Question 1

Let a be a positive constant .The equation $a ^ { x } = 1 + x$ has the solution x = 0, for all a.If a = 2.52, are there any solutions for x > 0?

Accepted Answer

A) True 
 B)False  True

Question 2

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 $V = 3 - 0.04 \cos ( 210 \pi t )$ , where V is the volume of the lungs in liters and t is the time in minutes.What is the period of this function in seconds?

Accepted Answer

0.571

Question 3

Find the equation of the tangent line to the curve given by f(x)= x sin x at the point $x = 3 \pi / 4$ .

Accepted Answer

A)  $y + \frac { \pi \sqrt { 2 } } { 8 } = \frac { \sqrt { 2 } } { 2 } \left( \frac { 3 \pi } { 4 } - 1 \right) \left( x - \frac { 3 \pi } { 4 } \right)$ 
B)  $y - \frac { \pi \sqrt { 2 } } { 8 } = \frac { \sqrt { 2 } } { 2 } \left( \frac { 3 \pi } { 4 } + 1 \right) \left( x - \frac { 3 \pi } { 4 } \right)$ 
C)  $y - \frac { \pi \sqrt { 2 } } { 8 } = - \frac { \sqrt { 2 } } { 2 } \left( \frac { 3 \pi } { 4 } + 1 \right) \left( x - \frac { 3 \pi } { 4 } \right)$ 
D)  $y - \frac { \pi \sqrt { 2 } } { 8 } = - \frac { \sqrt { 2 } } { 2 } \left( \frac { 3 \pi } { 4 } - 1 \right) \left( x - \frac { 3 \pi } { 4 } \right)$ 
A)  $y + \frac { \pi \sqrt { 2 } } { 8 } = \frac { \sqrt { 2 } } { 2 } \left( \frac { 3 \pi } { 4 } - 1 \right) \left( x - \frac { 3 \pi } { 4 } \right)$ 
B)  $y - \frac { \pi \sqrt { 2 } } { 8 } = \frac { \sqrt { 2 } } { 2 } \left( \frac { 3 \pi } { 4 } + 1 \right) \left( x - \frac { 3 \pi } { 4 } \right)$ 
C)  $y - \frac { \pi \sqrt { 2 } } { 8 } = - \frac { \sqrt { 2 } } { 2 } \left( \frac { 3 \pi } { 4 } + 1 \right) \left( x - \frac { 3 \pi } { 4 } \right)$ 
D)  $y - \frac { \pi \sqrt { 2 } } { 8 } = - \frac { \sqrt { 2 } } { 2 } \left( \frac { 3 \pi } { 4 } - 1 \right) \left( x - \frac { 3 \pi } { 4 } \right)$ D

Question 4

Find $y^t$ when $y=\frac{a}{b} \sqrt{x^{b}}$
 .

Accepted Answer

A)  $\frac{a}{2} \sqrt{x^{b-2}}$ 
B)  $\frac{a \sqrt{x^{3}}}{2 x}$ 
C)  $\frac{a x^{-b / 2-1}}{2}$ 
D) all of the above 
E) none of the above 
A)  $\frac{a}{2} \sqrt{x^{b-2}}$ 
B)  $\frac{a \sqrt{x^{3}}}{2 x}$ 
C)  $\frac{a x^{-b / 2-1}}{2}$ 
D) all of the above 
E) none of the above

Question 5

Find the derivative of $g ( x ) = \sqrt { e ^ { x } + e ^ { x ^ { 5 } } }$ .

