Question 1

Does $\lim _ { x \rightarrow - a }$ $\frac { x - a } { x ^ { 2 } - a ^ { 2 } }$ (where a ≠ 0) exist?
Enter your answer as either "yes" or "no" or "does not exist".

Accepted Answer

The answer of Does \(\lim _ { x \rightarrow -...

Question 2

Differentiate.
-h(x) = $\frac { 5 } { x ^ { 3 } - 4 x ^ { 2 } + 2 }$

Accepted Answer

A)  h'(x) = $\frac { 15 x ^ { 2 } - 40 x } { x ^ { 3 } - 4 x ^ { 2 } + 2 }$ 
B)  h'(x) = $\frac { 3 x ^ { 2 } - 8 x } { \left( x ^ { 3 } - 4 x ^ { 2 } + 2 \right) ^ { 2 } }$ 
C)  h'(x) = $\frac { - 15 x ^ { 2 } + 40 x } { \left( x ^ { 3 } - 4 x ^ { 2 } + 2 \right) ^ { 2 } }$ 
D)  h'(x) = $\frac { 5 } { \left( 3 x ^ { 2 } - 8 x \right) ^ { 2 } }$ 
A)  h'(x) = $\frac { 15 x ^ { 2 } - 40 x } { x ^ { 3 } - 4 x ^ { 2 } + 2 }$ 
B)  h'(x) = $\frac { 3 x ^ { 2 } - 8 x } { \left( x ^ { 3 } - 4 x ^ { 2 } + 2 \right) ^ { 2 } }$ 
C)  h'(x) = $\frac { - 15 x ^ { 2 } + 40 x } { \left( x ^ { 3 } - 4 x ^ { 2 } + 2 \right) ^ { 2 } }$ 
D)  h'(x) = $\frac { 5 } { \left( 3 x ^ { 2 } - 8 x \right) ^ { 2 } }$

Question 3

Calculate the following limit(s) if they exist.
-$\lim _ { x \rightarrow \infty }$ $\frac { x ^ { 3 } } { x ^ { 3 } - 1 }$ Enter just an integer or a fraction.

Accepted Answer

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

Question 4

Write the word or phrase that best completes each statement or answers the question. 
-y = $\frac { 2 } { \sqrt { 2 x + 1 } }$ Enter your answer exactly as: $\frac { d y } { d x } = a ( \mathrm { P } ( \mathrm { x } ) ) ^ { \mathrm { b } }$ where P(x) is a polynomial in standard form. a, b reduced fractions or integers. No parentheses on coefficients or powers.

Accepted Answer

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

Question 5

Find the derivative of f(x) = 4 $x ^ { 5 / 4 }$ .
Enter your answer in the form: $a x ^ { b }$ , where a, b are either fractions in lowest terms or integers. No parentheses.

Accepted Answer

The answer of Find the derivative of f(x) = 4...

Question 6

If the profit function is $P ( x ) = \frac { 5 x ^ { 2 } } { 3 } + \frac { 2 } { 3 x } + 2 x - 10$ for producing a product, find the marginal profit at a production level of 10 units.

Accepted Answer

A)  $\frac { 5299 } { 150 }$ 
B)  $\frac { 2651 } { 150 }$ 
C)  60 
D)  423 
E)  $\frac { 252 } { 75 }$ 
A)  $\frac { 5299 } { 150 }$ 
B)  $\frac { 2651 } { 150 }$ 
C)  60 
D)  423 
E)  $\frac { 252 } { 75 }$

Question 7

Find the derivative of f(x) = $x ^ { 6 / 5 }$ .
Enter your answer in the form: a $x ^ { b }$ , where a, b are either fractions of the form $\frac { \mathrm { c } } { \mathrm { d } }$ or integers. No parentheses.

