Question 1

Find the equation of the tangent line to the curve y = $x ^ { 3 }$ + 4 $x ^ { 2 }$ + 4 at (1, 9).
Enter your answer in standard slope-intercept form.

Accepted Answer

y = 11x - 2

Question 2

Write the word or phrase that best completes each statement or answers the question. 
-f(x) = $\sqrt { x }$ Enter your answer as just: a $x ^ { b }$ . No parentheses.

Accepted Answer

$\frac { 1 } { 2 }$ $x ^ { - 1 / 2 }$

Question 3

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

Accepted Answer

36

Question 4

Compute the third derivative of the following function: f(z) = $\frac { 3 } { 2 z - 2 }$ at z = $\frac { 5 } { 2 }$ .
Enter just a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

The answer of Compute the third derivative of the following...

Question 5

Let f(x) = $( 2 x + 1 ) ^ { 2 }$ . Compute $f ^ { \prime }$ (0) using limits.
Enter a reduced fraction or an integer.

Accepted Answer

The answer of Let f(x) = \(( 2 x +...

Question 6

Determine whether or not the following limit exists. If, so compute the limit. $\lim _ { x \rightarrow 3 }$ $\frac { \sqrt { x } - 4 } { x ^ { 3 } + 27 }$

Accepted Answer

A)  does not exist 
B)  0 
C)  $\frac { \sqrt { 3 } - 4 } { 54 }$ 
D)  $\sqrt { 3 }$ - 4 
E)  none of these 
A)  does not exist 
B)  0 
C)  $\frac { \sqrt { 3 } - 4 } { 54 }$ 
D)  $\sqrt { 3 }$ - 4 
E)  none of these

Question 7

Determine: $\frac { d y } { d x }$ if y = $\frac { x - 2 } { 5 }$ .
Enter your answer as just an integer or a reduced fraction.

Accepted Answer

The answer of Determine: \(\frac { d y } {...

Question 8

Find the equation of the following line: Perpendicular to 3y - $\frac { 5 } { 2 }$ x = 1; y-intercept is $\frac { 1 } { 3 }$ .
Enter your answer in slope-intercept form.

Accepted Answer

The answer of Find the equation of the following line:...

Question 9

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 point's instantaneous velocity when t = 5.
Enter just an integer.

Accepted Answer

The answer of A point P is moving along the...

Question 10

The graph below shows the total sales (in thousands of dollars) in a supermarket during a typical 24-hour period.   (i) Estimate the rate of sales during the period from 6 A.M. to 12 noon. 
(ii) Which 6-hour interval sees the highest rate of sales and what is this rate?

Accepted Answer

A)  (i) $167/hr
    (ii) 6pm to 12 noon 
B)  (i) $1167/hr
    (ii) noon to 6pm 
C)  (i) $667/hr
    (ii) 12 noon to 6 pm $1167/hr. 
D)  none of these 
A)  (i) $167/hr
    (ii) 6pm to 12 noon 
B)  (i) $1167/hr
    (ii) noon to 6pm 
C)  (i) $667/hr
    (ii) 12 noon to 6 pm $1167/hr. 
D)  none of these

Question 11

Find the slope of the graph of y = ( $x ^ { 2 }$ - 7 $)^ { 3 }$ at x = 3.
Enter just an integer.

Accepted Answer

The answer of Find the slope of the graph of...

Question 12

-Q

Accepted Answer

A)  large negative slope 
B)  small negative slope 
C)  zero slope 
D)  small positive slope 
A)  large negative slope 
B)  small negative slope 
C)  zero slope 
D)  small positive slope

Question 13

Find the second derivative of U = $( 3 u - 7 ) ^ { 4 }$ - $\mathbf { u } ^ { 3 }$ at u = $\frac { 5 } { 3 }$ .
Enter just a real number.

Accepted Answer

The answer of Find the second derivative of U =...

Question 14

Calculate the following limit(s) if they exist.
-$\lim _ { x \rightarrow - 1 }$ ( $x ^ { 3 }$ - 2x + 5).
Enter just an integer or "does not exist".

Accepted Answer

The answer of Calculate the following limit(s) if they exist.
-\(\lim...

Question 15

Find the derivative of f(x) = $\frac { 1 } { x ^ { 2 } }$ .
Enter your answer exactly in the form: a $x ^ { b }$

Accepted Answer

The answer of Find the derivative of f(x) = \(\frac...

