Exam 4: Applications of Derivatives

Find the value or values of $c$ that satisfy the equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$ in the conclusion of the Mean Value Theorem for the function and interval. - $f ( x ) = \tan ^ { - 1 } x , \left[ - \frac { \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right] \quad$ Round to the nearest thousandth.

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Question 1

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Determine all critical points for the function. - $f ( x ) = \frac { 3 x } { x + 2 }$

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Question 2

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Use the following function and a graphing calculator to answer the questions. $f ( x ) = \sqrt { 2 x } + 1.1 \sin x , [ 0,2 \pi ]$ $\text { a). Plot the function over the interval to see its general behavior there. Sketch the graph below. }$ $Use the following function and a graphing calculator to answer the questions. f ( x ) = \sqrt { 2 x } + 1.1 \sin x , [ 0,2 \pi ] \text { a). Plot the function over the interval to see its general behavior there. Sketch the graph below. } b). Find the interior points where f' = 0 (you may need to use the numerical equation solver to approximate a solution). You may wish to plot f' as well. List the points as ordered pairs (x, y). c). Find the interior points where f' does not exist. List the points as ordered pairs (x, y). d). Evaluate the function at the endpoints and list these points as ordered pairs (x, y). e). Find the function's absolute extreme values on the interval and identify where they occur.$ b). Find the interior points where f' = 0 (you may need to use the numerical equation solver to approximate a solution). You may wish to plot f' as well. List the points as ordered pairs (x, y). c). Find the interior points where f' does not exist. List the points as ordered pairs (x, y). d). Evaluate the function at the endpoints and list these points as ordered pairs (x, y). e). Find the function's absolute extreme values on the interval and identify where they occur.

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a).
$a). Solid line: \mathrm { f } ( \mathrm { x } ) ; dashed line: \mathrm { f } ^ { \prime } ( \mathrm { x } ) b). See figure above. f ^ { \prime } ( x ) = 0 at x = 4.4040 and x = 2.0379 . Critical points of f ( x ) are ( 4.4040,1.9197 ) and ( 2.0379,3.0010 ) . c). f ^ { \prime } ( x ) is undefined at the endpoint x = 0 . d). Endpoints are ( 0,0 ) and ( 2 \pi , 3.5449 ) . e). Absolute minimum: ( 0,0 ) ; absolute maximum ( 2 \pi , 3.5449 ) .$
Solid line: $\mathrm { f } ( \mathrm { x } )$ ; dashed line: $\mathrm { f } ^ { \prime } ( \mathrm { x } )$
b). See figure above. $f ^ { \prime } ( x ) = 0$ at $x = 4.4040$ and $x = 2.0379$ .
Critical points of $f ( x )$ are $( 4.4040,1.9197 )$ and $( 2.0379,3.0010 )$ .
c). $f ^ { \prime } ( x )$ is undefined at the endpoint $x = 0$ .
d). Endpoints are $( 0,0 )$ and $( 2 \pi , 3.5449 )$ .
e). Absolute minimum: $( 0,0 )$ ; absolute maximum $( 2 \pi , 3.5449 )$ .

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

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Question 4

Find the extreme values of the function and where they occur. - $y = \frac { 2 x } { x ^ { 2 } + 1 }$

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Question 5

Find the extreme values of the function and where they occur. - $y = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 8$

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Question 6

Solve the problem. -Find the graph that matches the given table. x (x) -1.5 0 2 7

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Question 7

Find the absolute extreme values of the function on the interval. - $f ( x ) = \tan x , - \frac { \pi } { 4 } \leq x \leq \frac { \pi } { 6 }$

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Question 8

Find the extreme values of the function and where they occur. - $y = \frac { \ln x } { x ^ { 2 } }$

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Question 9

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. - $y = \mid x - 3 \mid , 0 < x \leq 10$ $Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. - y = \mid x - 3 \mid , 0 < x \leq 10$

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Question 10

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

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Question 11

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 4 x - 3$

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Question 12

Solve the problem. -Find the table that matches the graph below.

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Question 13

Find the value or values of $c$ that satisfy the equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$ in the conclusion of the Mean Value Theorem for the function and interval. -If the derivative of an even function f(x) is zero at x = c, can anything be said about the value of f' at x = -c? Givereasons for your answer.

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Question 14

Find the absolute extreme values of the function on the interval. - $f ( x ) = - 6 e ^ { - x ^ { 2 } } , - \infty < x < \infty$

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Question 15

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = - x ^ { 3 } - 1.5 x ^ { 2 } + 18 x - 2$

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Question 16

Find the derivative at each critical point and determine the local extreme values. - $y = \left\{ \begin{array} { l l } - x ^ { 2 } - 5 x + 10 , & x \leq 1 \\- x ^ { 2 } + 15 x - 10 , & x > 1\end{array} \right.$

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Question 17

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 21 x ^ { 2 } + 74 x - 41$

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Question 18

Find the absolute extreme values of the function on the interval. -f(x) = 3x - 4, -2 $\leq$ x $\leq$ 3

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Question 19

Find all possible functions with the given derivative. - $y ^ { \prime } = 7 x ^ { 2 } + 1$

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Question 20

Showing 1 - 20 of 159

Determine all critical points for the function. - $f ( x ) = \frac { 3 x } { x + 2 }$

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

Find the extreme values of the function and where they occur. - $y = \frac { 2 x } { x ^ { 2 } + 1 }$

Find the extreme values of the function and where they occur. - $y = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 8$

Solve the problem. -Find the graph that matches the given table. x (x) -1.5 0 2 7

Find the absolute extreme values of the function on the interval. - $f ( x ) = \tan x , - \frac { \pi } { 4 } \leq x \leq \frac { \pi } { 6 }$

Find the extreme values of the function and where they occur. - $y = \frac { \ln x } { x ^ { 2 } }$

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. - $y = \mid x - 3 \mid , 0 < x \leq 10$ $Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. - y = \mid x - 3 \mid , 0 < x \leq 10$

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 4 x - 3$

Solve the problem. -Find the table that matches the graph below.

