Exam 5: Applications of Derivatives

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. - $y = \frac { x ^ { 2 } } { x ^ { 2 } + 13 }$

Use l'Hopital's Rule to evaluate the limit. - $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } - 7 x + 10 } { x - 2 }$

Solve the problem. -The graph below shows the position $s = f ( t )$ of a body moving back and forth on a coordinate line. (a) When is the body moving away from the origin? Toward the origin? At approximately what times is the (b) velocity equal to zero? (c) Acceleration equal to zero? (d) When is the acceleration positive? Negative? 8.5

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. - $y=x+\cos 2 x, 0 \leq x \leq \pi$

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. - $y = x \sqrt { 7 - x ^ { 2 } }$

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule. Show each step of your calculation. - $\lim _{ x \rightarrow \infty } \sqrt { x } ^ { 1 / x }$

Find the most general antiderivative. - $\int \left( 7 e ^ { 2 x } - 6 e ^ { - x } \right) d x$

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

Sketch the graph and show all local extrema and inflection points. - $= x \left( \frac { x } { 2 } - 5 \right) ^ { 4 }$

Use differentiation to determine whether the integral formula is correct. - $\int ( 5 x + 7 ) ^ { 4 } d x = \frac { ( 5 x + 7 ) ^ { 5 } } { 25 } + C$

Solve the initial value problem. - $\frac { d r } { d \theta } = - \frac { \pi } { 2 } \cos \frac { \pi } { 2 } \theta , r ( 0 ) = - 10$

Find the absolute extreme values of the function on the interval. - $F ( x ) = - \frac { 2 } { x ^ { 2 } } , 0.5 \leq x \leq 5$

Find the function with the given derivative whose graph passes through the point P. - $g ^ { \prime } ( x ) = \frac { 1 } { x ^ { 2 } } + 2 x , P ( - 2,2 )$

Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing. - $f ^ { \prime } ( x ) = ( 1 - x ) ( 9 - x )$

L'Hopital's rule does not help with the given limit. Find the limit some other way. - $\lim _ { x \rightarrow \theta ^ { + } } \frac { 1 } { \cot x \sin x }$

Find the derivative at each critical point and determine the local extreme values. - $y = \left\{ \begin{array} { l } 7 - 2 x , x \leq 1 \\x + 4 , x > 1\end{array} \right.$

Solve the problem. -A small frictionless cart, attached to the wall by a spring, is pulled $10 \mathrm {~cm}$ back from its rest position and released at time $t = 0$ to roll back and forth for $4 \mathrm { sec }$ . Its position at time $t$ is $\mathrm { s } = 1 - 10 \cos \pi \mathrm { t }$ . What is the cart's maximum speed? When is the cart moving that fast? What is the magnitude of of the acceleration then?

Estimate the limit by graphing the function for an appropriate domain. Confirm your estimate by using L'Hopital's rule. Show each step of your calculation. - $\lim _ { x \rightarrow 0 } \frac { 1 } { \sin x } - \frac { 1 } { \sqrt { x } }$

Graph the rational function. - =

Solve the initial value problem. - $\frac { \mathrm { dr } } { \mathrm { dt } } = 6 \mathrm { t } + \sec ^ { 2 } \mathrm { t } , \mathrm { r } ( - \pi ) = 5$