Accepted Answer

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

Question 8

A company is planning to manufacture a new blender. After conducting extensive market surveys, the research department estimates a weekly demand of 600 blenders at a price of $50 per blender and a weekly demand of 800 blenders at a price of $40 per blender. Assuming the demand equation is linear, use the research department's estimates to find the revenue equation in terms of the demand x.

Accepted Answer

A)  R(x) = 20x + $\frac { x ^ { 2 } } { 20 }$ 
B)  R(x) = 80x - $\frac { x ^ { 2 } } { 20 }$ 
C)  R(x) = 80x - 20 $x ^ { 2 }$ 
D)  R(x) = 80x - 20 
A)  R(x) = 20x + $\frac { x ^ { 2 } } { 20 }$ 
B)  R(x) = 80x - $\frac { x ^ { 2 } } { 20 }$ 
C)  R(x) = 80x - 20 $x ^ { 2 }$ 
D)  R(x) = 80x - 20

Question 9

Find the equation of the tangent line to the curve y = $\frac { 1 } { x + 2 }$ at $\left(2 , \frac { 1 } { 4 } \right)$ .
Enter your answer in standard slope-intercept form using reduced fractions .No parentheses.

Accepted Answer

The answer of Find the equation of the tangent line...

Question 10

Find the slope of the tangent line to the curve $y = \frac { 4 } { 3 \left( x ^ { 3 } - 2 x ^ { 2 } + 3 x - 5 \right) } \text { at } \left( 2 , \frac { 4 } { 3 } \right)$

Accepted Answer

A)  - $\frac { 8 } { 3 }$ 
B)  - $\frac { 28 } { 3 }$ 
C)  $\frac { 8 } { 3 }$ 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 2 } { 3 }$ 
A)  - $\frac { 8 } { 3 }$ 
B)  - $\frac { 28 } { 3 }$ 
C)  $\frac { 8 } { 3 }$ 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 2 } { 3 }$

Question 11

Find the equation of the tangent line to the graph of y = - $\frac { 3 } { 5 x + 2 }$ at x = 3.

Accepted Answer

A)  y = $\frac { 15 } { ( 5 x - 2 ) ^ { 2 } }$ 
B)  y = $\frac { 15 } { 169 }$ (x - 3) 
C)  y - 3 = 15(5x - 2 $) ^ { - 2 }$ 
D)  y = $\frac { 15 } { 289 }$ (x - 3) - $\frac { 3 } { 17 }$ 
A)  y = $\frac { 15 } { ( 5 x - 2 ) ^ { 2 } }$ 
B)  y = $\frac { 15 } { 169 }$ (x - 3) 
C)  y - 3 = 15(5x - 2 $) ^ { - 2 }$ 
D)  y = $\frac { 15 } { 289 }$ (x - 3) - $\frac { 3 } { 17 }$

Question 12

Find the first derivative of y= ( $x ^ { 3 }$ + 4x $) ^ { 5 }$ at x = 1.
Enter just an integer.

Accepted Answer

The answer of Find the first derivative of y= (...

Question 13

Find the slope of the tangent line to the curve y = $\frac { 3 } { ( x + 2 ) ^ { 2 } }$ at x = 3.
Enter a reduced fraction only.

Accepted Answer

The answer of Find the slope of the tangent line...

Question 14

Which of the following is/are true of the lines 2x - 5y = -15 and 5x + 2y = 6? (I) They are parallel.
(II) They are perpendicular.
(III) They cross the x-axis at the same point.
(IV) They cross the y-axis at the same point.

Accepted Answer

A)  I 
B)  II and III 
C)  II 
D)  II and IV 
E)  none of these 
A)  I 
B)  II and III 
C)  II 
D)  II and IV 
E)  none of these

Question 15

Consider the curve f(x) in the accompanying sketch.   Find the slope of the tangent line at the point where x = 2.
Enter your answer as just a fraction or an integer.

Accepted Answer

The answer of Consider the curve f(x) in the accompanying...

