Exam 1: The Derivative

Which of the following properties are satisfied by the following function: f(x) = $\left\{ \begin{array} { l l } x ^ { 2 } + 1 & \text { for } x < 0 \\1 & \text { for } x = 0 \\5 x + 1 & \text { for } x > 0\end{array} \right.$ (I) f(x) is continuous (II) f(x) is differentiable for all x (III) f(x) is differentiable at x = -2

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow 5 }$ $\frac { x ^ { 2 } - 8 x + 15 } { x ^ { 2 } - 7 x + 10 }$ Enter your answer as just a fraction in lowest terms or an integer, or the words "does not exist".

Find the slope of the tangent line to the graph of y = $x ^ { 2 }$ at the point $\left( \frac { 7 } { 6 } , \frac { 49 } { 36 } \right)$ and write the equation of the tangent line. Enter your answer exactly as just the equation in standard point-slope form. Use all fractions of form $\frac { a } { b }$ in lowest terms.

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

Let g(x) = 2 $x ^ { 3 }$ - $5 x ^ { 2 }$ + 1; find g'(3). Enter just an integer.

Find the second derivative of f(x) = 3 $x ^ { 4 }$ - 7 $x ^ { 2 }$ + 6.

Find the equation of the following line: $\left( \frac { 3 } { 2 } , - 4 \right.$ and $\left( - \frac { 3 } { 2 } , - 4 \right)$ on the line (in the xy -plane). Enter your equation in the simplest possible form.

Find the derivative of f(x) = 7x - 19 at x = 2.

Differentiate. -f(x) = 2 $x ^ { 4 }$ + 5 $x ^ { 3 }$ + 6

Use the graph of f to determine if f is continuous at x = $\frac { 3 } { 2 }$ .

Find the second derivative of y = π + $\sqrt { 3 }$ . Do not label your answer.

Find the derivative of f(x) = $x ^ { 5 / 2 }$ .

Find the equation of the following line: Parallel to y = - 9x + 4; (2, - 2) on line.

Compute $\frac { d ^ { 2 } y } { d y ^ { 2 } } \left( \frac { 9 t ^ { 2 / 3 } } { 2 } \right)$ $\mathrm { t } = 8$ . Enter just a reduced fraction of form $\frac { a } { b }$ .

A rock is thrown off a cliff. Its distance from the ground below at t seconds is $s ( t ) = - 16 t ^ { 2 } + 16 t + 96$ feet. How high was the cliff? Enter just an integer, no units.

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow - 1 }$ $\frac { x ^ { 2 } + x - 2 } { x ^ { 3 } + 1 }$ Enter either a fraction, integer, or "does not exist".

Find the second derivative of f(s) = $\frac { 4 } { 3 }$ π $s ^ { 3 }$ at s = $\frac { 1 } { \pi }$ . Enter just a real number.

Write the word or phrase that best completes each statement or answers the question. -F(x) = $\sqrt { 3 x + 1 }$ Enter your answer as just: a $\mathrm { P } ( \mathrm { x } ) ^ { \mathrm { b } }$ where P(x) is a polynomial in standard form.

For the following function f(x), determine whether or not $\lim _ { x \rightarrow 3 }$ f(x) exists. If so, give the limit. Enter either a real number or enter the words "does not exist".

Suppose that t hours after being placed in a freezer, the temperature of a piece of meat is given by $f ( t ) = 70 - 12 t + \frac { 4 } { t + 1 }$ What is the temperature of the meat after 3 hours? Enter just an integer, no units.