Question 1

Differentiate $y=25-\frac{7}{x^{4}}$
 .

Accepted Answer

A)  $\frac{28}{x^{5}}$ 
B)  $25-\frac{2.8}{x^{5}}$ 
C)  $\frac{7}{4 x^{3}}$ 
D)  $25-\frac{7}{4 x^{3}}$ 
E) None of the above 
A)  $\frac{28}{x^{5}}$ 
B)  $25-\frac{2.8}{x^{5}}$ 
C)  $\frac{7}{4 x^{3}}$ 
D)  $25-\frac{7}{4 x^{3}}$ 
E) None of the above

Question 2

If $f ( x ) = 2 \sin ^ { 2 } x$ and $g ( x ) = - 3 \cos ^ { 2 } x$ , is $f ^ { \prime } ( x ) = g ^ { \prime } ( x )$ ?

Accepted Answer

A) True 
 B)False

Question 3

Let f(x)and g(x)be two functions.Values of f(x), f '(x), g(x), and g'(x)for x = 0, 1, and 2 are given in the table below.Use the information in the table to find $H ^ { \prime } ( 0 )$ if $H ( x ) =$ e g(x)  
$$+ \pi$$ x.
$\begin{array} { c c c c c } 
\boldsymbol { x } & f ( x ) & f ^ { \prime } ( x ) & g ( x ) & g ^ { \prime } ( x ) \
0 & 1 & - 1 & 2 & 5 \
1 & - 1 & 2 & 4 & 0 \
2 & 7 & 3 & 11 & 0.5
\end{array}$

Accepted Answer

A)  $5 e ^ { 2 }$ 
B)  $5 e + \pi$ 
C)  $e ^ { 2 } + \pi$ 
D)  $5 e ^ { 2 } + \pi$ 
A)  $5 e ^ { 2 }$ 
B)  $5 e + \pi$ 
C)  $e ^ { 2 } + \pi$ 
D)  $5 e ^ { 2 } + \pi$

Question 4

Find the derivative of $f ( x ) = \frac { 1 + x } { 5 + 6 x + 7 x ^ { 2 } }$ .

Accepted Answer

A)  $- \frac { 1 + 7 x } { \left( 5 + 6 x + 7 x ^ { 2 } \right) ^ { 2 } }$ 
B)  $- \frac { 1 + 14 x + 7 x ^ { 2 } } { \left( 5 + 6 x + 7 x ^ { 2 } \right) ^ { 2 } }$ 
C)  $- \frac { 11 + 26 x + 21 x ^ { 2 } } { \left( 5 + 6 x + 7 x ^ { 2 } \right) ^ { 2 } }$ 
D)  $\frac { 1 } { 6 + 14 x }$ 
A)  $- \frac { 1 + 7 x } { \left( 5 + 6 x + 7 x ^ { 2 } \right) ^ { 2 } }$ 
B)  $- \frac { 1 + 14 x + 7 x ^ { 2 } } { \left( 5 + 6 x + 7 x ^ { 2 } \right) ^ { 2 } }$ 
C)  $- \frac { 11 + 26 x + 21 x ^ { 2 } } { \left( 5 + 6 x + 7 x ^ { 2 } \right) ^ { 2 } }$ 
D)  $\frac { 1 } { 6 + 14 x }$

Question 5

Show that the area of the triangle formed by the x-axis, the y-axis and the tangent line to the hyperbola $x y = 4$ is a constant for any point (a, b).What is the constant?

Accepted Answer

The equation of the tangent li

Question 6

On what intervals is the polynomial $p ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 70$ increasing? Decreasing?

Accepted Answer

Increasing

Question 7

If P dollars are invested at an annual rate of r%, then in t years this investment grows to F dollars, where $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$ .If you solve this equation for P and hold F and r constant, what will the sign of $\frac { d P } { d t }$ be?