Accepted Answer

A)  $\sqrt { e ^ { x } + 5 x ^ { 4 } e ^ { x ^ { 5 } } }$ 
B)  $\frac { 1 } { 2 \sqrt { e ^ { x } + e ^ { x ^ { 5 } } } }$ 
C)  $\frac { e ^ { x } + 5 x ^ { 4 } e ^ { x ^ { 5 } } } { 2 \sqrt { e ^ { x } + e ^ { x ^ { 5 } } } }$ 
D)  $\frac { e ^ { x } + x ^ { 5 } e ^ { x ^ { 5 } - 1 } } { 2 \sqrt { e ^ { x } + e ^ { x ^ { 5 } } } }$ 
A)  $\sqrt { e ^ { x } + 5 x ^ { 4 } e ^ { x ^ { 5 } } }$ 
B)  $\frac { 1 } { 2 \sqrt { e ^ { x } + e ^ { x ^ { 5 } } } }$ 
C)  $\frac { e ^ { x } + 5 x ^ { 4 } e ^ { x ^ { 5 } } } { 2 \sqrt { e ^ { x } + e ^ { x ^ { 5 } } } }$ 
D)  $\frac { e ^ { x } + x ^ { 5 } e ^ { x ^ { 5 } - 1 } } { 2 \sqrt { e ^ { x } + e ^ { x ^ { 5 } } } }$

Question 6

Let $f ( x ) = x ^ { 4 } + 4 x + 2$ .How many zeros does f have?

Accepted Answer

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

Question 7

Given $f(x)=3 x^{2}-x$
 and $g(x)=x^{3}+5 x^{2}-4$
 , find $\frac{d}{d x}[g(x)-3 f(x)]$
 .

Accepted Answer

The answer of Given $f(x)=3 x^{2}-x$
 and $g(x)=x^{3}+5 x^{2}-4$
 ,...

Question 8

The derivative of $\left( x ^ { 2 } + 1 \right) \arctan x$ is 1.

Accepted Answer

A) True 
 B)False

Question 9

Find $\frac { d } { d x } \left( \ln \left( x ^ { 3 } e ^ { x } \right) \right)$ .

Accepted Answer

A)  $\frac { 3 } { x } + 1$ 
B)  $\frac { 3 } { x }$ 
C)  $\frac { 1 } { x ^ { 3 } e ^ { x } }$ 
D)  $\frac { 3 x + e ^ { x } } { x ^ { 3 } e ^ { x } }$ 
A)  $\frac { 3 } { x } + 1$ 
B)  $\frac { 3 } { x }$ 
C)  $\frac { 1 } { x ^ { 3 } e ^ { x } }$ 
D)  $\frac { 3 x + e ^ { x } } { x ^ { 3 } e ^ { x } }$

Question 10

Determine the derivative rule for finding the derivative of the reciprocal function: $\frac { 1 } { g ( x ) }$

Accepted Answer

A)  $\frac { - g ^ { \prime } ( x ) } { [ g ( x ) ] ^ { 2 } }$ 
B)  $\frac { 1 } { g ^ { \prime } ( x ) }$ 
C)  $[ g ( x ) ] ^ { - 1 }$ 
D)  $\frac { g ^ { \prime } ( x ) } { 2 g ( x ) }$ 
A)  $\frac { - g ^ { \prime } ( x ) } { [ g ( x ) ] ^ { 2 } }$ 
B)  $\frac { 1 } { g ^ { \prime } ( x ) }$ 
C)  $[ g ( x ) ] ^ { - 1 }$ 
D)  $\frac { g ^ { \prime } ( x ) } { 2 g ( x ) }$

Question 11

Find the derivative of $g ( x ) = \cos ( \sin ( 2 x ) )$ .

Accepted Answer

A)  $- 2 \cos ( 2 x ) \sin ( \sin ( 2 x ) )$ 
B)  $2 \cos ( 2 x ) \sin ( \sin ( 2 x ) )$ 
C)  $- 2 \sin ( \sin ( 2 x ) )$ 
D)  $4 \sin ( \sin ( 2 x ) )$ 
A)  $- 2 \cos ( 2 x ) \sin ( \sin ( 2 x ) )$ 
B)  $2 \cos ( 2 x ) \sin ( \sin ( 2 x ) )$ 
C)  $- 2 \sin ( \sin ( 2 x ) )$ 
D)  $4 \sin ( \sin ( 2 x ) )$

Question 12

Differentiate $f ( w ) = \sqrt { w ^ { 2 } + 8 }$ .

