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17Calculus - Calculus 1 Final Exam Practice Problems

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

This page contains a set of practice problems to help you prepare for your calculus 1 final exam. The problems are arranged by topic with reference pages to direct you to what you need to review if you have trouble with the problems. They are mostly basic and intermediate level problems similar to what you might see on your final exam.
Work as many of these as you have time for. If you are short on time, pick a few from each section to test your skills. Of course, it is better to start early and work most or all of these problems. When you do, you will hone your skills, build confidence and make the final exam easier to work when the time comes.
However, these problems are not a substitute for working your homework problems. These solutions may be more terse and they assume that you have worked the practice problems in each section and that you understand them for the most part.
Enjoy!

These practice problems are from this excellent YouTube video. The problem numbers in the video solutions may differ from ours since we arrange them by type.

The Organic Chemistry Tutor - Calculus 1 Final Exam Review [2hrs-51min-44secs]

Deep Work: Rules for Focused Success in a Distracted World

Practice

Limits

Evaluate \(\displaystyle{ \lim_{x \to 3}{ \frac{x^2 + 2x + 15}{x^2-9} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 3}{ \frac{x^2 + 2x + 15}{x^2-9} } }\)

Final Answer

\(4/3\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 3}{ \frac{x^2 + 2x + 15}{x^2-9} } }\)

Solution

Calculus Topics In This Problem

Finite Limits

Limits - Substitution Technique

Limits - Factoring Technique

The Organic Chemistry Tutor - 3110 video solution

Final Answer

\(4/3\)

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Find the value of \(c\) that makes \(f(x)\) continuous using limits.
\(\displaystyle{f(x) = \left\{\begin{array}{ll} 2cx-6 & x \lt 3 \\ x^2 + cx & x \geq 3 \end{array} \right. }\)

Problem Statement

Find the value of \(c\) that makes \(f(x)\) continuous using limits.
\(\displaystyle{f(x) = \left\{\begin{array}{ll} 2cx-6 & x \lt 3 \\ x^2 + cx & x \geq 3 \end{array} \right. }\)

Final Answer

\( c = 5 \)

Problem Statement

Find the value of \(c\) that makes \(f(x)\) continuous using limits.
\(\displaystyle{f(x) = \left\{\begin{array}{ll} 2cx-6 & x \lt 3 \\ x^2 + cx & x \geq 3 \end{array} \right. }\)

Solution

Although he doesn't explicitly write out limits here, you will need to do so on your exam. So you need to make sure to write
\(\displaystyle{ \lim_{x \to 3^-}{f(x)} = \lim_{x \to 3^+}{f(x)} }\)
to show that you understand that this limit will make \(f(x)\) continuous at all values of \(x\).

Calculus Topics In This Problem

Finite Limits

Continuity

Limits - One-Sided Limits

The Organic Chemistry Tutor - 3112 video solution

Final Answer

\( c = 5 \)

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Evaluate \(\displaystyle{ \lim_{h \to 0}{ \frac{\sin(x+h) - \sin(x)}{h} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{h \to 0}{ \frac{\sin(x+h) - \sin(x)}{h} } }\)

Final Answer

\( \cos(x) \)

Problem Statement

Evaluate \(\displaystyle{ \lim_{h \to 0}{ \frac{\sin(x+h) - \sin(x)}{h} } }\)

Solution

Calculus Topics In This Problem

Limit Definition of the Derivative

The Organic Chemistry Tutor - 3116 video solution

Final Answer

\( \cos(x) \)

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Evaluate \(\displaystyle{ \lim_{x \to 4} { \frac{1/x - 1/4}{x-4} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 4} { \frac{1/x - 1/4}{x-4} } }\)

Final Answer

\(-1/16\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 4} { \frac{1/x - 1/4}{x-4} } }\)

Solution

Calculus Topics In This Problem

Finite Limits

Limits - Factoring

The Organic Chemistry Tutor - 3122 video solution

Final Answer

\(-1/16\)

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Evaluate \(\displaystyle{ \lim_{x \to 9} { \frac{x^2-81}{\sqrt{x}-3} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 9} { \frac{x^2-81}{\sqrt{x}-3} } }\)

Final Answer

\( 108 \)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 9} { \frac{x^2-81}{\sqrt{x}-3} } }\)

Solution

Calculus Topics In This Problem

Limits - Rationalizing

The Organic Chemistry Tutor - 3125 video solution

Final Answer

\( 108 \)

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Evaluate \(\displaystyle{ \lim_{x \to 0} { \frac{\tan(3x)}{5x} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 0} { \frac{\tan(3x)}{5x} } }\)

Final Answer

\( 3/5 \)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 0} { \frac{\tan(3x)}{5x} } }\)

Solution

Calculus Topics In This Problem

Finite Limits

Trig Limits

The Organic Chemistry Tutor - 3130 video solution

Final Answer

\( 3/5 \)

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Evaluate \(\displaystyle{ \lim_{x \to 0}{ [1-2x]^{1/x} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 0}{ [1-2x]^{1/x} } }\)

Final Answer

\( e^{-2} \)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 0}{ [1-2x]^{1/x} } }\)

Solution

Calculus Topics In This Problem

Finite Limits

L'Hopitals Rule

The Organic Chemistry Tutor - 3132 video solution

Final Answer

\( e^{-2} \)

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Evaluate \(\displaystyle{ \lim_{x \to 0}{ \frac{|x|}{x} } }\)

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 0}{ \frac{|x|}{x} } }\)

Final Answer

The limit does not exist.

Problem Statement

Evaluate \(\displaystyle{ \lim_{x \to 0}{ \frac{|x|}{x} } }\)

Solution

Calculus Topics In This Problem

Finite Limits

One-Sided Limits

Difference Between Limits at Infinity and Limits That Do Not Exist

The Organic Chemistry Tutor - 3136 video solution

Final Answer

The limit does not exist.