Accepted Answer

A) positive 
B) negative 
A) positive 
B) negative

Question 8

The curve graphed below has equation $2 y ^ { 3 } + y ^ { 2 } - y ^ { 5 } = x ^ { 4 } - 2 x ^ { 3 } + x ^ { 2 }$ .It is called the "bouncing wagon curve."
a)Use implicit differentiation to find $\frac { d y } { d x }$ .
b)Find the slope of the tangent line at the point (1, -1).

Accepted Answer

The answer of The curve graphed below has equation \(2...

Question 9

Find a function F(x)such that $F ^ { \prime } ( x ) = x ^ { 4 } + \sin x$ and F(0)= 5.

Accepted Answer

The answer of Find a function F(x)such that \(F ^...

Question 10

Find $\frac { d } { d \theta } \left( \frac { \cos \theta } { \sin \theta } \right)$ .

Accepted Answer

A)  $\frac { 1 } { \sin ^ { 2 } \theta }$ 
B)  $- \frac { 1 } { \sin ^ { 2 } \theta }$ 
C)  $- \frac { \sin \theta } { \cos \theta }$ 
D)  $1 - \frac { \cos ^ { 2 } \theta } { \sin ^ { 2 } \theta }$ 
A)  $\frac { 1 } { \sin ^ { 2 } \theta }$ 
B)  $- \frac { 1 } { \sin ^ { 2 } \theta }$ 
C)  $- \frac { \sin \theta } { \cos \theta }$ 
D)  $1 - \frac { \cos ^ { 2 } \theta } { \sin ^ { 2 } \theta }$

Question 11

If $g ( x ) = e ^ { ( 4 x + \sin x ) } \text {, then } g ^ { \prime } ( x ) = ( 4 + \cos x ) \cdot e ^ { ( 4 x + \sin x ) }$

Accepted Answer

A) True 
 B)False

Question 12

Find the second derivative of $f ( x ) = e ^ { - x ^ { 2 } }$ at x = 1.5.Round to three decimal places.

Accepted Answer

A) 0.738 
B) -0.105 
C) 0.316 
D) -1.159 
A) 0.738 
B) -0.105 
C) 0.316 
D) -1.159

Question 13

Differentiate $\frac { 5 x ^ { 2 } } { x ^ { 3 } + 1 }$ .

Accepted Answer

A)  $\frac { 25 x \left( 2 - x ^ { 3 } \right) } { \left( x ^ { 3 } + 1 \right) ^ { 2 } }$ 
B)  $\frac { 5 x \left( x ^ { 3 } - 1 \right) } { \left( x ^ { 3 } + 1 \right) ^ { 2 } }$ 
C)  $\frac { 5 x \left( 2 - x ^ { 3 } \right) } { \left( x ^ { 3 } + 1 \right) ^ { 2 } }$ 
D)  $\frac { 10 x } { 3 x ^ { 2 } + 1 }$ 
A)  $\frac { 25 x \left( 2 - x ^ { 3 } \right) } { \left( x ^ { 3 } + 1 \right) ^ { 2 } }$ 
B)  $\frac { 5 x \left( x ^ { 3 } - 1 \right) } { \left( x ^ { 3 } + 1 \right) ^ { 2 } }$ 
C)  $\frac { 5 x \left( 2 - x ^ { 3 } \right) } { \left( x ^ { 3 } + 1 \right) ^ { 2 } }$ 
D)  $\frac { 10 x } { 3 x ^ { 2 } + 1 }$

Question 14

Consider the function $f(x)=2 x^{4}-4 x^{3}+4$
 .Is f increasing or decreasing at the point x = 0.5?

Accepted Answer

The answer of Consider the function $f(x)=2 x^{4}-4 x^{3}+4$
 .Is...

Question 15

Differentiate $y = \frac { \cosh ( 4 x ) } { \cosh ^ { 2 } ( x ) }$ .(Do not simplify)

Accepted Answer

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

Question 16

Find the derivative of $y = \sinh ( \sin ( 7 x ) )$ .

