Exam 1: The Derivative

Find the slope of the line with the equation 2x + 4y = 5. Enter just a fraction of form $\frac { a } { b }$ in lowest terms.

Find the second derivative of $f ( x ) = \frac { 3 } { 2 x - 4 } \text { for } x \neq 0$

Find the derivative of f(x) = 2 $\sqrt { x }$ at x = 4.

-The average value of a certain type of automobile was $15,780 in 1994 and depreciated to $5400 in 1999. Let y be the average value of the automobile in the year x, where x = 0 represents 1994. Write a linear equation in slope-intercept form that models the value of the automobile, y, x years after 1994.

Let V = $\frac { 4 } { 3 }$ ? $r ^ { 3 }$ . Compute is $\frac { \mathrm { dV } } { \mathrm { dr } }$ $\mathrm { r } = 2$ .

Write the word or phrase that best completes each statement or answers the question. -y = $\frac { 5 } { x ^ { 3 } }$ Enter your answer in the form: $a x ^ { b }$ . No parentheses

If the revenue function is $R ( x ) = x ^ { 2 / 3 } + 2 x ^ { 1 / 2 } + 6 x + 2$ for producing a product, find the marginal revenue at a production level of 10 units.

For the following function g(x), determine whether or not $\lim _ { x \rightarrow - 1 }$ g(x) exists. If so, give the limit.

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow \infty }$ $\frac { 3 } { x ^ { 2 } + 1 }$ Enter just an integer or a fraction.

Find the slope of the curve y = $x ^ { 6 }$ at x = -3.

Find the slope of the tangent line to the graph of y = $x ^ { 2 }$ at the point (0.9, 0.81) and write the equation of the tangent line. Enter your answer exactly as just the equation in standard point-slope form.

In the graph of y = f (x), for which values of x is f(x) not continuous? Enter your answer exactly in the form: x = a where a is an integer. (If there is more than one value of x then enter: x = a, b, c)

Find the derivative of $f ( x ) = \frac { 1 } { \sqrt [ 3 ] { x } }$

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow 0 }$ $\frac { g ( x ) } { f ( x ) }$ where $\lim _ { x \rightarrow 0 } f ( x ) = - \frac { 1 } { 3 } \text { and } \lim _ { x \rightarrow 0 } g ( x ) = \frac { 2 } { 3 }$ Enter your answer as an integer, fraction in lowest terms, or the words " does not exist"

For the graphed function below, state the x-values for which the derivative does not exist.

Find all points on the graph of y = $x ^ { 3 }$ where the curve has slope 12. The slope of the tangent line to the graph is 3 $x ^ { 2 }$ . Enter your answer exactly in the form: (a, b), (c, d) where a > c.

A winter storm front moves through campus. At t hours after the onset of the storm, the temperature is given by $T ( t ) = 35 - 2 t ^ { 2 } + t$ . At what instantaneous rate is the water flowing when t = 4? Enter your answer as just a real number to one decimal place, no units.

Write the word or phrase that best completes each statement or answers the question. -y = $\frac { 4 } { 4 x + 1 }$ Enter your answer exactly as just: $\frac { \mathrm { P } ( \mathrm { x } ) } { ( \mathrm { Q } ( \mathrm { x } ) ) ^ { \mathrm { a } } }$ where P(x) and Q(x) are polynomials in standard form.

Which of the following is the equation of the line tangent to the curve $y = \sqrt { x }$ at $x = 9$ ? (I) y = -x - $\frac { 1 } { 6 }$ (II) y = $\frac { 1 } { 2 }$ x + 3 (III) y = $\frac { 1 } { 6 }$ x + $\frac { 3 } { 2 }$ (IV) y = $\frac { 1 } { 2 }$ $x ^ { - 1 / 2 }$ - $\frac { 1 } { 6 }$

During the month of February, a flu epidemic hit the University. The number of people sick at time t (measured in days) is given by the function P(t). The rate at which the epidemic is spreading on February 3 is 110 people per day. How is the information best represented mathematically?