Exam 2: A Preview of Calculus

Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the Problem seems to require calculus, use a graphical or numerical approach to estimate the Solution. A cyclist is riding on a path whose elevation is modeled by the function $f ( x ) = 0.08 \left( 16 x - x ^ { 2 } \right)$ where $x$ and $f ( x )$ are measured in miles. Find the rate of change of elevation when $x = 4$ .

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 3 } \frac { x - 3 } { x ^ { 2 } - 16 x + 39 }$ x 2.9 2.99 2.999 3.001 3.01 3.1 f(x)

Find the limit. $\lim _ { x \rightarrow 0 ^ { - } } \left( x ^ { 2 } - \frac { 1 } { x } \right)$

$\text { Consider the function } f ( x ) = 11 x - x ^ { 2 } \text { and the point } p ( 4,28 ) \text { on the graph of }$ $f$ . Find the slope of the secant line passing through $P ( 4,28 )$ and $Q ( x , f ( x ) )$ for $x = 5$ . Round your answer to one decimal place.

$\text { A petrol car is parked } 65 \text { feet from a long warehouse (see figure). The revolving }$ light on top of the car turns at a rate of $\frac { 1 } { 2 }$ revolution per second. The rate at which the light beam moves along the wall is $r = 65 \pi \sec ^ { 2 } \theta \mathrm { ft } / \mathrm { sec }$ . Find the limit of $r$ as $\theta \rightarrow ( \pi / 2 ) ^ { - }$ .

Find the limit (if it exists). $\lim _ { x \rightarrow - 8 } \frac { x + 8 } { x ^ { 2 } - 64 }$

$\text { Find the limit (if it exists). Note that } f ( x ) = ( | x | ) \text { represents the greatest integer }$ function. $\lim _ { x \rightarrow 5 ^ { + } } ( 2 x - [ | x | ] )$

Determine the following limit. (Hint: Use the graph to calculate the limit.) $\lim _ { x \rightarrow 2 } \frac { 1 } { x - 2 }$

$\text { Let } f ( x ) = 4 x - 2 \text { and } g ( x ) = x ^ { 3 } \text {. Find the limit. }$ $\lim _ { x \rightarrow 1 } g ( f ( x ) )$

$\text { Find all the vertical asymptotes (if any) of the graph of the function } f ( x ) = \frac { x ^ { 3 } + 8 } { x + 2 } \text {. }$

Complete the table and use the result to estimate the limit. $\lim _ { x \rightarrow 7 } \frac { \frac { 1 } { x - 3 } - \frac { 1 } { 4 } } { x - 7 }$ x 6.9 6.99 6.999 7.001 7.01 7.1 f(x)

$\text { Use a graphing utility to graph the function } f ( x ) = \frac { x ^ { 2 } - 2 x + 4 } { x ^ { 3 } + 8 } \text { and determine the }$ one-sided limit $\lim _ { x \rightarrow - 2 ^ { + } } f ( x )$ .

$\text { Find the vertical asymptotes (if any) of the function } f ( x ) = \tan ( 15 x ) \text {. }$

$\text { Find the } x \text {-values (if any) at which } f ( x ) = \frac { | x - 3 | } { x - 3 } \text { is not continuous. }$

Find the limit (if it exists). $\lim _ { x \rightarrow 11 ^ { + } } \frac { 11 - x } { x ^ { 2 } - 121 }$

Determine the following limit. (Hint: Use the graph to calculate the limit.) $\lim _ { x \rightarrow 1 } ( 5 - x )$

Find the lmit. $\lim _ { x \rightarrow \pi } \tan \left( \frac { x } { 3 } \right)$

$\text { Suppose that } \lim _ { x \rightarrow c } f ( x ) = 5 \text {. Find the following limit. }$ $\lim _ { x \rightarrow c } \left[ f ( x ) ^ { 3 } \right]$

Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If theProblem seems to require calculus, use a graphical or numerical approach to estimate the Solution. Find the area of the shaded region.

$\text { Suppose that } \lim _ { x \rightarrow c } f ( x ) = - 5 \text {. Find the following limit. }$ $\lim _ { x \rightarrow c } 3 f ( x )$