Accepted Answer

A)  $- 7 \cosh ( \cos ( 7 x ) )$ 
B)  $7 \cosh ( \cos ( 7 x ) )$ 
C)  $- 7 \cos ( 7 x ) \cdot \cosh ( \sin ( 7 x ) )$ 
D)  $7 \cos ( 7 x ) \cdot \cosh ( \sin ( 7 x ) )$ 
A)  $- 7 \cosh ( \cos ( 7 x ) )$ 
B)  $7 \cosh ( \cos ( 7 x ) )$ 
C)  $- 7 \cos ( 7 x ) \cdot \cosh ( \sin ( 7 x ) )$ 
D)  $7 \cos ( 7 x ) \cdot \cosh ( \sin ( 7 x ) )$

Question 17

Find the slope of the curve $y = e ^ { 05 x ^ { 3 } }$ at x = 2 to the nearest whole number.

Accepted Answer

The answer of Find the slope of the curve \(y...

Question 18

Consider the equation $\ln | y / ( 1 - y ) | = 0.34 x + \ln 4$ .Assuming this implicitly defines y as a function of x, find $\frac { d y } { d x }$ .

Accepted Answer

A)  $0.34 y ( 1 - y )$ 
B)  $0.34 y / ( 1 - y )$ 
C)  $0.34 y$ 
D)  $- 0.34 y ( 1 - y )$ 
A)  $0.34 y ( 1 - y )$ 
B)  $0.34 y / ( 1 - y )$ 
C)  $0.34 y$ 
D)  $- 0.34 y ( 1 - y )$

Question 19

When hyperventilating, a person breathes in and out extremely rapidly.A spirogram is a machine that draws a graph of the volume of air in a person's lungs as a function of time.During hyperventilation, the spirogram trace may be represented by $V = 3 - 0.06 \cos ( 200 \pi t )$ , where V is the volume of the lungs in liters and t is the time in minutes.What is the maximum volume of air in the lungs?

Accepted Answer

The answer of When hyperventilating, a person breathes in and...

Question 20

If $f ( x ) = ( \ln 6 ) ^ { x }$ , then f '0(6)is

Accepted Answer

A) 6 ln 6 
B)  $( \ln 6 ) ^ { 3 }$ 
C) approximately 19.297 
D) approximately 1.792 
A) 6 ln 6 
B)  $( \ln 6 ) ^ { 3 }$ 
C) approximately 19.297 
D) approximately 1.792

Differentiate $y=25-\frac{7}{x^{4}}$ .

If $f ( x ) = 2 \sin ^ { 2 } x$ and $g ( x ) = - 3 \cos ^ { 2 } x$ , is $f ^ { \prime } ( x ) = g ^ { \prime } ( x )$ ?

Let f(x)and g(x)be two functions.Values of f(x), f '(x), g(x), and g'(x)for x = 0, 1, and 2 are given in the table below.Use the information in the table to find $H ^ { \prime } ( 0 )$ if $H ( x ) =$ e ^g⁽^x⁾ $+ \pi$ x. f(x) (x) g(x) (x) 0 1 -1 2 5 1 -1 2 4 0 2 7 3 11 0.5

Find the derivative of $f ( x ) = \frac { 1 + x } { 5 + 6 x + 7 x ^ { 2 } }$ .

Show that the area of the triangle formed by the x-axis, the y-axis and the tangent line to the hyperbola $x y = 4$ is a constant for any point (a, b).What is the constant?

On what intervals is the polynomial $p ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 70$ increasing? Decreasing?

If P dollars are invested at an annual rate of r%, then in t years this investment grows to F dollars, where $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$ .If you solve this equation for P and hold F and r constant, what will the sign of $\frac { d P } { d t }$ be?

Find a function F(x)such that $F ^ { \prime } ( x ) = x ^ { 4 } + \sin x$ and F(0)= 5.