Question 16

Write the word or phrase that best completes each statement or answers the question. 
-y = $x ^ { 15 / 19 }$ Enter your answer exactly in the form: a $x ^ { b }$ where a, b are either fractions or integers. No parentheses. No labels.

Accepted Answer

The answer of Write the word or phrase that best...

Question 17

Find the first derivative of z = 4t + (3 - $\sqrt { 2 t + 1 }$ $) ^ { 3 }$ at t = $\frac { 3 } { 2 }$ .
Enter just a fraction of form $\frac { a } { b }$ in lowest terms.

Accepted Answer

The answer of Find the first derivative of z =...

Question 18

Find the equation of the following line: Perpendicular to - 8x + y - 3 = 0; (4, 3) on line.

Accepted Answer

A)  y = - $\frac { 1 } { 8 }$ x - $\frac { 7 } { 2 }$ 
B)  y = - 8x - 28 
C)  y = - $\frac { 1 } { 8 }$ x + $\frac { 7 } { 2 }$ 
D)  y = $\frac { 1 } { 8 }$ x - $\frac { 7 } { 2 }$ 
A)  y = - $\frac { 1 } { 8 }$ x - $\frac { 7 } { 2 }$ 
B)  y = - 8x - 28 
C)  y = - $\frac { 1 } { 8 }$ x + $\frac { 7 } { 2 }$ 
D)  y = $\frac { 1 } { 8 }$ x - $\frac { 7 } { 2 }$

Question 19

A manufacturer's profit from producing x units of a product is given by $P ( x ) = 0.002 x ^ { 3 } - 0.01 x ^ { 2 } + 0.5 x$ What is the marginal profit when the production level is at 50 units? Enter your answer as a real number to two decimal places, no units.

Accepted Answer

The answer of A manufacturer's profit from producing x units...

Question 20

Find the second derivative of $f ( x ) = \frac { 1 } { 3 } x ^ { 3 / 2 } - \frac { 4 } { 3 } x ^ { 1 / 4 } + 5 x - 2$

Accepted Answer

A)  $\frac { 1 } { 4 }$ $x - 1 / 2$ + $\frac { 1 } { 4 }$ $x _ {  } ^ { 7 / 4 }$ 
B)  $\frac { 1 } { 4 }$ $x - 1 / 2$ - $\frac { 1 } { 4 }$ $x ^ { - 7 / 4 }$ 
C)  $\frac { 2 } { 9 }$ $\times 1 / 2$ - $\frac { 16 } { 3 }$ $x - 3 / 4$ + 5 
D)  $\times 1 / 2$ + $\frac { 4 } { 9 }$ $x ^ { - 7 / 4 }$ 
E)  $\frac { 1 } { 2 }$ $\times 1 / 2$ - $\frac { 1 } { 3 }$ $x - 3 / 4$ + 5 
A)  $\frac { 1 } { 4 }$ $x - 1 / 2$ + $\frac { 1 } { 4 }$ $x _ {  } ^ { 7 / 4 }$ 
B)  $\frac { 1 } { 4 }$ $x - 1 / 2$ - $\frac { 1 } { 4 }$ $x ^ { - 7 / 4 }$ 
C)  $\frac { 2 } { 9 }$ $\times 1 / 2$ - $\frac { 16 } { 3 }$ $x - 3 / 4$ + 5 
D)  $\times 1 / 2$ + $\frac { 4 } { 9 }$ $x ^ { - 7 / 4 }$ 
E)  $\frac { 1 } { 2 }$ $\times 1 / 2$ - $\frac { 1 } { 3 }$ $x - 3 / 4$ + 5

Find the equation of the tangent line to the curve y = $x ^ { 3 }$ + 4 $x ^ { 2 }$ + 4 at (1, 9). Enter your answer in standard slope-intercept form.

Write the word or phrase that best completes each statement or answers the question. -f(x) = $\sqrt { x }$ Enter your answer as just: a $x ^ { b }$ . No parentheses.

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

Compute the third derivative of the following function: f(z) = $\frac { 3 } { 2 z - 2 }$ at z = $\frac { 5 } { 2 }$ . Enter just a reduced fraction of form $\frac { a } { b }$ .

Let f(x) = $( 2 x + 1 ) ^ { 2 }$ . Compute $f ^ { \prime }$ (0) using limits. Enter a reduced fraction or an integer.

