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Exam 13: Limits: a Preview of Calculus

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Question 31

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Find the slope of the tangent line of the graph of $f ( x ) = \frac { x ^ { 3 } } { 2 }$ at the point $\left( 1 , \frac { 1 } { 2 } \right)$ .

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Question 32

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Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 4 x ^ { 3 } + 1$ over the interval $0 \leq x \leq 6$ .

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Question 33

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A particle moves in a straight line with its displacement of motion described by equation $s ( t ) = t ^ { 2 } - 3 t + 5$ , where $s$ is measured in feet and $t$ is measured in seconds. Find the velocity of $s$ at $t = a$ , $t = 2$ , $t = 4$ , and $t = 6$ .

Use a table of values to estimate the value of $\lim _ { x \rightarrow 0 } \frac { 4 ^ { x } + 2 ^ { x } - 2 } { x }$ . Use a graphing device to confirm your answer.

For the function g whose graph is given, state the value of the given quantity if it exists. a) $\lim _ { x \rightarrow 0 } g ( t )$ b) $g ( 0 )$

Use a table of values to estimate the value of $\lim _ { x \rightarrow - ] } \frac { 81 - x ^ { 4 } } { 3 + x }$ . Use a graphing device to confirm your answer graphically.

A stone is dropped into a pond causing a circular ripple in the water. Find the rate of change of the area $A$ of the circle with respect to the radius when $r= 3.5$ ft.

Determine whether the sequence $\alpha _ { n } = \frac { 1 } { e ^ { n } }$ converges or diverges. If it converges, find the limit.

Find the equation of the tangent line to the curve $y =$ $x - 2 x ^ { 2 }$ at the point $( 2 , - 6 )$ .

Use the definition of area as a limit to find the area of the region that lies under the graph of $f ( x ) = 3 x + 1$ over the interval $0 \leq x \leq 4$ .

If $f ( x ) = \frac { 2 } { \sqrt { x } }$ , find $f ^ { \prime } ( a )$ .

Find the value of $\lim _ { x \rightarrow 1 ^ { + } } \frac { | 1 - x | } { x - x ^ { 2 } }$ , if it exists. If the limit does not exist, explain why.

Find the slope of the tangent line of the graph of $f ( x ) = \frac { x ^ { 2 } } { 2 } + 1$ at the point $( 4,9 )$ .

For the function $f$ whose graph is given, state the value of the given quantity. A) $\lim _ { x \rightarrow - 2 ^ { - } } f ( x )$ B) $\lim _ { x \rightarrow - 2 ^ { + } } f ( x )$ C) $\lim _ { x \rightarrow - 2 } f ( x )$ D) $f ( - 2 )$

Find $\lim _ { x \rightarrow \infty } \frac { 5 - 4 x } { 4 x - 5 }$ .

Determine whether the sequence $a _ { n } = \cos ( n ( 1 + 2 \pi ) )$ converges or diverges. If it converges, find the limit.

Evaluate $\lim _ { x \rightarrow - 2 } ( x + 1 ) ^ { 5 } \sqrt { 2 x ^ { 4 } + x ^ { 2 } }$ , and justify each step by indicating the appropriate Limit Law(s).

Determine whether the sequence $a _ { n } = \frac { ( - 1 ) ^ { n.1 } n } { n + 1 }$ converges or diverges. If it converges, find the limit.

If $f ( x ) = \frac { x + 1 } { x }$ , find $f ^ { \prime } ( a )$ .

Use a table of values to estimate the value of $\lim _ { x \rightarrow \infty } ( \sqrt { x + 1 } - \sqrt { x - 1 } )$ . Then use a graphing device to confirm your result graphically.