Find $\frac { d } { d \theta } \left( \frac { \cos \theta } { \sin \theta } \right)$ .

If $g ( x ) = e ^ { ( 4 x + \sin x ) } \text {, then } g ^ { \prime } ( x ) = ( 4 + \cos x ) \cdot e ^ { ( 4 x + \sin x ) }$

Find the second derivative of $f ( x ) = e ^ { - x ^ { 2 } }$ at x = 1.5.Round to three decimal places.

Differentiate $\frac { 5 x ^ { 2 } } { x ^ { 3 } + 1 }$ .

Consider the function $f(x)=2 x^{4}-4 x^{3}+4$ .Is f increasing or decreasing at the point x = 0.5?

Differentiate $y = \frac { \cosh ( 4 x ) } { \cosh ^ { 2 } ( x ) }$ .(Do not simplify)

Find the derivative of $y = \sinh ( \sin ( 7 x ) )$ .

Find the slope of the curve $y = e ^ { 05 x ^ { 3 } }$ at x = 2 to the nearest whole number.

Consider the equation $\ln | y / ( 1 - y ) | = 0.34 x + \ln 4$ .Assuming this implicitly defines y as a function of x, find $\frac { d y } { d x }$ .

If $f ( x ) = ( \ln 6 ) ^ { x }$ , then f '0(6)is

A Library of Functions

Key Concept: the Derivative

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 3: Short-Cuts to Differentiation

Differentiate y=25−7x4y=25-\frac{7}{x^{4}}y=25−x47​ .

If f(x)=2sin⁡2xf ( x ) = 2 \sin ^ { 2 } xf(x)=2sin2x and g(x)=−3cos⁡2xg ( x ) = - 3 \cos ^ { 2 } xg(x)=−3cos2x , is f′(x)=g′(x)f ^ { \prime } ( x ) = g ^ { \prime } ( x )f′(x)=g′(x) ?

Let f(x)and g(x)be two functions.Values of f(x), f '(x), g(x), and g'(x)for x = 0, 1, and 2 are given in the table below.Use the information in the table to find H′(0)H ^ { \prime } ( 0 )H′(0) if H(x)=H ( x ) =H(x)= e g(x) +π+ \pi+π x. f(x) (x) g(x) (x) 0 1 -1 2 5 1 -1 2 4 0 2 7 3 11 0.5

Find the derivative of f(x)=1+x5+6x+7x2f ( x ) = \frac { 1 + x } { 5 + 6 x + 7 x ^ { 2 } }f(x)=5+6x+7x21+x​ .

Show that the area of the triangle formed by the x-axis, the y-axis and the tangent line to the hyperbola xy=4x y = 4xy=4 is a constant for any point (a, b).What is the constant?

On what intervals is the polynomial p(x)=2x3−15x2+36x+70p ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 70p(x)=2x3−15x2+36x+70 increasing? Decreasing?

Find a function F(x)such that F′(x)=x4+sin⁡xF ^ { \prime } ( x ) = x ^ { 4 } + \sin xF′(x)=x4+sinx and F(0)= 5.

Find ddθ(cos⁡θsin⁡θ)\frac { d } { d \theta } \left( \frac { \cos \theta } { \sin \theta } \right)dθd​(sinθcosθ​) .

If g(x)=e(4x+sin⁡x), then g′(x)=(4+cos⁡x)⋅e(4x+sin⁡x)g ( x ) = e ^ { ( 4 x + \sin x ) } \text {, then } g ^ { \prime } ( x ) = ( 4 + \cos x ) \cdot e ^ { ( 4 x + \sin x ) }g(x)=e(4x+sinx), then g′(x)=(4+cosx)⋅e(4x+sinx)

Find the second derivative of f(x)=e−x2f ( x ) = e ^ { - x ^ { 2 } }f(x)=e−x2 at x = 1.5.Round to three decimal places.

