Question 1

If f has a local minimum at x = 2, then what can you say about $f ^ { \prime } ( 2 )$ ? What if you also know that f is differentiable at x = 2?

Accepted Answer

Either @#LAT-DLM& is undefined

Question 2

Sketch labeled graphs of each function $f ( x ) = x ^ { 2 } + 5 x$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f$

Accepted Answer

The answer of Sketch labeled graphs of each function \(f...

Question 3

Find the location and values of any global extrema of $f ( x ) = x - 2 \cos x$ on $\left[ - \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$

Accepted Answer

Global maximum: @#LAT-DLM& /2,

Question 4

Use the derivative $f ^ { \prime } ( x ) = \frac { 2 } { x }$ to find the local extrema and inflection points of $f$

Accepted Answer

No local e

Question 5

Find an equation of a possible function with a local minimum at x = 2 that is continuous but not differentiable at x = 2.

Accepted Answer

A)  $f ( x ) = | x | + 2$ 
B)  $f ( x ) = | x + 2 |$ 
C)  $f ( x ) = | x | - 2$ 
D)  $f ( x ) = | x - 2 |$ 
A)  $f ( x ) = | x | + 2$ 
B)  $f ( x ) = | x + 2 |$ 
C)  $f ( x ) = | x | - 2$ 
D)  $f ( x ) = | x - 2 |$

Question 6

Find the critical points of $f ( x ) = ( 2 x + 1 ) ^ { 3 }$

Accepted Answer

A)  $1 / 2$ 
B)  - $1 / 2$ 
C)  1 
D)  2 
A)  $1 / 2$ 
B)  - $1 / 2$ 
C)  1 
D)  2

Question 7

Find $\lim _ { x \rightarrow 0 } \frac { 2 \cos x - 2 } { \sin 2 x }$

Accepted Answer

A)  1 
B)  2 
C)  -1 
D)  0 
A)  1 
B)  2 
C)  -1 
D)  0

Question 8

Does $f ( x ) = \sqrt { x + 4 }$ satisfy the hypothesis of the Mean Value Theorem on the interval [0, 5]. If it does, then find the exact values of all $c \in ( 0,5 )$ that satisfy the conclusion of the Mean Value Theorem.

Accepted Answer

A)  9/2 
B)  $3 / 4$ 
C)  9/4 
D)  Does not satisfy M.V.T. 
A)  9/2 
B)  $3 / 4$ 
C)  9/4 
D)  Does not satisfy M.V.T.

Question 9

Use the first derivative test to determine the local extrema of $f ( x ) = \frac { ( x - 2 ) ^ { 2 } } { x + 1 }$

Accepted Answer

A)  f has a local maximum at x = 2 and a local minimum at x = 4. 
B)  f has a local maximum at x = 2 and a local minimum at x = -4. 
C)  f has a local maximum at x = -4 and a local minimum at x = 2. 
D)  f has a local maximum at x = 4 and a local minimum at x = 2. 
A)  f has a local maximum at x = 2 and a local minimum at x = 4. 
B)  f has a local maximum at x = 2 and a local minimum at x = -4. 
C)  f has a local maximum at x = -4 and a local minimum at x = 2. 
D)  f has a local maximum at x = 4 and a local minimum at x = 2.

Question 10

Determine the intervals on which $f ( x ) = e ^ { x } ( x - 3 )$ is increasing and decreasing.

Accepted Answer

f increase

Question 11

Sketch the graph of a continuous function, if possible, such that $f  0 \text { on } ( 0 , \infty )$ , and $f ^ { \prime } > 0 \text { on } ( - \infty , 1 / 3 ) \text { and } ( 1 , \infty )$ $f ^ { \prime }  0 \text { on } ( 2 / 3 , \infty )$ and $f ^ { \prime \prime } < 0 \text { on } ( - \infty , 2 / 3 )$

Accepted Answer

For instan

Question 12

Determine the intervals on which $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ is increasing and decreasing.

Accepted Answer

f increases on (-1,

Question 13

Use the derivative $f ^ { \prime } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x$ to find the local extrema and inflection points of $f$

Accepted Answer

f has a local minimum at x = 0

Question 14

Find $\lim _ { x \rightarrow 0 } \frac { x e ^ { x } } { 1 - e ^ { x } }$

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  -1 
A)  0 
B)  1 
C)  2 
D)  -1

Question 15

If a continuous and differentiable function f is equal to - 3 at $x = - 2$ and $x = 2$ , what can you say about $f ^ { \prime }$ on [-2, 2]?