Find the absolute extreme values of the function on the interval. - $f ( x ) = - 6 e ^ { - x ^ { 2 } } , - \infty < x < \infty$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = - x ^ { 3 } - 1.5 x ^ { 2 } + 18 x - 2$

Find the derivative at each critical point and determine the local extreme values. - $y = \left\{ \begin{array} { l l } - x ^ { 2 } - 5 x + 10 , & x \leq 1 \\- x ^ { 2 } + 15 x - 10 , & x > 1\end{array} \right.$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 21 x ^ { 2 } + 74 x - 41$

Find the absolute extreme values of the function on the interval. -f(x) = 3x - 4, -2 $\leq$ x $\leq$ 3

Find all possible functions with the given derivative. - $y ^ { \prime } = 7 x ^ { 2 } + 1$

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Exam 4: Applications of Derivatives

Determine all critical points for the function. - f(x)=3xx+2f ( x ) = \frac { 3 x } { x + 2 }f(x)=x+23x​

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

Find the extreme values of the function and where they occur. - y=2xx2+1y = \frac { 2 x } { x ^ { 2 } + 1 }y=x2+12x​

Find the extreme values of the function and where they occur. - y=x3−3x2+6x−8y = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 8y=x3−3x2+6x−8

Solve the problem. -Find the graph that matches the given table. x (x) -1.5 0 2 7

Find the absolute extreme values of the function on the interval. - f(x)=tan⁡x,−π4≤x≤π6f ( x ) = \tan x , - \frac { \pi } { 4 } \leq x \leq \frac { \pi } { 6 }f(x)=tanx,−4π​≤x≤6π​

Find the extreme values of the function and where they occur. - y=ln⁡xx2y = \frac { \ln x } { x ^ { 2 } }y=x2lnx​

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. - y=∣x−3∣,0<x≤10y = \mid x - 3 \mid , 0 < x \leq 10y=∣x−3∣,0<x≤10

Determine from the graph whether the function has any absolute extreme values on the interval [a, b]. -

Identify the function's local and absolute extreme values, if any, saying where they occur. - f(x)=x3+4x2+4x−3f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 4 x - 3f(x)=x3+4x2+4x−3

Solve the problem. -Find the table that matches the graph below.

Find the absolute extreme values of the function on the interval. - f(x)=−6e−x2,−∞<x<∞f ( x ) = - 6 e ^ { - x ^ { 2 } } , - \infty < x < \inftyf(x)=−6e−x2,−∞<x<∞

Identify the function's local and absolute extreme values, if any, saying where they occur. - f(x)=−x3−1.5x2+18x−2f ( x ) = - x ^ { 3 } - 1.5 x ^ { 2 } + 18 x - 2f(x)=−x3−1.5x2+18x−2

Find the derivative at each critical point and determine the local extreme values. - y={−x2−5x+10,x≤1−x2+15x−10,x>1y = \left\{ \begin{array} { l l } - x ^ { 2 } - 5 x + 10 , & x \leq 1 \\- x ^ { 2 } + 15 x - 10 , & x > 1\end{array} \right.y={−x2−5x+10,−x2+15x−10,​x≤1x>1​

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - f(x)=x4−3x3−21x2+74x−41f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 21 x ^ { 2 } + 74 x - 41f(x)=x4−3x3−21x2+74x−41

Find the absolute extreme values of the function on the interval. -f(x) = 3x - 4, -2 ≤ \leq ≤ x ≤ \leq ≤ 3

Find all possible functions with the given derivative. - y′=7x2+1y ^ { \prime } = 7 x ^ { 2 } + 1y′=7x2+1

Functions

Limits and Derivatives

Differentiation

Integration

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

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Determine all critical points for the function. - $f ( x ) = \frac { 3 x } { x + 2 }$

Find the extreme values of the function and where they occur. - $y = \frac { 2 x } { x ^ { 2 } + 1 }$

Find the extreme values of the function and where they occur. - $y = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 8$

Find the absolute extreme values of the function on the interval. - $f ( x ) = \tan x , - \frac { \pi } { 4 } \leq x \leq \frac { \pi } { 6 }$

Find the extreme values of the function and where they occur. - $y = \frac { \ln x } { x ^ { 2 } }$

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. - $y = \mid x - 3 \mid , 0 < x \leq 10$ $Sketch the graph of the function and determine whether it has any absolute extreme values on its domain. - y = \mid x - 3 \mid , 0 < x \leq 10$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 4 x - 3$

Find the absolute extreme values of the function on the interval. - $f ( x ) = - 6 e ^ { - x ^ { 2 } } , - \infty < x < \infty$

Identify the function's local and absolute extreme values, if any, saying where they occur. - $f ( x ) = - x ^ { 3 } - 1.5 x ^ { 2 } + 18 x - 2$

Find the derivative at each critical point and determine the local extreme values. - $y = \left\{ \begin{array} { l l } - x ^ { 2 } - 5 x + 10 , & x \leq 1 \\- x ^ { 2 } + 15 x - 10 , & x > 1\end{array} \right.$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 21 x ^ { 2 } + 74 x - 41$

Find the absolute extreme values of the function on the interval. -f(x) = 3x - 4, -2 $\leq$ x $\leq$ 3

Find all possible functions with the given derivative. - $y ^ { \prime } = 7 x ^ { 2 } + 1$