Accepted Answer

A)  $\frac { 8 w } { \sqrt { w ^ { 2 } + 8 } }$ 
B)  $\sqrt { 2 w }$ 
C)  $\frac { w } { \sqrt { w ^ { 2 } + 8 } }$ 
D)  $\frac { 1 } { 2 \sqrt { w ^ { 2 } + 8 } }$ 
A)  $\frac { 8 w } { \sqrt { w ^ { 2 } + 8 } }$ 
B)  $\sqrt { 2 w }$ 
C)  $\frac { w } { \sqrt { w ^ { 2 } + 8 } }$ 
D)  $\frac { 1 } { 2 \sqrt { w ^ { 2 } + 8 } }$

Question 13

Find the equation of the tangent line to the graph of $f ( \alpha ) = \frac { \tan \alpha } { \alpha }$ at the point $\alpha = \pi / 4$ .

Accepted Answer

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

Question 14

If $f ( x ) = \frac { x ^ { 2 } + 2 } { x ^ { 2 } - 2 }$ , then $f ^ { \prime } ( x ) = - \frac { 4 x } { \left( x ^ { 2 } - 2 \right) ^ { 2 } }$ .

Accepted Answer

A) True 
 B)False

Question 15

The derivative of $6 ^ { e } - \frac { 6 } { x } + \sqrt { 1 - x }$ is $\frac { 6 } { x ^ { 2 } } + \frac { 1 } { 2 \sqrt { 1 - x } }$ .

Accepted Answer

A) True 
 B)False

Question 16

Given $f ( x ) = \frac { x ^ { 3 } } { 2 x + 1 }$ , $g ( x ) = \frac { x ^ { 2 } + 2 } { 3 x ^ { 2 } }$ , and h(x)= f(x)g(x), find h'(1).Round to 2 decimal places.

Accepted Answer

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

Question 17

Find a formula for the slope of the tangent line to $y=(x-9)^{2}$

Accepted Answer

A) 2x - 18 
B) 2x - 9 
C) 2(x - 9)3 
D) (x - 9)3/3 
E) none of the above 
A) 2x - 18 
B) 2x - 9 
C) 2(x - 9)3 
D) (x - 9)3/3 
E) none of the above

Question 18

Suppose that $x y ^ { 2 } + \sin y + x ^ { 3 } = 1$ .Find $\frac { d y } { d x }$ .

Accepted Answer

A)  $- \frac { 1 } { 2 x y + \cos y }$ 
B)  $- \frac { 3 x ^ { 2 } } { 2 x y + \cos y }$ 
C)  $- \frac { 3 x ^ { 2 } + y ^ { 2 } } { 2 x y + \cos y }$ 
D)  $- \frac { y ^ { 2 } } { 2 x y + \cos y }$ 
A)  $- \frac { 1 } { 2 x y + \cos y }$ 
B)  $- \frac { 3 x ^ { 2 } } { 2 x y + \cos y }$ 
C)  $- \frac { 3 x ^ { 2 } + y ^ { 2 } } { 2 x y + \cos y }$ 
D)  $- \frac { y ^ { 2 } } { 2 x y + \cos y }$

Question 19

The derivative of $\arctan \left( \frac { 67 } { x } \right)$ is ________________________.Do not simplify.
.

Accepted Answer

The answer of The derivative of \(\arctan \left( \frac {...

Question 20

Differentiate $r ( t ) = \frac { t } { 3 + e ^ { t } }$ .

Accepted Answer

The answer of Differentiate \(r ( t ) = \frac...

Let a be a positive constant .The equation $a ^ { x } = 1 + x$ has the solution x = 0, for all a.If a = 2.52, are there any solutions for x > 0?

Find the equation of the tangent line to the curve given by f(x)= x sin x at the point $x = 3 \pi / 4$ .

Find $y^t$ when $y=\frac{a}{b} \sqrt{x^{b}}$ .

Find the derivative of $g ( x ) = \sqrt { e ^ { x } + e ^ { x ^ { 5 } } }$ .

Let $f ( x ) = x ^ { 4 } + 4 x + 2$ .How many zeros does f have?

Given $f(x)=3 x^{2}-x$ and $g(x)=x^{3}+5 x^{2}-4$ , find $\frac{d}{d x}[g(x)-3 f(x)]$ .

The derivative of $\left( x ^ { 2 } + 1 \right) \arctan x$ is 1.

