Question 1

Find $\frac { d y } { d x }$ if $y = \sqrt [ 3 ] { u }$ and $$u = x ^ { 4 } - 3 x ^ { 3 } - 7$$

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 2

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 1 is:

Accepted Answer

A)  y = 9x - 8 
B)  y = 9x 
C)  y = 2x + 1 
D)  y = 9x - 1

Question 3

The equation for the tangent line to the curve $x ^ { 2 } + 2 x y = y ^ { 3 }$ at the point (1, -1) is y = -1.

Accepted Answer

A) True 
 B)False

Question 4

If $f ( x ) = \frac { 2 - 3 x ^ { 2 } } { x ^ { 3 } + x - 1 }$ , what is $f ^ { \prime } ( x )$ ?

Accepted Answer

The answer of If \(f ( x ) = \frac...

Question 5

The gross national product (GNP) of a certain country is $N ( t ) = t ^ { 2 } + 3 t + 121$ billion dollars where t is the number of years after 1990. At what percentage rate will the GNP be changing with respect to time in 1995? Round your answer to one hundredth of a percent, if necessary.

Accepted Answer

The answer of The gross national product (GNP) of a...

Question 6

Differentiate: $f ( x ) = \sqrt { x } + \frac { 9 } { \sqrt { x } }$

Accepted Answer

The answer of Differentiate: \(f ( x ) = \sqrt...

Question 7

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

Accepted Answer

A)  $\frac { x ^ { 2 } - 4 x } { ( x - 2 ) ^ { 2 } }$ 
B)  $\frac { 3 x ^ { 2 } + 4 x } { ( x - 2 ) ^ { 2 } }$ 
C)  2x 
D)  -x

Question 8

Find the equation of the normal line to $f ( x ) = 4 x ^ { 3 } - 8 x + 49$ at the point with x-coordinate -2.

Accepted Answer

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

Question 9

You measure the side of a cube to be 12 centimeters long and conclude that the volume of the cube is $12 ^ { 3 } = 1,728$ cubic centimeters. If your measurement of the side is accurate to within 6%, approximately how accurate is your calculation of this volume? Round to two decimal places, if necessary.

Accepted Answer

The answer of You measure the side of a cube...

Question 10

Differentiate: $f ( x ) = \sqrt [ 7 ] { x } - \frac { 3 } { \sqrt { x } }$

Accepted Answer

The answer of Differentiate: \(f ( x ) = \sqrt...

Question 11

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 3$ that passes through (1, 4) is y = 2x + 3.

Accepted Answer

A) True 
 B)False

Question 12

It is estimated that t years from now, the population of a certain community will be $p ( t ) = 14 - \frac { 6 } { 3 + t }$ thousand. An environmental study indicates that the average daily level of carbon monoxide in the air will be $C ( p ) = 0.5 \sqrt { p ^ { 2 } + 2 p + 30 }$ units when the population is p thousand. The rate at which the level of carbon monoxide will be changing 3 years from now is

Accepted Answer

A)  -0.078 ppm per thousand people 
B)  0.078 ppm per thousand people 
C)  1.000 ppm per thousand people 
D)  -1.000 ppm per thousand people

Question 13

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 3$ at the point (3, 12).

Accepted Answer

A)  y = 12 
B)  y = 6x - 6 
C)  x = 3 
D)  Not defined

Question 14

If $f ( x ) = x \sqrt { 2 - x }$ , then $f ^ { \prime \prime } ( x ) = 0$ at x = 0 and x = 2.

Accepted Answer

A) True 
 B)False

Question 15

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 2$ that passes through (25, 7) is y = 2x + 2.

Accepted Answer

A) True 
 B)False

Question 16

Find the equation of the line that is tangent to the curve $f ( x ) = 5 + 3 x ^ { 2 } - x ^ { 5 }$ at the point (1, 7).

