Exam 3: Applications of the Derivative

Sketch labeled graphs of each function $f ( x ) = \frac { - 1 } { x ^ { 2 } + 2 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime\prime }$

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Question 1

Correct Answer:

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$f < 0 \text { for all } x$ $f ^ { \prime } > 0 \text { on } ( 0 , \infty ) \text { and } f ^ { \prime } < 0 \text { on } ( - \infty , 0 )$ $f ^ { \prime \prime } < 0 \text { on } \left( - \infty , - \sqrt { \frac { 2 } { 3 } } \right) \text { and } \left( \sqrt { \frac { 2 } { 3 } } , \infty \right) ; f ^ { \prime \prime } > 0 \text { on } \left( - \sqrt { \frac { 2 } { 3 } } , \sqrt { \frac { 2 } { 3 } } \right)$

Find the critical points of $f ( x ) = \frac { \ln 4 x } { 2 x }$

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Question 2

Correct Answer:

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C

Find $\lim _ { x \rightarrow 0 ^ { + } } x ^ { \sin x }$

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Question 3

Correct Answer:

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B

Find $\lim _ { x \rightarrow 0 ^ { + } } \left( e ^ { 3 x } - 1 \right) ^ { x }$

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Question 4

Melissa wants to make a rectangular box with a square base and cover its top and bottom faces by velvet which will cost her $3 per square inch and the sides by silk which will cost her $5 per square inch. The box should have a volume of 1600 cubic inches. Find the dimensions of the box that will cost her the least amount of money.

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Question 5

Sketch labeled graphs of each function $f ( x ) = \frac { x } { x ^ { 2 } + 4 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime\prime }$

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Question 6

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

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Question 7

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

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Question 8

Determine whether or not the function $f ( x ) = \cos 2 x$ satisfies the hypothesis of Rolle's Theorem on the interval $\left[ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right]$ If it does, find the exact values of all values of $c \in \left( \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right)$ that satisfy the conclusion of Rolle's Theorem.

(Multiple Choice)

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Question 9

Determine the intervals on which $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 1$ is increasing and decreasing.

(Short Answer)

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Question 10

Find the critical points of $f ( x ) = \sin x$

(Multiple Choice)

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Question 11

Find the critical points of $f ( x ) = e ^ { x } \left( x ^ { 2 } - 2 x \right)$

(Multiple Choice)

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Question 12

Find $\lim _ { x \rightarrow \infty } \frac { 2 ^ { x } } { 1 + 3 ^ { x } }$

(Multiple Choice)

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Question 13

Use a sign chart to determine the intervals on which $f ( x ) = x ^ { 3 } - 3 x + 1$ is concave up and concave down, and identify the locations of any inflection points.

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Question 14

Given $u = u ( t ) , \quad v = v ( t ) , \quad w = w ( t )$ are functions of t, calculate the derivative of the functions: (a) $f ( t ) = ( u + v ) ^ { 2 } + 2 t w$ (b) $f ( t ) = \frac { 3 } { u ^ { 2 } v }$

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Question 15

Sketch labeled graphs of each function $f ( x ) = x ^ { 3 } + 2 x ^ { 2 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f ^ { \prime\prime }$

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Question 16

Use a sign chart to determine the intervals on which $f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }$ is concave up and concave down, and identify the locations of any inflection points.

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Question 17

Use a sign chart to determine the intervals on which $f ( x ) = e ^ { 2 x } \left( 1 - e ^ { x } \right)$ is concave up and concave down, and identify the locations of any inflection points.

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Question 18

A function f that is defined on [-1, 3] with $f ( - 1 ) = f ( 3 ) = 2$ such that f is continuous everywhere, differentiable everywhere except at $x = 1$ but fails the conclusion of Rolle's Theorem. Explain why it doesn't satisfy the Rolle's Theorem?

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Question 19

Find the location and values of any global extrema of $f ( x ) = 3 x ^ { 4 } - 4 x ^ { 3 } - 6 x ^ { 2 } + 12 x$ on the intervals: (a) (-2, 1](b) [-2, 1]

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Question 20

Sketch labeled graphs of each function $f ( x ) = \frac { - 1 } { x ^ { 2 } + 2 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime\prime }$

Find the critical points of $f ( x ) = \frac { \ln 4 x } { 2 x }$

Find $\lim _ { x \rightarrow 0 ^ { + } } x ^ { \sin x }$

Find $\lim _ { x \rightarrow 0 ^ { + } } \left( e ^ { 3 x } - 1 \right) ^ { x }$

Sketch labeled graphs of each function $f ( x ) = \frac { x } { x ^ { 2 } + 4 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime\prime }$

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

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

Determine the intervals on which $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 1$ is increasing and decreasing.

