Exam 4: Applications of Differentiation

Consider $f ( x ) = x ^ { 4 } - 6 x ^ { 2 }$ .(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.

One model of a company's costs is given by the short-run Cobb-Douglas cost curve $C ( x ) = K x ^ { \frac { 1 } { a } } + F$ , where a is a positive constant, F is the fixed cost, and K measures the available technology.(a) What is the marginal cost function? (b) For what values of a will C be concave downward?

Use Newton's Method to approximate all real roots of $x ^ { 4 } + x ^ { 3 } - 3 x ^ { 2 } + 4 x - 28 = 0$ .

Let $f ( x ) = \frac { x } { x ^ { 2 } + 25 }$ .(a) Use your graphing calculator to graph f using the default viewing rectangle .(b) Find the critical points of f and use them to determine an appropriate viewing rectangle (or rectangles) for f. The graph f again using the viewing rectangle.(c) Compare the graphs in parts (a) and (b)

Find all critical numbers for the function $f ( x ) = x ^ { 3 } - 12 x ^ { 2 }$ .

A rectangle with base on the x-axis has its upper vertices on the curve $y = 12 - x ^ { 2 }$ . Find the maximum area of such a rectangle.

A ball is dropped from a tower. When it strikes the ground it is traveling downward at a rate of 60 feet per second. How tall is the tower?

Find the range of the function $f ( x ) = e ^ { \cos x }$ .

Find all critical numbers for the function $f ( x ) = \sqrt [ 3 ] { x } ( 4 - x )$ .

Find each of the following limits or explain why it does not exist.(a) $\lim _ { x \rightarrow \infty } \frac { \ln \left( a + b e ^ { x } \right) } { x }$ (b) $\lim _ { x \rightarrow \infty } x \sin \left( \frac { 1 } { x } \right)$ (c) $\lim _ { x \rightarrow 0 ^ { + } } \left( \frac { x } { 1 + x } \right) ^ { x }$

Let $f ( x ) = x ^ { 6 } + 192 x + 17$ . Find the intervals on which f is decreasing.

A rectangle with base on the x-axis has its upper vertices on the curve $y = 4 - \frac { 1 } { 2 } x ^ { 2 }$ . Find the maximum perimeter of such a rectangle.

Find two positive numbers whose sum is 12 and the product of one number and the square of the other number is a maximum.

Find the absolute maximum values of $f ( x ) = x ^ { 3 } - 3 x$ over the set of all real numbers x such that $x ^ { 2 } + 6 \leq 5 x$

Sketch the graph of a function f that is continuous on [0,4] and has the following properties: Absolute maximum at 1, absolute minimum at 3.

Find the maximum and minimum values of $f ( x ) = x - \ln x ^ { 100 } \text { where } 1 \leq x \leq 100$

At what rate is the surface area of a cube increasing if its edges are 2 inches and are increasing at a rate 3 inches per minute?

Explain why L'Hospital's Rule is not applicable to evaluate $\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \cos x } { x }$ , then find the limit.

An astronaut stands on a platform 3 meters above the moon's surface and throws a rock directly upward with an initial velocity of 32 m/s. Given that the acceleration due to gravity on the moon's surface is 1.6 m/s $2$ , how high above the surface of the moon will the rock travel?

Find the largest interval on which the function $f ( \theta ) = \theta + \sin \theta$ is concave upward.