Accepted Answer

There is s

Question 16

Determine the intervals on which $f ( x ) = \ln \left( x ^ { 2 } + 2 \right)$ is increasing and decreasing.

Accepted Answer

f increase

Question 17

Determine the intervals on which $f ( x ) = \sin ^ { 2 } x$ is increasing and decreasing.

Accepted Answer

f increase

Question 18

Sketch the graph of a continuous function, if possible, such that $f  0 \text { on } ( - \infty , - 2 )$ , $f ^ { \prime }  0 \text { on } ( - \infty , - 1 )$ , but $f ^ { \prime \prime } < 0 \text { on } ( - 1 , \infty )$

Accepted Answer

For instan

Question 19

Determine the intervals on which $f ( x ) = \frac { 2 x + 1 } { x ^ { 2 } - 4 }$ is increasing and decreasing.

Accepted Answer

f decrease

Question 20

Find the point(s) on the curve $y = x ^ { 2 }$ that is closest to the point (3, 0).

Accepted Answer

A)  (2, 2) 
B)  (-1, 1) 
C)  (1, 1) 
D)  (3, 3) 
A)  (2, 2) 
B)  (-1, 1) 
C)  (1, 1) 
D)  (3, 3)

If f has a local minimum at x = 2, then what can you say about $f ^ { \prime } ( 2 )$ ? What if you also know that f is differentiable at x = 2?

Sketch labeled graphs of each function $f ( x ) = x ^ { 2 } + 5 x$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f$

Find the location and values of any global extrema of $f ( x ) = x - 2 \cos x$ on $\left[ - \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$

Use the derivative $f ^ { \prime } ( x ) = \frac { 2 } { x }$ to find the local extrema and inflection points of $f$

Find an equation of a possible function with a local minimum at x = 2 that is continuous but not differentiable at x = 2.

Find the critical points of $f ( x ) = ( 2 x + 1 ) ^ { 3 }$

Find $\lim _ { x \rightarrow 0 } \frac { 2 \cos x - 2 } { \sin 2 x }$

Does $f ( x ) = \sqrt { x + 4 }$ satisfy the hypothesis of the Mean Value Theorem on the interval [0, 5]. If it does, then find the exact values of all $c \in ( 0,5 )$ that satisfy the conclusion of the Mean Value Theorem.

Use the first derivative test to determine the local extrema of $f ( x ) = \frac { ( x - 2 ) ^ { 2 } } { x + 1 }$

Determine the intervals on which $f ( x ) = e ^ { x } ( x - 3 )$ is increasing and decreasing.

Determine the intervals on which $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ is increasing and decreasing.

Use the derivative $f ^ { \prime } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x$ to find the local extrema and inflection points of $f$

Find $\lim _ { x \rightarrow 0 } \frac { x e ^ { x } } { 1 - e ^ { x } }$

If a continuous and differentiable function f is equal to - 3 at $x = - 2$ and $x = 2$ , what can you say about $f ^ { \prime }$ on [-2, 2]?

Determine the intervals on which $f ( x ) = \ln \left( x ^ { 2 } + 2 \right)$ is increasing and decreasing.

Determine the intervals on which $f ( x ) = \sin ^ { 2 } x$ is increasing and decreasing.

Determine the intervals on which $f ( x ) = \frac { 2 x + 1 } { x ^ { 2 } - 4 }$ is increasing and decreasing.

Find the point(s) on the curve $y = x ^ { 2 }$ that is closest to the point (3, 0).

Limits

Derivatives

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

Exam 3: Applications of the Derivative

If f has a local minimum at x = 2, then what can you say about f′(2)f ^ { \prime } ( 2 )f′(2) ? What if you also know that f is differentiable at x = 2?

Sketch labeled graphs of each function f(x)=x2+5xf ( x ) = x ^ { 2 } + 5 xf(x)=x2+5x by hand. As part of your work make sign charts for the signs, roots and undefined points of f,f′ and ff , f ^ { \prime } \text { and } ff,f′ and f

Find the location and values of any global extrema of f(x)=x−2cos⁡xf ( x ) = x - 2 \cos xf(x)=x−2cosx on [−π4,π2]\left[ - \frac { \pi } { 4 } , \frac { \pi } { 2 } \right][−4π​,2π​]

Use the derivative f′(x)=2xf ^ { \prime } ( x ) = \frac { 2 } { x }f′(x)=x2​ to find the local extrema and inflection points of fff

Find an equation of a possible function with a local minimum at x = 2 that is continuous but not differentiable at x = 2.