Differentiate 5x2x3+1\frac { 5 x ^ { 2 } } { x ^ { 3 } + 1 }x3+15x2​ .

Consider the function f(x)=2x4−4x3+4f(x)=2 x^{4}-4 x^{3}+4f(x)=2x4−4x3+4 .Is f increasing or decreasing at the point x = 0.5?

Differentiate y=cosh⁡(4x)cosh⁡2(x)y = \frac { \cosh ( 4 x ) } { \cosh ^ { 2 } ( x ) }y=cosh2(x)cosh(4x)​ .(Do not simplify)

Find the derivative of y=sinh⁡(sin⁡(7x))y = \sinh ( \sin ( 7 x ) )y=sinh(sin(7x)) .

Find the slope of the curve y=e05x3y = e ^ { 05 x ^ { 3 } }y=e05x3 at x = 2 to the nearest whole number.

Consider the equation ln⁡∣y/(1−y)∣=0.34x+ln⁡4\ln | y / ( 1 - y ) | = 0.34 x + \ln 4ln∣y/(1−y)∣=0.34x+ln4 .Assuming this implicitly defines y as a function of x, find dydx\frac { d y } { d x }dxdy​ .

If f(x)=(ln⁡6)xf ( x ) = ( \ln 6 ) ^ { x }f(x)=(ln6)x , then f '0(6)is

A Library of Functions

Key Concept: the Derivative

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Differentiate $y=25-\frac{7}{x^{4}}$ .

If $f ( x ) = 2 \sin ^ { 2 } x$ and $g ( x ) = - 3 \cos ^ { 2 } x$ , is $f ^ { \prime } ( x ) = g ^ { \prime } ( x )$ ?

Let f(x)and g(x)be two functions.Values of f(x), f '(x), g(x), and g'(x)for x = 0, 1, and 2 are given in the table below.Use the information in the table to find $H ^ { \prime } ( 0 )$ if $H ( x ) =$ e ^g⁽^x⁾ $+ \pi$ x. f(x) (x) g(x) (x) 0 1 -1 2 5 1 -1 2 4 0 2 7 3 11 0.5

Find the derivative of $f ( x ) = \frac { 1 + x } { 5 + 6 x + 7 x ^ { 2 } }$ .

Show that the area of the triangle formed by the x-axis, the y-axis and the tangent line to the hyperbola $x y = 4$ is a constant for any point (a, b).What is the constant?

On what intervals is the polynomial $p ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 70$ increasing? Decreasing?

Find a function F(x)such that $F ^ { \prime } ( x ) = x ^ { 4 } + \sin x$ and F(0)= 5.

Find $\frac { d } { d \theta } \left( \frac { \cos \theta } { \sin \theta } \right)$ .

If $g ( x ) = e ^ { ( 4 x + \sin x ) } \text {, then } g ^ { \prime } ( x ) = ( 4 + \cos x ) \cdot e ^ { ( 4 x + \sin x ) }$

Find the second derivative of $f ( x ) = e ^ { - x ^ { 2 } }$ at x = 1.5.Round to three decimal places.

Differentiate $\frac { 5 x ^ { 2 } } { x ^ { 3 } + 1 }$ .

Consider the function $f(x)=2 x^{4}-4 x^{3}+4$ .Is f increasing or decreasing at the point x = 0.5?

Differentiate $y = \frac { \cosh ( 4 x ) } { \cosh ^ { 2 } ( x ) }$ .(Do not simplify)

Find the derivative of $y = \sinh ( \sin ( 7 x ) )$ .

Find the slope of the curve $y = e ^ { 05 x ^ { 3 } }$ at x = 2 to the nearest whole number.

Consider the equation $\ln | y / ( 1 - y ) | = 0.34 x + \ln 4$ .Assuming this implicitly defines y as a function of x, find $\frac { d y } { d x }$ .

If $f ( x ) = ( \ln 6 ) ^ { x }$ , then f '0(6)is