Determine whether or not the following limit exists. If, so compute the limit. $\lim _ { x \rightarrow 3 }$ $\frac { \sqrt { x } - 4 } { x ^ { 3 } + 27 }$

Determine: $\frac { d y } { d x }$ if y = $\frac { x - 2 } { 5 }$ . Enter your answer as just an integer or a reduced fraction.

Find the equation of the following line: Perpendicular to 3y - $\frac { 5 } { 2 }$ x = 1; y-intercept is $\frac { 1 } { 3 }$ . Enter your answer in slope-intercept form.

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 point's instantaneous velocity when t = 5. Enter just an integer.

The graph below shows the total sales (in thousands of dollars) in a supermarket during a typical 24-hour period. (i) Estimate the rate of sales during the period from 6 A.M. to 12 noon. (ii) Which 6-hour interval sees the highest rate of sales and what is this rate?

Find the slope of the graph of y = ( $x ^ { 2 }$ - 7 $)^ { 3 }$ at x = 3. Enter just an integer.

-Q

Find the second derivative of U = $( 3 u - 7 ) ^ { 4 }$ - $\mathbf { u } ^ { 3 }$ at u = $\frac { 5 } { 3 }$ . Enter just a real number.

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow - 1 }$ ( $x ^ { 3 }$ - 2x + 5). Enter just an integer or "does not exist".

Find the derivative of f(x) = $\frac { 1 } { x ^ { 2 } }$ . Enter your answer exactly in the form: a $x ^ { b }$

Write the word or phrase that best completes each statement or answers the question. -y = $x ^ { 15 / 19 }$ Enter your answer exactly in the form: a $x ^ { b }$ where a, b are either fractions or integers. No parentheses. No labels.

Find the first derivative of z = 4t + (3 - $\sqrt { 2 t + 1 }$ $) ^ { 3 }$ at t = $\frac { 3 } { 2 }$ . Enter just a fraction of form $\frac { a } { b }$ in lowest terms.

Find the equation of the following line: Perpendicular to - 8x + y - 3 = 0; (4, 3) on line.

A manufacturer's profit from producing x units of a product is given by $P ( x ) = 0.002 x ^ { 3 } - 0.01 x ^ { 2 } + 0.5 x$ What is the marginal profit when the production level is at 50 units? Enter your answer as a real number to two decimal places, no units.

Find the second derivative of $f ( x ) = \frac { 1 } { 3 } x ^ { 3 / 2 } - \frac { 4 } { 3 } x ^ { 1 / 4 } + 5 x - 2$

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 1: The Derivative

Find the equation of the tangent line to the curve y = x3x ^ { 3 }x3 + 4 x2x ^ { 2 }x2 + 4 at (1, 9). Enter your answer in standard slope-intercept form.

Write the word or phrase that best completes each statement or answers the question. -f(x) = x\sqrt { x }x​ Enter your answer as just: a xbx ^ { b }xb . No parentheses.

Find the slope of the tangent line to the curve y = 2 (x3\left( x ^ { 3 } \right.(x3 + 4 )2)^2)2 at x = -1. Enter just an integer.

Compute the third derivative of the following function: f(z) = 32z−2\frac { 3 } { 2 z - 2 }2z−23​ at z = 52\frac { 5 } { 2 }25​ . Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

Let f(x) = (2x+1)2( 2 x + 1 ) ^ { 2 }(2x+1)2 . Compute f′f ^ { \prime }f′ (0) using limits. Enter a reduced fraction or an integer.

Determine whether or not the following limit exists. If, so compute the limit. lim⁡x→3\lim _ { x \rightarrow 3 }limx→3​ x−4x3+27\frac { \sqrt { x } - 4 } { x ^ { 3 } + 27 }x3+27x​−4​

Determine: dydx\frac { d y } { d x }dxdy​ if y = x−25\frac { x - 2 } { 5 }5x−2​ . Enter your answer as just an integer or a reduced fraction.

Find the equation of the following line: Perpendicular to 3y - 52\frac { 5 } { 2 }25​ x = 1; y-intercept is 13\frac { 1 } { 3 }31​ . Enter your answer in slope-intercept form.

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=t3−4t2+3tx = t ^ { 3 } - 4 t ^ { 2 } + 3 tx=t3−4t2+3t . Determine the point's instantaneous velocity when t = 5. Enter just an integer.

The graph below shows the total sales (in thousands of dollars) in a supermarket during a typical 24-hour period. (i) Estimate the rate of sales during the period from 6 A.M. to 12 noon. (ii) Which 6-hour interval sees the highest rate of sales and what is this rate?

