Exam 1: The Derivative

An automobile's brakes are applied at time t = 0 when the vehicle is traveling at 48 ft/sec. The brakes cause the automobile to decelerate so that after t sec the velocity is given by $\mathrm { v } ( \mathrm { t } ) = 48 - 16 \mathrm { t }$ How long will it take for the vehicle to come to a complete stop? Enter just an integer, no units.

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow - 1 }$ [ $( x + 1 ) ^ { 3 }$ $\left( x ^ { 2 } \right.$ + 1) $\sqrt { 2 x + 1 }$ ] Enter your answer as an integer, or " does not exist"

Find the derivative of f(x) = $x ^ { 4 / 5 }$ at x = 32. Enter just a reduced fraction or an integer.

Compute the third derivative of f(x) = 3 $x ^ { 4 }$ - 4 $x ^ { 3 }$ + 5x + 1. Enter your answer as an unlabeled polynomial in x in standard form.

Use limits to compute f'(5) where f(x) = $\sqrt { 3 x + 1 }$ . Enter just an integer or a fraction in lowest terms.

Compute f ''(2) when f(t) = $\frac { 3 } { ( 3 t - 1 ) ^ { 2 } }$ . Enter just a reduced fraction of form $\frac { a } { b }$ .

Each of the lines (I), (II), (III), and (IV) in the figure below is the graph of one of the equations (i), (ii), (iii), and (iv). Match each equation with its graph. (a) x + y = 3 (b) x - y = 3 (c) x + y = -3 (d) x - y = -3

Determine: $\frac { d y } { d x }$ if y = 4 - 6x. Enter your answer as just an integer.

Find the equation of the following line: Perpendicular to y = $\frac { 1 } { 5 }$ x + 8; (2, 3) on line.

What is the difference between the actual increase in profit as production increases from 5 to 6 units, and the marginal profit at a production level of 5 units where the profit function is given by $P ( x ) = 3 x ^ { 2 } - 5 x + 2 ?$

Find the first derivative of w(t) = 7 $t ^ { 2 }$ - 19 $\sqrt { t }$ + 23

Find the slope of the tangent line to the curve y = 3 $x ^ { 4 }$ + 2 $x ^ { 3 }$ at x = 1. Enter just an integer.

An automobile's brakes are applied at time t = 0 when the vehicle is traveling at 48 ft/sec. The brakes cause the automobile to decelerate so that after t sec the velocity is given by v(t ) = 48 - 16t. At what rate is the vehicle decelerating after 1 sec? Enter just an integer, no units.

Find the second derivative of $f ( x ) = \frac { 2 x } { 3 x ^ { 2 } + 4 x }$ for x ? 0.

A ball is thrown straight up. Its height, in feet, at time t, in seconds, is represented by the equation $h ( t ) = 20 t - 16 t ^ { 2 } + 10$ . Determine the maximum height of the ball. (Hint: Consider the velocity of the ball at the moment the ball reaches its maximum height.)Enter just a reduced fraction of form $\frac { a } { b }$ . No units.

Write the word or phrase that best completes each statement or answers the question. -f(x) = $\frac { 1 } { x ^ { 2 } + 5 }$ 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.

Find the slope of the graph of y = $\sqrt { 2 x ^ { 2 } + 1 }$ at the point (2, 3).

Suppose that t hours after being placed in a freezer, the temperature of a piece of meat is given by $f ( t ) = 70 - 12 t + \frac { 4 } { t + 1 }$ . How fast is the temperature of the meat falling 3 hours after being placed in the freezer? Enter your answer as a reduced fraction of the form $\frac { a } { b }$ , no units.

Differentiate. -y = - $\frac { 2 } { 3 x ^ { 4 } }$

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow 0 }$ $\frac { 7 } { 2 f ( x ) + g ( x ) }$ 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 a fraction, integer, or the words "does not exist".