Exam 1: Functions and Models

Suppose the distance $s$ (in feet) covered by a car moving along a straight road after $t$ sec is given by the function $s = f ( t ) = 3 t ^ { 2 } + 13 t$ . Calculate the (instantaneous) velocity of the car when $t = 35$ . Select the correct answer.

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $y = 4 + 2 x - x ^ { 2 }$ Select the correct answer.

Find the function $f \cdot g$ and its domain if $f ( x ) = \sqrt { x + 7 }$ and $g ( x ) = \sqrt { x - 7 }$ .

The graph of the function $f ( x ) = x ^ { 2 } - 11 x + 7$ has been stretched horizontally by a factor of 2 . Find the function for the transformed graph.

Use the graph of $f ( x ) = \frac { x ^ { 2 } + x - 2 } { x + 2 }$ to guess at the limit $\lim _ { x \rightarrow - 2 } \frac { x ^ { 2 } + x - 2 } { x + 2 }$ , if it exists.

Find the value of the limit. $\lim _ { x \rightarrow 0 } 3 \frac { \tan 4 x - 4 x } { x ^ { 3 } }$

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

Scientists have discovered that a linear relationship exists between the amount of flobberworm mucus secretions and the air temperature. When the temperature is $65 ^ { \circ } \mathrm { F }$ , the flobberworms each secrete 16 grams of mucus a day; when the temperature is $95 ^ { \circ } \mathrm { F }$ , they each secrete 22 grams of mucus a day. Find a function $M ( t )$ that gives the amount of mucus secreted on a given day, where $t$ is the temperature of that day in degrees Fahrenheit.

Use the graph of the function to state the value of $\lim _ { x \rightarrow 0 } f ( x )$ , if it exists. $f ( x ) = \frac { x ^ { 2 } + x } { 2 \sqrt { x ^ { 3 } + x ^ { 2 } } }$

The curve with the equation $x ^ { 2 / 3 } + y ^ { 2 / 3 } = 25$ is called an asteroid. Find an equation of the tangent to the curve at the point $( 48 \sqrt { 6 } , 1 )$ .

Suppose that the graph of is given $f$ is given. Describe how the graph of the function $y = f ( x - 5 ) - 5$ can be obtained from the graph of $f$ .

If $f ( x ) = x ^ { 2 } - x + 6$ , evaluate the difference quotient $\frac { f ( a + h ) - f ( a ) } { h }$ .

By graphing the function $f ( x ) = \frac { ( \cos x - \cos 5 x ) } { x ^ { 2 } }$ and zooming in toward the point where the graph crosses the $y$ -axis, estimate the value of $\lim _ { x \rightarrow 0 } f ( x )$ .

Evaluate the limit. $\lim _ { x \rightarrow 3 } \left( \frac { x ^ { 3 } - 5 } { x ^ { 2 } - 6 } \right)$

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $y = 4 + 2 x - x ^ { 2 }$

Find the limit. $\lim _ { \theta \rightarrow 0 } 4 \frac { \sin ( \sin 4 \theta ) } { \sec 4 \theta }$

The monthly cost of driving a car depends on the number of miles driven. Julia found that in October it cost her $\$ 200$ to drive $300 \mathrm { mi }$ and in July it cost her $\$ 350$ to drive $600 \mathrm { mi }$ . Express the monthly cost $C$ as a function of the distance driven $d$ assuming that a linear relationship gives a suitable model.

Find a function $g$ that agrees with $f$ for $x \neq 25$ and is continuous on $\mathfrak { R }$ . $f ( x ) = \frac { 5 - \sqrt { x } } { 25 - x }$