Find $\frac { d } { d x } \left( \ln \left( x ^ { 3 } e ^ { x } \right) \right)$ .

Determine the derivative rule for finding the derivative of the reciprocal function: $\frac { 1 } { g ( x ) }$

Find the derivative of $g ( x ) = \cos ( \sin ( 2 x ) )$ .

Differentiate $f ( w ) = \sqrt { w ^ { 2 } + 8 }$ .

Find the equation of the tangent line to the graph of $f ( \alpha ) = \frac { \tan \alpha } { \alpha }$ at the point $\alpha = \pi / 4$ .

If $f ( x ) = \frac { x ^ { 2 } + 2 } { x ^ { 2 } - 2 }$ , then $f ^ { \prime } ( x ) = - \frac { 4 x } { \left( x ^ { 2 } - 2 \right) ^ { 2 } }$ .

The derivative of $6 ^ { e } - \frac { 6 } { x } + \sqrt { 1 - x }$ is $\frac { 6 } { x ^ { 2 } } + \frac { 1 } { 2 \sqrt { 1 - x } }$ .

Given $f ( x ) = \frac { x ^ { 3 } } { 2 x + 1 }$ , $g ( x ) = \frac { x ^ { 2 } + 2 } { 3 x ^ { 2 } }$ , and h(x)= f(x)g(x), find h'(1).Round to 2 decimal places.

Find a formula for the slope of the tangent line to $y=(x-9)^{2}$

Suppose that $x y ^ { 2 } + \sin y + x ^ { 3 } = 1$ .Find $\frac { d y } { d x }$ .

The derivative of $\arctan \left( \frac { 67 } { x } \right)$ is ________________________.Do not simplify. .

Differentiate $r ( t ) = \frac { t } { 3 + e ^ { t } }$ .

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

Let a be a positive constant .The equation ax=1+xa ^ { x } = 1 + xax=1+x has the solution x = 0, for all a.If a = 2.52, are there any solutions for x > 0?

Find the equation of the tangent line to the curve given by f(x)= x sin x at the point x=3π/4x = 3 \pi / 4x=3π/4 .

Find yty^tyt when y=abxby=\frac{a}{b} \sqrt{x^{b}}y=ba​xb​ .

Find the derivative of g(x)=ex+ex5g ( x ) = \sqrt { e ^ { x } + e ^ { x ^ { 5 } } }g(x)=ex+ex5​ .

Let f(x)=x4+4x+2f ( x ) = x ^ { 4 } + 4 x + 2f(x)=x4+4x+2 .How many zeros does f have?

Given f(x)=3x2−xf(x)=3 x^{2}-xf(x)=3x2−x and g(x)=x3+5x2−4g(x)=x^{3}+5 x^{2}-4g(x)=x3+5x2−4 , find ddx[g(x)−3f(x)]\frac{d}{d x}[g(x)-3 f(x)]dxd​[g(x)−3f(x)] .

The derivative of (x2+1)arctan⁡x\left( x ^ { 2 } + 1 \right) \arctan x(x2+1)arctanx is 1.

Find ddx(ln⁡(x3ex))\frac { d } { d x } \left( \ln \left( x ^ { 3 } e ^ { x } \right) \right)dxd​(ln(x3ex)) .

Determine the derivative rule for finding the derivative of the reciprocal function: 1g(x)\frac { 1 } { g ( x ) }g(x)1​

Find the derivative of g(x)=cos⁡(sin⁡(2x))g ( x ) = \cos ( \sin ( 2 x ) )g(x)=cos(sin(2x)) .

Differentiate f(w)=w2+8f ( w ) = \sqrt { w ^ { 2 } + 8 }f(w)=w2+8​ .

Find the equation of the tangent line to the graph of f(α)=tan⁡ααf ( \alpha ) = \frac { \tan \alpha } { \alpha }f(α)=αtanα​ at the point α=π/4\alpha = \pi / 4α=π/4 .

If f(x)=x2+2x2−2f ( x ) = \frac { x ^ { 2 } + 2 } { x ^ { 2 } - 2 }f(x)=x2−2x2+2​ , then f′(x)=−4x(x2−2)2f ^ { \prime } ( x ) = - \frac { 4 x } { \left( x ^ { 2 } - 2 \right) ^ { 2 } }f′(x)=−(x2−2)24x​ .

