Exam 5: Applications of Derivatives

Provide an appropriate response. -Decide if the statement is true or false. If false, explain. The points $( - 1 , - 1 )$ and $( 1,1 )$ lie on the graph of $f ( x ) = \frac { 1 } { x }$ . Therefore, the Mean Value Theorem says that there exists some value $x = c$ on $( - 1,1 )$ for which $f ^ { \prime } ( x ) = \frac { 1 - ( - 1 ) } { 1 - ( - 1 ) } = 1$

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. -Find the error in the following incorrect application of L'Hôpital's Rule. $\lim _ { x \rightarrow 0 } \frac { \sin x } { x + x ^ { 2 } } = \lim _ { x \rightarrow 0 } \frac { \cos x } { 1 + 2 x } = \lim _ { x \rightarrow 0 } \frac { - \sin x } { 2 } = 0 \text {. }$

Find an antiderivative of the given function. - $\frac { 4 } { 7 } x ^ { 6 / 7 }$

Solve the initial value problem. - $\frac { d y } { d x } = \frac { 3 } { 1 + x ^ { 2 } } + \frac { 5 } { \sqrt { 1 - x ^ { 2 } } } , \quad y ( 0 ) = 6$

Solve the problem. -For $x > 0$ , sketch a curve $y = f ( x )$ that has $f ( 1 ) = 0$ and $f ^ { \prime } ( x ) = - \frac { 1 } { x }$ . Can anything be said about the concavity of such a curve? Give reasons for your answer.

Use l'H^opital's rule to find the limit. - $\lim _ { x \rightarrow \infty } \frac { 7 x ^ { 2 } - 5 x + 1 } { 3 x ^ { 2 } + 3 x - 8 }$

Answer the problem. -Use the following function and a graphing calculator to answer the questions. $f ( x ) = \sqrt { 3 x } + 0.9 \sin x , [ 0,2 \pi ]$ a). Plot the function over the interval to see its general behavior there. Sketch the graph below. b). Find the interior points where f = 0 (you may need to use the numerical equation solver to approximate a solution). You may wish to plot f as well. List the points as ordered pairs (x, y). c). Find the interior points where f does not exist. List the points as ordered pairs (x, y). d). Evaluate the function at the endpoints and list these points as ordered pairs (x, y). e). Find the function's absolute extreme values on the interval and identify where they occur.

Solve the problem. -A particle moves on a coordinate line with acceleration $a = d ^ { 2 } s / d t ^ { 2 } = ( 5 / \sqrt { t } ) + 9 \sqrt { t }$ , subject to the conditions that $\mathrm { ds } / \mathrm { dt } = 3$ and $\mathrm { s } = 1$ when $\mathrm { t } = 1$ . Find the velocity $\mathrm { v } = \mathrm { ds } / \mathrm { dt }$ in terms of $\mathrm { t }$ and the position $s$ in terms of $t$ .

Answer each question appropriately. -Suppose the velocity of a body moving along the s-axis is $\frac { \mathrm { ds } } { \mathrm { dt } } = 9.8 \mathrm { t } - 3$ . Find the body's displacement over the time interval from $t = 3$ to $t = 7$ given that $s = 3$ when $t = 0$ .

Solve the problem. -A rocket lifts off the surface of Earth with a constant acceleration of $30 \mathrm {~m} / \mathrm { sec } ^ { 2 }$ . How fast will the rocket be going $2.5$ minutes later?

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. - $f ( \theta ) = \left\{ \begin{array} { c l } \frac { \cos \theta } { \theta } , & - \pi \leq \theta < 0 \\0 , & \theta = 0\end{array} \right.$

Use l'Hopital's Rule to evaluate the limit. - $\lim _ { x \rightarrow \frac { \pi } { 3 } } \frac { \cos x - \frac { 1 } { 2 } } { x - \frac { \pi } { 3 } }$

Find the open intervals on which the function is increasing and decreasing. Identify the function's local and absolute extreme values, if any, saying where they occur. -

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. - $s ( t ) = \sqrt { t ( 3 - t ) } , \quad [ - 1,5 ]$

Find the absolute extreme values of the function on the interval. - $f ( \theta ) = \sin \left( \theta + \frac { \pi } { 2 } \right) , 0 \leq \theta \leq \frac { 7 \pi } { 4 }$

Graph the function, then find the extreme values of the function on the interval and indicate where they occur. -

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

Use l'Hopital's Rule to evaluate the limit. - $\lim _ { x \rightarrow \infty }\frac { x ^ { 2 } + 2 x + 14 } { x ^ { 3 } + 2 x ^ { 2 } + 9 }$

Find the limit. - $\lim _ { x \rightarrow \theta ^ { + } } x \tan \left( \frac { 3 \pi } { 2 } + x \right)$

Find all possible functions with the given derivative. - $y ^ { \prime } = \csc ^ { 2 } 3 \theta$