Find the critical points of $f ( x ) = \sin x$

Find the critical points of $f ( x ) = e ^ { x } \left( x ^ { 2 } - 2 x \right)$

Find $\lim _ { x \rightarrow \infty } \frac { 2 ^ { x } } { 1 + 3 ^ { x } }$

Use a sign chart to determine the intervals on which $f ( x ) = x ^ { 3 } - 3 x + 1$ is concave up and concave down, and identify the locations of any inflection points.

Given $u = u ( t ) , \quad v = v ( t ) , \quad w = w ( t )$ are functions of t, calculate the derivative of the functions: (a) $f ( t ) = ( u + v ) ^ { 2 } + 2 t w$ (b) $f ( t ) = \frac { 3 } { u ^ { 2 } v }$

Sketch labeled graphs of each function $f ( x ) = x ^ { 3 } + 2 x ^ { 2 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f ^ { \prime\prime }$

Use a sign chart to determine the intervals on which $f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }$ is concave up and concave down, and identify the locations of any inflection points.

Use a sign chart to determine the intervals on which $f ( x ) = e ^ { 2 x } \left( 1 - e ^ { x } \right)$ is concave up and concave down, and identify the locations of any inflection points.

A function f that is defined on [-1, 3] with $f ( - 1 ) = f ( 3 ) = 2$ such that f is continuous everywhere, differentiable everywhere except at $x = 1$ but fails the conclusion of Rolle's Theorem. Explain why it doesn't satisfy the Rolle's Theorem?

Find the location and values of any global extrema of $f ( x ) = 3 x ^ { 4 } - 4 x ^ { 3 } - 6 x ^ { 2 } + 12 x$ on the intervals: (a) (-2, 1](b) [-2, 1]

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

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Exam 3: Applications of the Derivative

Sketch labeled graphs of each function f(x)=−1x2+2f ( x ) = \frac { - 1 } { x ^ { 2 } + 2 }f(x)=x2+2−1​ by hand. As part of your work make sign charts for the signs, roots and undefined points of f,f′ and f′′f , f ^ { \prime } \text { and } f^ { \prime\prime }f,f′ and f′′

Find the critical points of f(x)=ln⁡4x2xf ( x ) = \frac { \ln 4 x } { 2 x }f(x)=2xln4x​

Find lim⁡x→0+xsin⁡x\lim _ { x \rightarrow 0 ^ { + } } x ^ { \sin x }limx→0+​xsinx

Find lim⁡x→0+(e3x−1)x\lim _ { x \rightarrow 0 ^ { + } } \left( e ^ { 3 x } - 1 \right) ^ { x }limx→0+​(e3x−1)x

Sketch labeled graphs of each function f(x)=xx2+4f ( x ) = \frac { x } { x ^ { 2 } + 4 }f(x)=x2+4x​ by hand. As part of your work make sign charts for the signs, roots and undefined points of f,f′ and f′′f , f ^ { \prime } \text { and } f^ { \prime\prime }f,f′ and f′′

Use the derivative f′(x)=x4−2f ^ { \prime } ( x ) = x ^ { 4 } - 2f′(x)=x4−2 to find the local extrema and inflection points of fff

Sketch labeled graphs of each function f(x)=e−x2f ( x ) = e ^ { - x ^ { 2 } }f(x)=e−x2 by hand. As part of your work make sign charts for the signs, roots and undefined points of f,f′ and f′′f , f ^ { \prime } \text { and } f^ { \prime\prime }f,f′ and f′′

Determine the intervals on which f(x)=x3−4x2+1f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 1f(x)=x3−4x2+1 is increasing and decreasing.

Find the critical points of f(x)=sin⁡xf ( x ) = \sin xf(x)=sinx

Find the critical points of f(x)=ex(x2−2x)f ( x ) = e ^ { x } \left( x ^ { 2 } - 2 x \right)f(x)=ex(x2−2x)

Find lim⁡x→∞2x1+3x\lim _ { x \rightarrow \infty } \frac { 2 ^ { x } } { 1 + 3 ^ { x } }limx→∞​1+3x2x​

Use a sign chart to determine the intervals on which f(x)=x3−3x+1f ( x ) = x ^ { 3 } - 3 x + 1f(x)=x3−3x+1 is concave up and concave down, and identify the locations of any inflection points.

