Exam 12: Limits and An Introduction To Calculus

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = \frac { 1 } { x + 8 } , \quad \left( 0 , \frac { 1 } { 8 } \right)$

Use the position function $s ( t ) = 16 t ^ { 2 } - 102$ to find the velocity in feet/second at time $t = 2$ seconds.The velocity at time $t = c$ seconds is given by $\lim \frac { [ s ( c ) - s ( t ) ] } { ( c - t )_{c?t} }$ .

Find $\lim _ { n \rightarrow \infty }S ( n )$ . $\sum _ { i = 1 } ^ { n } \frac { i ^ { 3 } } { n ^ { 4 } }$

Graph the function.Determine the limit (if it exists)by evaluating the corresponding one-sided limits. $\lim _ { x \rightarrow 5 } \frac { 1 } { x ^ { 2 } + 5 }$

Select the correct graph for the following function using a graphing utility. $f ( x ) = \frac { 5 x } { \tan 10 x }$

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

Evaluate $\sum _ { j = 1 } ^ { 25 } \left( j ^ { 3 } - 3 j ^ { 2 } \right)$ using the summation formulas and properties.

Select correct graph of the following function. $y = 2 - \frac { 8 } { x ^ { 2 } }$

Select the correct function for the graph using oblique asymptotes as aids.

Determine $\lim _ { x \rightarrow 6 } \frac { | x - 6 | } { x ^ { 2 } - 36 }$ (if it exists)by evaluating the corresponding one-sided limits.

Find the first five terms of the sequence. $\alpha _ { n } = \frac { n } { 2 n + 2 }$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the specified interval. Function Interval f(x)=8x- [0,1]

Approximate the area of the indicated region under the given curve using five rectangles. $f ( x ) = 3 - x ^ { 2 }$

Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 6 + y } - \sqrt { 6 } } { y }$ .

Determine $\lim _ { x \rightarrow 10 } \frac { | x - 10 | } { x ^ { 2 } - 100 }$ (if it exists)by evaluating the corresponding one-sided limits.

Find a formula for the slope of the graph of $f$ at the point $( x , f ( x ) )$ .Then use it to find the slope at the given point. $f ( x ) = x ^ { 3 } , \quad ( 2,8 )$

Find the limit by direct substitution. $\lim _ { x \rightarrow - 2 } \left( \frac { 11 x - 9 } { 3 x + 9 } \right)$

Use the limit process to find the area of the region between $f ( x ) = \frac { 1 } { 8 } \left( x ^ { 2 } + 8 x \right)$ and the x-axis on the interval [1,8].Round your answer upto 1 decimal places.

Find $\lim _ { x \rightarrow 9 } \frac { x - 7 } { x ^ { 2 } + 11 x + 18 }$ by direct substitution.

Use the slope $m = 24$ to find an equation of the tangent line to the graph at the given point. $f ( x ) = x ^ { 2 } - 1$