Exam 3: Differentiation Rules

A side of a square field is measured to be 144 feet with a possible error of 1 inch.(a) Use differentials to estimate the maximum error in the calculated area of the field.(b) What is the relative error?

If $f ( x ) = \frac { x } { \sin x }$ , find $f ^ { \prime } \left( \frac { \pi } { 4 } \right)$ .

If $f ( x ) = \left( \frac { x + 1 } { x - 1 } \right) ^ { 3 } , \text { find } f ^ { \prime } ( 3 )$

Find $\frac { d y } { d x } .$ (a) $y = \left( \tan \frac { \pi } { 4 } - 3 \right) ^ { 4 }$ (b) $y = \tan x ^ { 3 }$ (c) $y = \tan ^ { 3 } x$ (d) $y = \left( 1 + e ^ { \sin x } \right) ^ { 3 }$

Let $f ( x ) = \sqrt { x } \ln x$ . Find the interval on which $f$ is concave upward.

Each of the following is a derivative of a function obtained by using the Product Rule. Determine the original function: (a) $f ^ { \prime } ( x ) = 2 x e ^ { x } + x ^ { 2 } e ^ { x }$ (b) $g ^ { \prime } ( x ) = - 6 x \left( 5 x ^ { 3 } - 2 x \right) + \left( 1 - 3 x ^ { 2 } \right) \left( 15 x ^ { 2 } - 2 \right)$

Let $f ( x ) = x ^ { 2 / 3 }.$ (a) Find a linear approximation of f at x = 8: (b) Use this linear approximation to estimate the value of the function at 7, 9, 7.99, and 8.01.(c) Make a table comparing your estimates with the actual function values. Discuss what this tells you about the linear approximation.(d) Graph the original function and its linear approximation over the interval [7; 9]. What does the graph tell you about the size of the difference between the function values and the linear approximation values?

If $f ( x ) = \frac { e ^ { x } } { x } , \text { find } f ^ { \prime \prime } ( x )$