Accepted Answer

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

Question 17

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

Accepted Answer

A)  $\frac { x ^ { 2 } - 4 x } { ( x - 2 ) ^ { 2 } }$ 
B)  $\frac { x ^ { 2 } + 4 x } { ( x - 2 ) ^ { 2 } }$ 
C)  2x 
D)  -x

Question 18

Find the equation of the tangent line to the given curve at the specified point. $x ^ { 3 } y ^ { 2 } - 3 x y = 4 x + y - 19$ ; (0, 19)

Accepted Answer

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

Question 19

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 } { x } - x$ at the point where x = 1.

Accepted Answer

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

Question 20

You measure the side of a cube to be 14 centimeters long and conclude that the volume of the cube is $14 ^ { 3 } = 2,744$ cubic centimeters. If your measurement of the side is accurate to within 5%, approximately how accurate is your calculation of this volume? Round to two decimal places, if necessary.

Accepted Answer

A)  Maximum error in volume is about ±29.4 cm3 
B)  Maximum error in volume is about ±411.6 cm3 
C)  Maximum error in volume is about ±2.1 cm3 
D)  Maximum error in volume is about ±5,762.4 cm3

Exam 2: Differentiation: Basic Concepts

Find $\frac { d y } { d x }$ if $y = \sqrt [ 3 ] { u }$ and $u = x ^ { 4 } - 3 x ^ { 3 } - 7$

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 1 is:

The equation for the tangent line to the curve $x ^ { 2 } + 2 x y = y ^ { 3 }$ at the point (1, -1) is y = -1.

If $f ( x ) = \frac { 2 - 3 x ^ { 2 } } { x ^ { 3 } + x - 1 }$ , what is $f ^ { \prime } ( x )$ ?

The gross national product (GNP) of a certain country is $N ( t ) = t ^ { 2 } + 3 t + 121$ billion dollars where t is the number of years after 1990. At what percentage rate will the GNP be changing with respect to time in 1995? Round your answer to one hundredth of a percent, if necessary.

Differentiate: $f ( x ) = \sqrt { x } + \frac { 9 } { \sqrt { x } }$

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

Find the equation of the normal line to $f ( x ) = 4 x ^ { 3 } - 8 x + 49$ at the point with x-coordinate -2.

Differentiate: $f ( x ) = \sqrt [ 7 ] { x } - \frac { 3 } { \sqrt { x } }$

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 3$ that passes through (1, 4) is y = 2x + 3.

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 3$ at the point (3, 12) .

If $f ( x ) = x \sqrt { 2 - x }$ , then $f ^ { \prime \prime } ( x ) = 0$ at x = 0 and x = 2.

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 2$ that passes through (25, 7) is y = 2x + 2.

Find the equation of the line that is tangent to the curve $f ( x ) = 5 + 3 x ^ { 2 } - x ^ { 5 }$ at the point (1, 7).

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

Find the equation of the tangent line to the given curve at the specified point. $x ^ { 3 } y ^ { 2 } - 3 x y = 4 x + y - 19$ ; (0, 19)

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 } { x } - x$ at the point where x = 1.

Exam 2: Differentiation: Basic Concepts

Find dydx\frac { d y } { d x }dxdy​ if y=u3y = \sqrt [ 3 ] { u }y=3u​ and u=x4−3x3−7u = x ^ { 4 } - 3 x ^ { 3 } - 7u=x4−3x3−7

An equation for the tangent line to the curve y=(x2+x−1) 3y = \left( x ^ { 2 } + x - 1 \right) ^ { 3 }y=(x2+x−1) 3 at the point where x = 1 is:

The equation for the tangent line to the curve x2+2xy=y3x ^ { 2 } + 2 x y = y ^ { 3 }x2+2xy=y3 at the point (1, -1) is y = -1.

If f(x)=2−3x2x3+x−1f ( x ) = \frac { 2 - 3 x ^ { 2 } } { x ^ { 3 } + x - 1 }f(x)=x3+x−12−3x2​ , what is f′(x)f ^ { \prime } ( x )f′(x) ?

