Exam 1: The Derivative

Use the graph of f to determine if f is continuous at x = -3.

Write the word or phrase that best completes each statement or answers the question. -y = $\frac { 1 } { \sqrt { 2 x + 1 } }$ Enter your answer as just: $\frac { \mathrm { P } ( \mathrm { x } ) } { ( \mathrm { Q } ( \mathrm { x } ) ) ^ { \mathrm { a } } }$ where P(x) and Q(x) are polynomials in standard form.

Using the graph below, find the given limit. - $\lim _ { x \rightarrow 0 }$ $[ \mathrm { f(x) }]^2$

Write the word or phrase that best completes each statement or answers the question. -y = $\frac { 3 } { 4 x ^ { 2 } + 1 }$ 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.

-R

Let y = (-4 + 3 $\sqrt { x }$ $) ^ { 4 }$ . Compute $\frac { d y } { d x }$ $x = 4$ .

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow 1 }$ $\frac { x ^ { 6 } - 1 } { x ^ { 3 } - 1 }$ Enter your answer as just an integer or a fraction in lowest terms, or the words "does not exist".

Find the second derivative of f(x) = 3 $x ^ { 4 }$ - 4 $x ^ { 3 }$ + 5x + 1. Enter just a polynomial in standard form, unlabeled.

-P

Find $\frac { \mathrm { d } } { \mathrm { ds } }$ ( $p ^ { 2 }$ $\mathrm { s } ^ { 4 }$ - $q ^ { 4 }$ $r ^ { 3 }$ $s ^ { 2 }$ )

Compute $\frac { \mathrm { d } ^ { 2 } } { \mathrm { dt } ^ { 2 } } ( 2 \sqrt { 2 \mathrm { t } } |$ $\mathrm { t } = 4$ .

If f(x) = 2 $x ^ { 2 }$ + 5x, find $\frac { f ( x + h ) - f ( x ) } { h }$ and simplify.

If f(t) = 5t - 4, find $\frac { f ( t + h ) - f ( t ) } { h }$ where h ≠ 0 and simplify. Enter your answer as just an integer.

Let f(x) = $\frac { x ^ { 3 } - 9 x } { 2 x + 6 }$ . Does $\lim _ { x \rightarrow - 3 }$ f(x) exist?

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow 0 }$ $\frac { ( x + 1 ) ^ { 1 / 2 } - 1 } { x }$ Enter your answer as just a fraction or an integer, or the words " does not exist"

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow 0 }$ $\frac { 1 } { x ^ { 3 } - 1 }$ + 1 Enter either a fraction in lowest terms, an integer, or "does not exist".

The tangent line to the curve y = $\frac { 1 } { 6 }$ $x ^ { 3 }$ - $\frac { 1 } { 4 }$ $x ^ { 2 }$ - x + 4 is perpendicular to the line $18 x + 9 y = - 37$ at two points on the curve. Find the two points.

Find second derivative of y = $( 2 x - 4 ) ^ { 5 }$ at x = -1. Enter just an integer.

Find the average rate of change for $f ( x ) = - 2 x ^ { 2 } + 5 x - 2$ on [-1, 1]

If f(t) = $\frac { 1 } { x ^ { 2 / 3 } }$ , then $\lim _ { h \rightarrow 0 }$ $\frac { f ( - 8 + h ) - f ( - 8 ) } { h }$ equals