Exam 4: Applications of Differentiation

Find each of the following limits or explain why it does not exist.(a) $\lim _ { x \rightarrow 1 ^ { + } } \frac { \ln ( 2 x ) } { \ln x }$ (b) $\lim _ { x \rightarrow 0 } \frac { e ^ { - 2 x } - 1 } { x ^ { 2 } - x }$ (c) $\lim _ { x \rightarrow \frac { \pi } { 2 } } ( \sec x - \tan x )$

Find the value of the limit $\lim _ { x \rightarrow 0 ^ { + } } ( 1 + x ) ^ { \frac { 1 } { x ^ { 2 } } }$ . Use L'Hospital's Rule, if applicable.

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

Let $f ( x ) = \frac { 2 x ^ { 2 } + 3 x + 6 } { x + 2 }$ . Use the graphs of f and $f ^ { \prime }$ to estimate, to two decimal places, (a) the interval(s) where f is decreasing.(b) the interval(s) where f is concave upward.(c) the critical numbers of f .

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

Let $f ( x ) = x ^ { 4 } - 4 x ^ { 3 }$ .(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)

Use the graph of f to estimate the value of c that satisfy the Mean Value Theorem for the interval $[ 0,8 ]$ .

Find the value of the limit $\lim _ { x \rightarrow 0 } \left( 1 + x ^ { 2 } \right) ^ { \frac { 1 } { x } }$ . Use L'Hospital's Rule, if applicable.

The graph of f is given below. (a) Where is f increasing? (b) Where is f decreasing? (c) Where is f concave upward? (d) Where is f concave downward?

Amy is 2 miles offshore in a boat and wishes to reach a coastal village 6 miles down a straight shoreline from the point nearest to the boat. She can row 2 miles per hour and can walk 5 miles per hour. Where should she land her boat to reach the village in the least amount of time? How long will it take Amy to reach the village?

Find the x-coordinate of the point of inflection of the function $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 1$ .

Find the distance between the two critical numbers of the function $f ( x ) = x ^ { 3 } - 3 x + 27$ .

A particle moves along the curve $y = \frac { 1 } { x }$ . If x increases at a rate 8 units per minute, at what rate is y changing when $x = 2$ ?

A drinking cup is made in the shape of a right circular cylinder. For a fixed volume, we wish to make the total material used (the circular bottom and the cylindrical side) as small as possible. Under this condition, what is the ratio of the height to the diameter?

Find all critical numbers for the function $f ( x ) = \left| x ^ { 2 } - 9 \right|$ .

The graph of the derivative of $f ^ { \prime } ( x )$ of a function f(x) is given below. Sketch a possible graph of $y = f ( x )$ on the interval $[ - 2,2 ]$ such that $f ( - 2 ) = 0$ . On your graph, label the critical point(s) and inflection point(s).

Find the value of the limit $\lim _ { x \rightarrow \infty } \frac { \frac { \pi } { 2 } - \arctan x } { \ln \left( \frac { 2 x - 1 } { 1 + 2 x } \right) }$ .

Sketch the graph of $f ( x ) = 2 x ^ { 3 } + x ^ { 2 } + x - 1$ on $[ - 3,3 ]$ . For each initial approximation given below, determine graphically what happens if Newton's Method is used to approximate the roots of $f ( x ) = 0$ .(a) $x _ { 1 } = - 2$ (b) $x _ { 1 } = - 1$ (c) $x _ { 1 } = 0$ (d) $x _ { 1 } = 1$ (e) $x _ { 1 } = - 1.56$

Find the maximum and minimum values of $f ( x ) = x \sqrt { 2 - x }$ and sketch its graph.

What is the length of the largest interval on which the function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ is increasing?