Differentiate: f(x)=x+9xf ( x ) = \sqrt { x } + \frac { 9 } { \sqrt { x } }f(x)=x​+x​9​

Differentiate: f(x) =x2x−2f ( x ) = \frac { x ^ { 2 } } { x - 2 }f(x) =x−2x2​

Find the equation of the normal line to f(x)=4x3−8x+49f ( x ) = 4 x ^ { 3 } - 8 x + 49f(x)=4x3−8x+49 at the point with x-coordinate -2.

Differentiate: f(x)=x7−3xf ( x ) = \sqrt [ 7 ] { x } - \frac { 3 } { \sqrt { x } }f(x)=7x​−x​3​

The equation of the line tangent to the graph of f(x)=x+3f ( x ) = \sqrt { x } + 3f(x)=x​+3 that passes through (1, 4) is y = 2x + 3.

Find the equation of the tangent line to the graph of f(x) =x2+3f ( x ) = x ^ { 2 } + 3f(x) =x2+3 at the point (3, 12) .

If f(x)=x2−xf ( x ) = x \sqrt { 2 - x }f(x)=x2−x​ , then f′′(x)=0f ^ { \prime \prime } ( x ) = 0f′′(x)=0 at x = 0 and x = 2.

The equation of the line tangent to the graph of f(x)=x+2f ( x ) = \sqrt { x } + 2f(x)=x​+2 that passes through (25, 7) is y = 2x + 2.

Find the equation of the line that is tangent to the curve f(x)=5+3x2−x5f ( x ) = 5 + 3 x ^ { 2 } - x ^ { 5 }f(x)=5+3x2−x5 at the point (1, 7).

Differentiate: f(x) =x2x−2f ( x ) = \frac { x ^ { 2 } } { x - 2 }f(x) =x−2x2​

Find the equation of the tangent line to the given curve at the specified point. x3y2−3xy=4x+y−19x ^ { 3 } y ^ { 2 } - 3 x y = 4 x + y - 19x3y2−3xy=4x+y−19 ; (0, 19)

Find the equation of the tangent line to the curve f(x)=4x−xf ( x ) = \frac { 4 } { x } - xf(x)=x4​−x at the point where x = 1.

Find $\frac { d y } { d x }$ if $y = \sqrt [ 3 ] { u }$ and $u = x ^ { 4 } - 3 x ^ { 3 } - 7$

An equation for the tangent line to the curve $y = \left( x ^ { 2 } + x - 1 \right) ^ { 3 }$ at the point where x = 1 is:

The equation for the tangent line to the curve $x ^ { 2 } + 2 x y = y ^ { 3 }$ at the point (1, -1) is y = -1.

If $f ( x ) = \frac { 2 - 3 x ^ { 2 } } { x ^ { 3 } + x - 1 }$ , what is $f ^ { \prime } ( x )$ ?

Differentiate: $f ( x ) = \sqrt { x } + \frac { 9 } { \sqrt { x } }$

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

Find the equation of the normal line to $f ( x ) = 4 x ^ { 3 } - 8 x + 49$ at the point with x-coordinate -2.

Differentiate: $f ( x ) = \sqrt [ 7 ] { x } - \frac { 3 } { \sqrt { x } }$

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 3$ that passes through (1, 4) is y = 2x + 3.

Find the equation of the tangent line to the graph of $f ( x ) = x ^ { 2 } + 3$ at the point (3, 12) .

If $f ( x ) = x \sqrt { 2 - x }$ , then $f ^ { \prime \prime } ( x ) = 0$ at x = 0 and x = 2.

The equation of the line tangent to the graph of $f ( x ) = \sqrt { x } + 2$ that passes through (25, 7) is y = 2x + 2.

Find the equation of the line that is tangent to the curve $f ( x ) = 5 + 3 x ^ { 2 } - x ^ { 5 }$ at the point (1, 7).

Differentiate: $f ( x ) = \frac { x ^ { 2 } } { x - 2 }$

Find the equation of the tangent line to the given curve at the specified point. $x ^ { 3 } y ^ { 2 } - 3 x y = 4 x + y - 19$ ; (0, 19)

Find the equation of the tangent line to the curve $f ( x ) = \frac { 4 } { x } - x$ at the point where x = 1.