Question 16

Find all points at which the tangent line to the graph of y = $x ^ { 3 }$ is parallel to the line $y = 27 x + 5$ .
Enter your answer in the form: (a, b), (c, d)where a > c

Accepted Answer

The answer of Find all points at which the tangent...

Question 17

A winter storm front moves through campus. At t hours after the onset of the storm, the temperature is given by $T ( t ) = 35 - 2 t ^ { 2 } + t$ . What is the average rate at which water pours into the tub over the first 4 minutes?
Enter just a real number to one decimal place, no units.

Accepted Answer

The answer of A winter storm front moves through campus....

Question 18

Determine whether or not the following limit exists. If, so compute the limit. $\lim _ { x \rightarrow - 1 }$ $\frac { x ^ { 2 } - 1 } { x + 1 }$

Accepted Answer

A)  -2 
B)  1 
C)  does not exist 
D)  0 
E)  none of these 
A)  -2 
B)  1 
C)  does not exist 
D)  0 
E)  none of these

Question 19

Find the second derivative of y = $\frac { 1 } { 2 x }$ + 3.

Accepted Answer

A)  $\frac { 1 } { 6 x ^ { 4 } }$ 
B)  - $\frac { 1 } { 6 x ^ { 4 } }$ 
C)  2 $x ^ { 0 }$ 
D)  -6 $x ^ { 4 }$ 
E)  none of these 
A)  $\frac { 1 } { 6 x ^ { 4 } }$ 
B)  - $\frac { 1 } { 6 x ^ { 4 } }$ 
C)  2 $x ^ { 0 }$ 
D)  -6 $x ^ { 4 }$ 
E)  none of these

Question 20

Using the graph below, find the given limit.  
-$\lim _ { x \rightarrow \infty }$ (2f(x) - 1)

Accepted Answer

A)  6 
B)  0 
C)  5 
D)  2 
A)  6 
B)  0 
C)  5 
D)  2

Does $\lim _ { x \rightarrow - a }$ $\frac { x - a } { x ^ { 2 } - a ^ { 2 } }$ (where a ≠ 0) exist? Enter your answer as either "yes" or "no" or "does not exist".

Differentiate. -h(x) = $\frac { 5 } { x ^ { 3 } - 4 x ^ { 2 } + 2 }$

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow \infty }$ $\frac { x ^ { 3 } } { x ^ { 3 } - 1 }$ Enter just an integer or a fraction.

Find the derivative of f(x) = 4 $x ^ { 5 / 4 }$ . Enter your answer in the form: $a x ^ { b }$ , where a, b are either fractions in lowest terms or integers. No parentheses.

If the profit function is $P ( x ) = \frac { 5 x ^ { 2 } } { 3 } + \frac { 2 } { 3 x } + 2 x - 10$ for producing a product, find the marginal profit at a production level of 10 units.

Find the derivative of f(x) = $x ^ { 6 / 5 }$ . Enter your answer in the form: a $x ^ { b }$ , where a, b are either fractions of the form $\frac { \mathrm { c } } { \mathrm { d } }$ or integers. No parentheses.

Find the equation of the tangent line to the curve y = $\frac { 1 } { x + 2 }$ at $\left(2 , \frac { 1 } { 4 } \right)$ . Enter your answer in standard slope-intercept form using reduced fractions .No parentheses.

Find the slope of the tangent line to the curve $y = \frac { 4 } { 3 \left( x ^ { 3 } - 2 x ^ { 2 } + 3 x - 5 \right) } \text { at } \left( 2 , \frac { 4 } { 3 } \right)$

Find the equation of the tangent line to the graph of y = - $\frac { 3 } { 5 x + 2 }$ at x = 3.

Find the first derivative of y= ( $x ^ { 3 }$ + 4x $) ^ { 5 }$ at x = 1. Enter just an integer.

Find the slope of the tangent line to the curve y = $\frac { 3 } { ( x + 2 ) ^ { 2 } }$ at x = 3. Enter a reduced fraction only.