Given u=u(t),v=v(t),w=w(t)u = u ( t ) , \quad v = v ( t ) , \quad w = w ( t )u=u(t),v=v(t),w=w(t) are functions of t, calculate the derivative of the functions: (a) f(t)=(u+v)2+2twf ( t ) = ( u + v ) ^ { 2 } + 2 t wf(t)=(u+v)2+2tw (b) f(t)=3u2vf ( t ) = \frac { 3 } { u ^ { 2 } v }f(t)=u2v3​

Sketch labeled graphs of each function f(x)=x3+2x2f ( x ) = x ^ { 3 } + 2 x ^ { 2 }f(x)=x3+2x2 by hand. As part of your work make sign charts for the signs, roots and undefined points of f,f′ and f′′f , f ^ { \prime } \text { and } f ^ { \prime\prime }f,f′ and f′′

Use a sign chart to determine the intervals on which f(x)=1x2+4f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }f(x)=x2+41​ is concave up and concave down, and identify the locations of any inflection points.

Use a sign chart to determine the intervals on which f(x)=e2x(1−ex)f ( x ) = e ^ { 2 x } \left( 1 - e ^ { x } \right)f(x)=e2x(1−ex) is concave up and concave down, and identify the locations of any inflection points.

A function f that is defined on [-1, 3] with f(−1)=f(3)=2f ( - 1 ) = f ( 3 ) = 2f(−1)=f(3)=2 such that f is continuous everywhere, differentiable everywhere except at x=1x = 1x=1 but fails the conclusion of Rolle's Theorem. Explain why it doesn't satisfy the Rolle's Theorem?

Find the location and values of any global extrema of f(x)=3x4−4x3−6x2+12xf ( x ) = 3 x ^ { 4 } - 4 x ^ { 3 } - 6 x ^ { 2 } + 12 xf(x)=3x4−4x3−6x2+12x on the intervals: (a) (-2, 1](b) [-2, 1]

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

Sketch labeled graphs of each function $f ( x ) = \frac { - 1 } { x ^ { 2 } + 2 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime\prime }$

Find the critical points of $f ( x ) = \frac { \ln 4 x } { 2 x }$

Find $\lim _ { x \rightarrow 0 ^ { + } } x ^ { \sin x }$

Find $\lim _ { x \rightarrow 0 ^ { + } } \left( e ^ { 3 x } - 1 \right) ^ { x }$

Sketch labeled graphs of each function $f ( x ) = \frac { x } { x ^ { 2 } + 4 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f^ { \prime\prime }$

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

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

Determine the intervals on which $f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 1$ is increasing and decreasing.

Find the critical points of $f ( x ) = \sin x$

Find the critical points of $f ( x ) = e ^ { x } \left( x ^ { 2 } - 2 x \right)$

Find $\lim _ { x \rightarrow \infty } \frac { 2 ^ { x } } { 1 + 3 ^ { x } }$

Use a sign chart to determine the intervals on which $f ( x ) = x ^ { 3 } - 3 x + 1$ is concave up and concave down, and identify the locations of any inflection points.

Given $u = u ( t ) , \quad v = v ( t ) , \quad w = w ( t )$ are functions of t, calculate the derivative of the functions: (a) $f ( t ) = ( u + v ) ^ { 2 } + 2 t w$ (b) $f ( t ) = \frac { 3 } { u ^ { 2 } v }$

Sketch labeled graphs of each function $f ( x ) = x ^ { 3 } + 2 x ^ { 2 }$ by hand. As part of your work make sign charts for the signs, roots and undefined points of $f , f ^ { \prime } \text { and } f ^ { \prime\prime }$

Use a sign chart to determine the intervals on which $f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }$ is concave up and concave down, and identify the locations of any inflection points.

Use a sign chart to determine the intervals on which $f ( x ) = e ^ { 2 x } \left( 1 - e ^ { x } \right)$ is concave up and concave down, and identify the locations of any inflection points.

A function f that is defined on [-1, 3] with $f ( - 1 ) = f ( 3 ) = 2$ such that f is continuous everywhere, differentiable everywhere except at $x = 1$ but fails the conclusion of Rolle's Theorem. Explain why it doesn't satisfy the Rolle's Theorem?

Find the location and values of any global extrema of $f ( x ) = 3 x ^ { 4 } - 4 x ^ { 3 } - 6 x ^ { 2 } + 12 x$ on the intervals: (a) (-2, 1](b) [-2, 1]