Exam 1: The Derivative

Find the equation of the following line: Slope is 3; y-intercept is 5. Enter your answer in slope-intercept form.

Which of the following is the best description of f'(t)?

A point P is moving along the x-axis. At any time t, the location of P on the x-axis is described by $x = t ^ { 3 } - 4 t ^ { 2 } + 3 t$ . Determine the instantaneous acceleration at time $t = 5$ of the point P. Enter just an integer.

Find the slope of the line with equation 2x + 4 = 2(2y + 3). Enter just a fraction of form $\frac { a } { b }$ in lowest terms.

Find the equation of the tangent line to the curve y = 1 + 3x - $x ^ { 2 }$ at x = 5. Enter your answer in standard slope-intercept form.

Consider 2 lines each having slope $\frac { 4 } { 3 }$ , one passing through the point (-1, 2) and the other through (3, 5). Which of the following is/are true? (I) They are parallel. (II) They are the same line. (III) They cross the x-axis at the same point. (IV) They intersect at the point (0, 3).

If f(x)= - $x ^ { 2 }$ + 2x - 1, find $\frac { f ( x + h ) - f ( x ) } { h }$ , h ? 0.

Find the derivative of f(x) = $\frac { 1 } { \sqrt { x } }$ . Enter your answer exactly in the form: a $x ^ { b }$ . Do not use parentheses. No parentheses.

The function i(x) = $\frac { x ^ { 2 } + 2 x - 35 } { x + 7 }$ is continuous everywhere except at x = -7. If possible, define f(x) at $x = - 7$ in a way that makes f(x) continuous for all x. Enter your answer exactly as just: f(-7) = a where a is an integer or enter the words "not possible".

Find the equation of the following line: $\left( \frac { 1 } { 2 } , 1 \right)$ and (2, 0) on line. Enter your answer in point-slope form using $\left( \frac { 1 } { 2 } , 1 \right)$ .

Which of the following lines is/are parallel to the tangent line of the graph of $y = - x ^ { 3 }$ at the point where $x = 1$ and the slope of the tangent line is $- 3 x ^ { 2 }$ . (I) y = -3x + 1 (II) y = 3x + 1 (III) y = -3x - 4 (IV) y = 3x + 1

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

A ball is thrown straight up. Its height, in feet, at time t, in seconds, is represented by the equation $h ( t ) = 30 t - 16 t ^ { 2 } + 6$ . Determine the instantaneous velocity of the ball at $t = 2$ . Enter just an integer (no units)

If g(x) = 2 $x ^ { 2 }$ - x + 3, compute g(1) and g'(1). Enter your answer exactly in the form: g(1) = a, g'(1) = b where a,b are integers

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. What is the velocity of the rock when it slams into the ground? Enter just an integer.

Let f(t) = $t ^ { 3 }$ - $\frac { 9 } { t }$ . Compute $\frac { d ^ { 2 } \mathrm { f } } { \mathrm { dt } ^ { 2 } } \mid$ $\mathrm { t } = 3$ . Enter just a fraction $\frac { a } { b }$ in lowest terms.

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow - 1 }$ $\frac { x ^ { 2 } - 4 x - 5 } { x + 1 }$ Enter your answer as a fraction, integer, or "does not exist".

Find the derivative of f(x) = $\frac { 5 } { x ^ { 2 } }$ . Enter your answer exactly in the form: $a x^b$ where a, b are integers. Do not use parentheses.

For the following function f(x), determine whether or not $\lim _ { x \rightarrow 0 }$ f(x) exists. If so, give the limit.

For the following function f(x), determine whether or not $\lim _ { x \rightarrow 2 }$ f(x) exists. If so, give the limit. Enter either a real number or the words "does not exist".