Exam 11: Limits and an Introduction to Calculus

Use the graph below to find $\lim _ { x \rightarrow - 10 } \frac { | x + 10 | } { x + 10 }$ , if it exists.

Find the derivative of $f ( x ) = 4 x ^ { 2 } + 2 x + 3$ .

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

Evaluate $\sum _ { j = 1 } ^ { 10 } \left( j ^ { 3 } - 4 j ^ { 2 } \right)$ using the summation formulas and properties.

Use the limit process to find the slope of the graph of $\sqrt { x + 12 }$ at $( 4,4 )$ .

Find $\lim _ { x \rightarrow 6 } \frac { 4 x } { \sqrt { x + 4 } }$ by direct substitution.

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 4 - 6 x$ .

Find $\lim _ { x \rightarrow 3 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 3 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 2 } } { 5 x ^ { 2 } }$ .

Find the following limit of the sequence as $n$ approaches infinity, if it exists. $a _ { n } = \frac { 1 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 6 } \right] \right)$

The cost function for a certain graphing calculator is given by $C = 11.35 x + 53,000$ where $C$ is in dollars and $x$ is the number of calculators produced. a. Write a model for the average cost per unit produced. b. Find the average cost per unit when $x = 900$ . c. Determine the limit of the average cost function as $x$ approaches $\infty$ .

Find $\lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x ) } { h }$ for $f ( x ) = 2 + 6 x$ .

Find $\lim _ { x \rightarrow 7 } \frac { \sqrt { x + 3 } } { x - 3 }$ by direct substitution.

Find an equation of the tangent line to the graph of the following function at the point $( 3 , - 30 )$ . $- 3 x ^ { 2 } - 3$

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 6 x } { 1 + 8 x }$ (if it exists).

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ .

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

Graph $f ( x ) = \left\{ \begin{array} { l c } 4 x + 3 & x < 0 \\ x + 3 & x \geq 0 \end{array} \right.$ and find the limit of $f ( x )$ as $x$ approaches 0 .

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

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

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { \sqrt { x + 15 } - \sqrt { 15 } }$