Find the critical points of f(x)=(2x+1)3f ( x ) = ( 2 x + 1 ) ^ { 3 }f(x)=(2x+1)3

Find lim⁡x→02cos⁡x−2sin⁡2x\lim _ { x \rightarrow 0 } \frac { 2 \cos x - 2 } { \sin 2 x }limx→0​sin2x2cosx−2​

Does f(x)=x+4f ( x ) = \sqrt { x + 4 }f(x)=x+4​ satisfy the hypothesis of the Mean Value Theorem on the interval [0, 5]. If it does, then find the exact values of all c∈(0,5)c \in ( 0,5 )c∈(0,5) that satisfy the conclusion of the Mean Value Theorem.

Use the first derivative test to determine the local extrema of f(x)=(x−2)2x+1f ( x ) = \frac { ( x - 2 ) ^ { 2 } } { x + 1 }f(x)=x+1(x−2)2​

Determine the intervals on which f(x)=ex(x−3)f ( x ) = e ^ { x } ( x - 3 )f(x)=ex(x−3) is increasing and decreasing.

Determine the intervals on which f(x)=xx2+1f ( x ) = \frac { x } { x ^ { 2 } + 1 }f(x)=x2+1x​ is increasing and decreasing.

Use the derivative f′(x)=x3−4x2+4xf ^ { \prime } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 xf′(x)=x3−4x2+4x to find the local extrema and inflection points of fff

Find lim⁡x→0xex1−ex\lim _ { x \rightarrow 0 } \frac { x e ^ { x } } { 1 - e ^ { x } }limx→0​1−exxex​

If a continuous and differentiable function f is equal to - 3 at x=−2x = - 2x=−2 and x=2x = 2x=2 , what can you say about f′f ^ { \prime }f′ on [-2, 2]?

Determine the intervals on which f(x)=ln⁡(x2+2)f ( x ) = \ln \left( x ^ { 2 } + 2 \right)f(x)=ln(x2+2) is increasing and decreasing.

Determine the intervals on which f(x)=sin⁡2xf ( x ) = \sin ^ { 2 } xf(x)=sin2x is increasing and decreasing.

Determine the intervals on which f(x)=2x+1x2−4f ( x ) = \frac { 2 x + 1 } { x ^ { 2 } - 4 }f(x)=x2−42x+1​ is increasing and decreasing.

Find the point(s) on the curve y=x2y = x ^ { 2 }y=x2 that is closest to the point (3, 0).

Limits

Derivatives

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Functions and Precalculus

Filters

If f has a local minimum at x = 2, then what can you say about $f ^ { \prime } ( 2 )$ ? What if you also know that f is differentiable at x = 2?

Sketch labeled graphs of each function $f ( x ) = x ^ { 2 } + 5 x$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f$

Find the location and values of any global extrema of $f ( x ) = x - 2 \cos x$ on $\left[ - \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$

Use the derivative $f ^ { \prime } ( x ) = \frac { 2 } { x }$ to find the local extrema and inflection points of $f$

Find the critical points of $f ( x ) = ( 2 x + 1 ) ^ { 3 }$

Find $\lim _ { x \rightarrow 0 } \frac { 2 \cos x - 2 } { \sin 2 x }$

Does $f ( x ) = \sqrt { x + 4 }$ satisfy the hypothesis of the Mean Value Theorem on the interval [0, 5]. If it does, then find the exact values of all $c \in ( 0,5 )$ that satisfy the conclusion of the Mean Value Theorem.

Use the first derivative test to determine the local extrema of $f ( x ) = \frac { ( x - 2 ) ^ { 2 } } { x + 1 }$

Determine the intervals on which $f ( x ) = e ^ { x } ( x - 3 )$ is increasing and decreasing.

Determine the intervals on which $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ is increasing and decreasing.

Use the derivative $f ^ { \prime } ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 4 x$ to find the local extrema and inflection points of $f$

Find $\lim _ { x \rightarrow 0 } \frac { x e ^ { x } } { 1 - e ^ { x } }$

If a continuous and differentiable function f is equal to - 3 at $x = - 2$ and $x = 2$ , what can you say about $f ^ { \prime }$ on [-2, 2]?

Determine the intervals on which $f ( x ) = \ln \left( x ^ { 2 } + 2 \right)$ is increasing and decreasing.

Determine the intervals on which $f ( x ) = \sin ^ { 2 } x$ is increasing and decreasing.

Determine the intervals on which $f ( x ) = \frac { 2 x + 1 } { x ^ { 2 } - 4 }$ is increasing and decreasing.

Find the point(s) on the curve $y = x ^ { 2 }$ that is closest to the point (3, 0).