Exam 11: Calculus Practice Problems

The cost function for a certain model of a digital camera given by $C = 13.50 x + 48,950$ , where $C$ is the cost (in dollars) and $x$ is the number of cameras produced. Find the average cost per unit when $x = 200$ . Round your answer to the nearest cent.

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } , \quad f ^ { \prime } ( x ) = - 8 x ^ { 3 } + 8 x$

A union contract guarantees a $15 \%$ salary increase yearly for 3 years. For a current salary of $\$ 31,000$ , the salary $f ( t )$ (in thousands of dollars) for the next 3 years is given by $f ( t ) = \left\{ \begin{array} { l l } 31.00 , & 0 < t \leq 1 \\35.65 , & 1 < t \leq 2 \\41.00 , & 2 < t \leq 3\end{array} \right.$ where $t$ represents the time in years. Find the limit of $f$ as $t \rightarrow 1.00$ , if it exists.

Find $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ for $f(x)=-4+7 x$ .

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 3 x } { 1 + 7 x }$ (if it exists).

Use the graph to find $\lim _ { x \rightarrow - 1 } \frac { 1 } { x + 1 }$ , if it exists.

Find the slope of the graph of the following function at the point $( 1 , - 3 )$ . $- x ^ { 2 } - 2$

Consider the graph of the function and approximate $\lim _ { x \rightarrow 0 } \frac { \sin ( - 8 x ) } { x }$ , if it exists.

Find the following limit. Round your answer to two decimals. $\lim _ { x \rightarrow 3 } \frac { \sqrt { x ^ { 2 } + 7 } } { 2 x ^ { 2 } }$

Find the following limit, if it exists. $\lim _ { x \rightarrow - 3 } x ^ { 8 }$

Graph $f ( x ) = \left\{ \begin{array} { l c } 3 x + 2 & x < - 3 \\ x - 4 & x \geq - 3 \end{array} \right.$ and find the limit of $f ( x )$ as $x$ approaches $- 3$

Find $\lim _ { x \rightarrow - \infty } \left[ \frac { x } { ( 5 + x ) ^ { 2 } } + 8 \right]$ (if it exists).

Find the following limit, if it exists. $\lim _ { x \rightarrow 3 } x ^ { 7 }$

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f(x)=3 x^{4}-6 x^{2}, f^{\prime}(x)=12 x^{3}-12 x$

Use the figure below to approximate the slope of the curve at the point $( x , y )$ .

Complete the table and numerically estimate the limit as $x$ approaches infinity for $f ( x ) = x - \sqrt { x ^ { 2 } + 4 }$ . x 1 1 1 1 1 1 1 f(x)

Use the function below and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. $f ( x ) = - 3 x ^ { 4 } + 6 x ^ { 2 } , f ^ { \prime } ( x ) = - 12 x ^ { 3 } + 12 x$

Find $\lim _{y \rightarrow 0} \frac{\sqrt{15+y}-\sqrt{15}}{y}$ .

Use the limit process to find the slope of the graph of $4 x-8 x^{2}$ at $(3,-60)$ .

Use the graph to find $\lim _{x \rightarrow 0} 3 \sin \left(\frac{\pi}{3 x}\right)$