Exam 12: Review of Graphs, Equations, and Inequalities

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 $0 .$

Complete the table and use the result to estimate $\lim _ { x \rightarrow 5 } ( 2 x + 4 )$ numerically. x 4.9 4.99 4.999 5 5.001 5.01 5.1 f(x) ?

Find $\lim _ { x \rightarrow - 3 } \left( \frac { 1 } { 3 } x ^ { 3 } - 2 x \right)$ by direct substitution.

Use the limit process to find the slope of the graph of the following function at the point $( 3 , - 19 )$ . $g ( x ) = - 10 - 3 x$

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 the value of $y$ that corresponds to $x = - 2$ in the graph of the equation $2 x + 3 y = 20$ .

Solve the following equation (if possible) : $\frac { 1 } { x - 7 } + \frac { 1 } { x + 7 } = \frac { 12 } { x ^ { 2 } - 49 }$

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

Rewrite $\sum _ { i = 1 } ^ { n } \left[ \frac { 12 } { n } + \left( \frac { 11 i } { n ^ { 2 } } \right) \right] \left( \frac { 8 i } { n } \right)$ as a rational function $S ( n )$ and find $\lim _ { n \rightarrow \infty } S ( n )$ .

Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 8 n ^ { 3 } + 6 } { n + 3 }$ (assume $n$ begins with 1).

Evaluate $\sum _ { j = 1 } ^ { 15 } \left( j ^ { 3 } - 5 j ^ { 2 } \right)$ using the summation formulas and properties.

Find $\lim _ { x \rightarrow \infty } \frac { 1 - 7 x } { 1 + 9 x }$ (if it exists).

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

The point $A$ has coordinates $( - 2,7 )$ . If $A$ is moved 2 units upward 6 units to the left, what are the new coordinates of $A$ ?

$f ( x ) = \frac { 4 x ^ { 2 } } { x ^ { 2 } + 1 }$ Use asymptotes to match

Find an equation of the tangent line to the graph of the following function at the point $( - 1 , - 5 )$

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }$ (if it exists).

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

Complete the table and use the result to estimate $\lim _ { x \rightarrow - 2 } \frac { x + 2 } { x ^ { 2 } - x - 6 }$ numerically. x -2.1 -2.01 -2.001 -2 -1.999 -1.99 -1.9 f(x) ?

Use the limit process to find the slope of the graph of $\sqrt { x + 21 }$ at $( 4,5 )$ .