The derivative of 6e−6x+1−x6 ^ { e } - \frac { 6 } { x } + \sqrt { 1 - x }6e−x6​+1−x​ is 6x2+121−x\frac { 6 } { x ^ { 2 } } + \frac { 1 } { 2 \sqrt { 1 - x } }x26​+21−x​1​ .

Given f(x)=x32x+1f ( x ) = \frac { x ^ { 3 } } { 2 x + 1 }f(x)=2x+1x3​ , g(x)=x2+23x2g ( x ) = \frac { x ^ { 2 } + 2 } { 3 x ^ { 2 } }g(x)=3x2x2+2​ , and h(x)= f(x)g(x), find h'(1).Round to 2 decimal places.

Find a formula for the slope of the tangent line to y=(x−9)2y=(x-9)^{2}y=(x−9)2

Suppose that xy2+sin⁡y+x3=1x y ^ { 2 } + \sin y + x ^ { 3 } = 1xy2+siny+x3=1 .Find dydx\frac { d y } { d x }dxdy​ .

The derivative of arctan⁡(67x)\arctan \left( \frac { 67 } { x } \right)arctan(x67​) is ________________________.Do not simplify. .

Differentiate r(t)=t3+etr ( t ) = \frac { t } { 3 + e ^ { t } }r(t)=3+ett​ .

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

Let a be a positive constant .The equation $a ^ { x } = 1 + x$ has the solution x = 0, for all a.If a = 2.52, are there any solutions for x > 0?

Find the equation of the tangent line to the curve given by f(x)= x sin x at the point $x = 3 \pi / 4$ .

Find $y^t$ when $y=\frac{a}{b} \sqrt{x^{b}}$ .

Find the derivative of $g ( x ) = \sqrt { e ^ { x } + e ^ { x ^ { 5 } } }$ .

Let $f ( x ) = x ^ { 4 } + 4 x + 2$ .How many zeros does f have?

Given $f(x)=3 x^{2}-x$ and $g(x)=x^{3}+5 x^{2}-4$ , find $\frac{d}{d x}[g(x)-3 f(x)]$ .

The derivative of $\left( x ^ { 2 } + 1 \right) \arctan x$ is 1.

Find $\frac { d } { d x } \left( \ln \left( x ^ { 3 } e ^ { x } \right) \right)$ .

Determine the derivative rule for finding the derivative of the reciprocal function: $\frac { 1 } { g ( x ) }$

Find the derivative of $g ( x ) = \cos ( \sin ( 2 x ) )$ .

Differentiate $f ( w ) = \sqrt { w ^ { 2 } + 8 }$ .

Find the equation of the tangent line to the graph of $f ( \alpha ) = \frac { \tan \alpha } { \alpha }$ at the point $\alpha = \pi / 4$ .

If $f ( x ) = \frac { x ^ { 2 } + 2 } { x ^ { 2 } - 2 }$ , then $f ^ { \prime } ( x ) = - \frac { 4 x } { \left( x ^ { 2 } - 2 \right) ^ { 2 } }$ .

The derivative of $6 ^ { e } - \frac { 6 } { x } + \sqrt { 1 - x }$ is $\frac { 6 } { x ^ { 2 } } + \frac { 1 } { 2 \sqrt { 1 - x } }$ .

Given $f ( x ) = \frac { x ^ { 3 } } { 2 x + 1 }$ , $g ( x ) = \frac { x ^ { 2 } + 2 } { 3 x ^ { 2 } }$ , and h(x)= f(x)g(x), find h'(1).Round to 2 decimal places.

Find a formula for the slope of the tangent line to $y=(x-9)^{2}$

Suppose that $x y ^ { 2 } + \sin y + x ^ { 3 } = 1$ .Find $\frac { d y } { d x }$ .

The derivative of $\arctan \left( \frac { 67 } { x } \right)$ is ________________________.Do not simplify. .

Differentiate $r ( t ) = \frac { t } { 3 + e ^ { t } }$ .