Which of the following is/are true of the lines 2x - 5y = -15 and 5x + 2y = 6? (I) They are parallel. (II) They are perpendicular. (III) They cross the x-axis at the same point. (IV) They cross the y-axis at the same point.

Find all points at which the tangent line to the graph of y = $x ^ { 3 }$ is parallel to the line $y = 27 x + 5$ . Enter your answer in the form: (a, b), (c, d)where a > c

A winter storm front moves through campus. At t hours after the onset of the storm, the temperature is given by $T ( t ) = 35 - 2 t ^ { 2 } + t$ . What is the average rate at which water pours into the tub over the first 4 minutes? Enter just a real number to one decimal place, no units.

Determine whether or not the following limit exists. If, so compute the limit. $\lim _ { x \rightarrow - 1 }$ $\frac { x ^ { 2 } - 1 } { x + 1 }$

Find the second derivative of y = $\frac { 1 } { 2 x }$ + 3.

Using the graph below, find the given limit. $Using the graph below, find the given limit. - \lim _ { x \rightarrow \infty } (2f(x) - 1)$ - $\lim _ { x \rightarrow \infty }$ (2f(x) - 1)

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

Does lim⁡x→−a\lim _ { x \rightarrow - a }limx→−a​ x−ax2−a2\frac { x - a } { x ^ { 2 } - a ^ { 2 } }x2−a2x−a​ (where a ≠ 0) exist? Enter your answer as either "yes" or "no" or "does not exist".

Differentiate. -h(x) = 5x3−4x2+2\frac { 5 } { x ^ { 3 } - 4 x ^ { 2 } + 2 }x3−4x2+25​

Calculate the following limit(s) if they exist. - lim⁡x→∞\lim _ { x \rightarrow \infty }limx→∞​ x3x3−1\frac { x ^ { 3 } } { x ^ { 3 } - 1 }x3−1x3​ Enter just an integer or a fraction.

Find the derivative of f(x) = 4 x5/4x ^ { 5 / 4 }x5/4 . Enter your answer in the form: axba x ^ { b }axb , where a, b are either fractions in lowest terms or integers. No parentheses.

If the profit function is P(x)=5x23+23x+2x−10P ( x ) = \frac { 5 x ^ { 2 } } { 3 } + \frac { 2 } { 3 x } + 2 x - 10P(x)=35x2​+3x2​+2x−10 for producing a product, find the marginal profit at a production level of 10 units.

Find the derivative of f(x) = x6/5x ^ { 6 / 5 }x6/5 . Enter your answer in the form: a xbx ^ { b }xb , where a, b are either fractions of the form cd\frac { \mathrm { c } } { \mathrm { d } }dc​ or integers. No parentheses.

Find the equation of the tangent line to the curve y = 1x+2\frac { 1 } { x + 2 }x+21​ at (2,14)\left(2 , \frac { 1 } { 4 } \right)(2,41​) . Enter your answer in standard slope-intercept form using reduced fractions .No parentheses.

Find the slope of the tangent line to the curve y=43(x3−2x2+3x−5) at (2,43)y = \frac { 4 } { 3 \left( x ^ { 3 } - 2 x ^ { 2 } + 3 x - 5 \right) } \text { at } \left( 2 , \frac { 4 } { 3 } \right)y=3(x3−2x2+3x−5)4​ at (2,34​)

Find the equation of the tangent line to the graph of y = - 35x+2\frac { 3 } { 5 x + 2 }5x+23​ at x = 3.

Find the first derivative of y= ( x3x ^ { 3 }x3 + 4x )5) ^ { 5 })5 at x = 1. Enter just an integer.

Find the slope of the tangent line to the curve y = 3(x+2)2\frac { 3 } { ( x + 2 ) ^ { 2 } }(x+2)23​ at x = 3. Enter a reduced fraction only.

