frac { 1 } { 2000 } \mathrm { radian } / \mathrm { s }$ when the balloon reaches a height of 3000 meters. How fast is the balloon rising at that moment?

Sketch the graph of $f ( x ) = x ^ { 4 } + x - 3$ on the interval $[ - 3,3 ]$ . Suppose that Newton's method is used to approximate the positive root of f with initial approximation $x _ { 1 } = 2$ .(a) On your sketch, draw the tangent lines that you would use to find $x _ { 2 }$ and $x _ { 3 }$ , and estimate the numerical values of $x _ { 2 }$ and $x _ { 3 }$ .(b) To approximate the negative root of f, use $x _ { 1 } = - 2$ as the starting approximation. As before, draw the tangent lines that you would use to find $x _ { 2 }$ and $x _ { 3 }$ , and estimate the numerical values of $x _ { 2 }$ and $x _ { 3 }$ .(c) Suppose that you had used $x _ { 1 } = 0$ as the starting point for approximating the negative root. Discuss what would happen.

Find the critical number(s) (if any) and test them to determine if $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 16$ has a local maximum or local minimum or neither at each critical number.

Find all critical numbers for the function $f ( x ) = 2 \sin x - x \text { for } 0 < x < \pi$ .

Find the minimum value of the function $f ( x ) = \frac { e ^ { x } } { x } \text { on } ( 0 , \infty )$ .

Find the value of the limit $\lim _ { x \rightarrow \infty } \sqrt { x ^ { 2 } + 20 x } - x$ . Use L'Hospital's Rule, if applicable.

Find the critical number(s) (if any) and test them to determine if $f ( x ) = \frac { x } { 1 - x ^ { 2 } }$ has a local maximum or local minimum or neither at each critical number.

Find the most general antiderivative of the function.(a) $f ( x ) = 12 x ^ { 2 } + 6 x - 5$ (b) $f ( x ) = \left( 1 - \frac { 1 } { \sqrt { x } } \right) \left( 1 + \frac { 1 } { \sqrt { x } } \right)$ (c) $f ( x ) = \frac { 1 } { x ^ { 2 } } - \frac { 2 } { x ^ { 3 } }$ (d) $f ( t ) = \frac { t ^ { 3 } + t ^ { 2 } } { \sqrt { t } }$

A direction field for a function f is given below. Use this to sketch the antiderivative F that satisfies $F ( 0 ) = 2$ .

Is it true that $\int \ln x d x = \frac { 1 } { x } + C$ ? Explain why or why not.

Use Newton's method to find $\sqrt { 89 }$ correct to four decimal places.

A company has cost function $C ( x ) = 1000 + 20 x + x ^ { 2 }$ . Find: (a) The average cost function.(b) The marginal cost function.(c) The production level x that will minimize the average cost.(d) The minimum average cost.

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 foot of the ladder being pulled away from the wall when the foot of the ladder is 8 feet away from the wall?

Choose the correct statement: If f(x) is not continuous on [1, 3] then (i) f(x) has no critical point over the interval.(ii) f(x) has no extreme value over the interval.(iii) f(x) might have extreme value over the interval.