Find the slope of the graph of y = ( x2x ^ { 2 }x2 - 7 )3)^ { 3 })3 at x = 3. Enter just an integer.

-Q

Find the second derivative of U = (3u−7)4( 3 u - 7 ) ^ { 4 }(3u−7)4 - u3\mathbf { u } ^ { 3 }u3 at u = 53\frac { 5 } { 3 }35​ . Enter just a real number.

Calculate the following limit(s) if they exist. - lim⁡x→−1\lim _ { x \rightarrow - 1 }limx→−1​ ( x3x ^ { 3 }x3 - 2x + 5). Enter just an integer or "does not exist".

Find the derivative of f(x) = 1x2\frac { 1 } { x ^ { 2 } }x21​ . Enter your answer exactly in the form: a xbx ^ { b }xb

Write the word or phrase that best completes each statement or answers the question. -y = x15/19x ^ { 15 / 19 }x15/19 Enter your answer exactly in the form: a xbx ^ { b }xb where a, b are either fractions or integers. No parentheses. No labels.

Find the first derivative of z = 4t + (3 - 2t+1\sqrt { 2 t + 1 }2t+1​ )3) ^ { 3 })3 at t = 32\frac { 3 } { 2 }23​ . Enter just a fraction of form ab\frac { a } { b }ba​ in lowest terms.

Find the equation of the following line: Perpendicular to - 8x + y - 3 = 0; (4, 3) on line.

Find the second derivative of f(x)=13x3/2−43x1/4+5x−2f ( x ) = \frac { 1 } { 3 } x ^ { 3 / 2 } - \frac { 4 } { 3 } x ^ { 1 / 4 } + 5 x - 2f(x)=31​x3/2−34​x1/4+5x−2

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Find the equation of the tangent line to the curve y = $x ^ { 3 }$ + 4 $x ^ { 2 }$ + 4 at (1, 9). Enter your answer in standard slope-intercept form.

Write the word or phrase that best completes each statement or answers the question. -f(x) = $\sqrt { x }$ Enter your answer as just: a $x ^ { b }$ . No parentheses.

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

Compute the third derivative of the following function: f(z) = $\frac { 3 } { 2 z - 2 }$ at z = $\frac { 5 } { 2 }$ . Enter just a reduced fraction of form $\frac { a } { b }$ .

Let f(x) = $( 2 x + 1 ) ^ { 2 }$ . Compute $f ^ { \prime }$ (0) using limits. Enter a reduced fraction or an integer.

Determine whether or not the following limit exists. If, so compute the limit. $\lim _ { x \rightarrow 3 }$ $\frac { \sqrt { x } - 4 } { x ^ { 3 } + 27 }$

Determine: $\frac { d y } { d x }$ if y = $\frac { x - 2 } { 5 }$ . Enter your answer as just an integer or a reduced fraction.

Find the equation of the following line: Perpendicular to 3y - $\frac { 5 } { 2 }$ x = 1; y-intercept is $\frac { 1 } { 3 }$ . Enter your answer in slope-intercept form.

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 point's instantaneous velocity when t = 5. Enter just an integer.

Find the slope of the graph of y = ( $x ^ { 2 }$ - 7 $)^ { 3 }$ at x = 3. Enter just an integer.

Find the second derivative of U = $( 3 u - 7 ) ^ { 4 }$ - $\mathbf { u } ^ { 3 }$ at u = $\frac { 5 } { 3 }$ . Enter just a real number.

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow - 1 }$ ( $x ^ { 3 }$ - 2x + 5). Enter just an integer or "does not exist".

Find the derivative of f(x) = $\frac { 1 } { x ^ { 2 } }$ . Enter your answer exactly in the form: a $x ^ { b }$

Write the word or phrase that best completes each statement or answers the question. -y = $x ^ { 15 / 19 }$ Enter your answer exactly in the form: a $x ^ { b }$ where a, b are either fractions or integers. No parentheses. No labels.

Find the first derivative of z = 4t + (3 - $\sqrt { 2 t + 1 }$ $) ^ { 3 }$ at t = $\frac { 3 } { 2 }$ . Enter just a fraction of form $\frac { a } { b }$ in lowest terms.

Find the second derivative of $f ( x ) = \frac { 1 } { 3 } x ^ { 3 / 2 } - \frac { 4 } { 3 } x ^ { 1 / 4 } + 5 x - 2$