Which of the following is/are true of the lines 2x - 5y = -15 and 5x + 2y = 6? (I) They are parallel. (II) They are perpendicular. (III) They cross the x-axis at the same point. (IV) They cross the y-axis at the same point.

Consider the curve f(x) in the accompanying sketch. Find the slope of the tangent line at the point where x = 2. Enter your answer as just a fraction or an integer.

Find all points at which the tangent line to the graph of y = x3x ^ { 3 }x3 is parallel to the line y=27x+5y = 27 x + 5y=27x+5 . Enter your answer in the form: (a, b), (c, d)where a > c

Determine whether or not the following limit exists. If, so compute the limit. lim⁡x→−1\lim _ { x \rightarrow - 1 }limx→−1​ x2−1x+1\frac { x ^ { 2 } - 1 } { x + 1 }x+1x2−1​

Find the second derivative of y = 12x\frac { 1 } { 2 x }2x1​ + 3.

Using the graph below, find the given limit. - lim⁡x→∞\lim _ { x \rightarrow \infty }limx→∞​ (2f(x) - 1)

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

Does $\lim _ { x \rightarrow - a }$ $\frac { x - a } { x ^ { 2 } - a ^ { 2 } }$ (where a ≠ 0) exist? Enter your answer as either "yes" or "no" or "does not exist".

Differentiate. -h(x) = $\frac { 5 } { x ^ { 3 } - 4 x ^ { 2 } + 2 }$

Calculate the following limit(s) if they exist. - $\lim _ { x \rightarrow \infty }$ $\frac { x ^ { 3 } } { x ^ { 3 } - 1 }$ Enter just an integer or a fraction.

Find the derivative of f(x) = 4 $x ^ { 5 / 4 }$ . Enter your answer in the form: $a x ^ { b }$ , where a, b are either fractions in lowest terms or integers. No parentheses.

If the profit function is $P ( x ) = \frac { 5 x ^ { 2 } } { 3 } + \frac { 2 } { 3 x } + 2 x - 10$ for producing a product, find the marginal profit at a production level of 10 units.

Find the derivative of f(x) = $x ^ { 6 / 5 }$ . Enter your answer in the form: a $x ^ { b }$ , where a, b are either fractions of the form $\frac { \mathrm { c } } { \mathrm { d } }$ or integers. No parentheses.

Find the equation of the tangent line to the curve y = $\frac { 1 } { x + 2 }$ at $\left(2 , \frac { 1 } { 4 } \right)$ . Enter your answer in standard slope-intercept form using reduced fractions .No parentheses.

Find the slope of the tangent line to the curve $y = \frac { 4 } { 3 \left( x ^ { 3 } - 2 x ^ { 2 } + 3 x - 5 \right) } \text { at } \left( 2 , \frac { 4 } { 3 } \right)$

Find the equation of the tangent line to the graph of y = - $\frac { 3 } { 5 x + 2 }$ at x = 3.

Find the first derivative of y= ( $x ^ { 3 }$ + 4x $) ^ { 5 }$ at x = 1. Enter just an integer.

Find the slope of the tangent line to the curve y = $\frac { 3 } { ( x + 2 ) ^ { 2 } }$ at x = 3. Enter a reduced fraction only.

Find all points at which the tangent line to the graph of y = $x ^ { 3 }$ is parallel to the line $y = 27 x + 5$ . Enter your answer in the form: (a, b), (c, d)where a > c

Determine whether or not the following limit exists. If, so compute the limit. $\lim _ { x \rightarrow - 1 }$ $\frac { x ^ { 2 } - 1 } { x + 1 }$

Find the second derivative of y = $\frac { 1 } { 2 x }$ + 3.

Using the graph below, find the given limit. $Using the graph below, find the given limit. - \lim _ { x \rightarrow \infty } (2f(x) - 1)$ - $\lim _ { x \rightarrow \infty }$ (2f(x) - 1)