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Derivatives

Find the equation of the tangent line to the curve \( x^3 + 4xy^2 + y^3 = 107 \) at the point \( (2, 3) \) using implicit differentiation. Give your answer in standard form, \( Ax + By = C \).

Problem Statement

Find the equation of the tangent line to the curve \( x^3 + 4xy^2 + y^3 = 107 \) at the point \( (2, 3) \) using implicit differentiation. Give your answer in standard form, \( Ax + By = C \).

Final Answer

\( 16x + 25y = 107 \)

Problem Statement

Find the equation of the tangent line to the curve \( x^3 + 4xy^2 + y^3 = 107 \) at the point \( (2, 3) \) using implicit differentiation. Give your answer in standard form, \( Ax + By = C \).

Solution

Calculus Topics In This Problem

Implicit Differentiation

Derivatives - Tangent Lines

The Organic Chemistry Tutor - 3115 video solution

Final Answer

\( 16x + 25y = 107 \)

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Evaluate \(\displaystyle{ \frac{d}{dx}\left[ x^6 + \frac{3}{x} - \sqrt{x} \right] }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx}\left[ x^6 + \frac{3}{x} - \sqrt{x} \right] }\)

Final Answer

\(\displaystyle{ 6x^5 - \frac{3}{x^2} + \frac{1}{2\sqrt{x}} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx}\left[ x^6 + \frac{3}{x} - \sqrt{x} \right] }\)

Solution

Calculus Topics In This Problem

Derivatives

Derivatives - Power Rule

The Organic Chemistry Tutor - 3111 video solution

Final Answer

\(\displaystyle{ 6x^5 - \frac{3}{x^2} + \frac{1}{2\sqrt{x}} }\)

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Evaluate \(\displaystyle{ \frac{d}{dx} [ e^{4x} \ln(2x+5) ] }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} [ e^{4x} \ln(2x+5) ] }\)

Final Answer

\(\displaystyle{ 4e^{4x} \ln(2x+5) + \frac{2e^{4x}}{2x+5} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} [ e^{4x} \ln(2x+5) ] }\)

Solution

Calculus Topics In This Problem

Derivatives

Derivatives - Exponentials

Derivatives - Logarithms

Product Rule

Chain Rule

The Organic Chemistry Tutor - 3113 video solution

Final Answer

\(\displaystyle{ 4e^{4x} \ln(2x+5) + \frac{2e^{4x}}{2x+5} }\)

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Evaluate \(\displaystyle{ \frac{d}{dx} \left[ 2x^3 - 7x^2 \right]^8 }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} \left[ 2x^3 - 7x^2 \right]^8 }\)

Final Answer

\( 16x(3x-7)[2x^3-7x^2]^7 \)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} \left[ 2x^3 - 7x^2 \right]^8 }\)

Solution

Although this answer is correct, it is not completely factored. We would probably pull out a \( (x^2)^7 = x^{14} \) from the last term to get the completely factored answer
\( 16x^{15}(3x-7)[2x-7]^7 \)

Calculus Topics In This Problem

Derivatives - Chain Rule

The Organic Chemistry Tutor - 3121 video solution

Final Answer

\( 16x(3x-7)[2x^3-7x^2]^7 \)

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Evaluate \(\displaystyle{ \frac{d}{dx} [ x^{\sin x} ] }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} [ x^{\sin x} ] }\)

Final Answer

\(\displaystyle{ x^{\sin x} \left[ \cos x \ln x + \frac{\sin x}{x} \right] }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} [ x^{\sin x} ] }\)

Solution

Calculus Topics In This Problem

Logarithmic Differentiation

The Organic Chemistry Tutor - 3124 video solution

Final Answer

\(\displaystyle{ x^{\sin x} \left[ \cos x \ln x + \frac{\sin x}{x} \right] }\)

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Evaluate \(\displaystyle{ \frac{d}{dx}[ e^{8x} \ln x \sin x ] }\). Give your answer completely factored.

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx}[ e^{8x} \ln x \sin x ] }\). Give your answer completely factored.

Final Answer

\( e^{8x}[ 8\ln x \sin x + (\sin x)/x + \ln x \cos x ] \)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx}[ e^{8x} \ln x \sin x ] }\). Give your answer completely factored.

Solution

Calculus Topics In This Problem

Product Rule

Trig Derivatives

Derivatives of Logarithms

Derivatives of Exponentials

The Organic Chemistry Tutor - 3131 video solution

Final Answer

\( e^{8x}[ 8\ln x \sin x + (\sin x)/x + \ln x \cos x ] \)

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Evaluate \(\displaystyle{ \frac{d}{dx} \left[ \frac{x^2+3}{x^3-4} \right] }\) giving your answer in completely factored form.

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} \left[ \frac{x^2+3}{x^3-4} \right] }\) giving your answer in completely factored form.

Final Answer

\(\displaystyle{ \frac{-x(x^3+9x+8)}{(x^3-4)^2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} \left[ \frac{x^2+3}{x^3-4} \right] }\) giving your answer in completely factored form.

Solution

Calculus Topics In This Problem

Derivatives - Power Rule

Derivatives - Quotient Rule

The Organic Chemistry Tutor - 3133 video solution

Final Answer

\(\displaystyle{ \frac{-x(x^3+9x+8)}{(x^3-4)^2} }\)

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Integrals

Evaluate \(\displaystyle{ \int{ \frac{4x^5+x^4-3x^2}{x^2} ~ dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{4x^5+x^4-3x^2}{x^2} ~ dx } }\)

Final Answer

\(\displaystyle{ x^4 + \frac{1}{3}x^3 - 3x + C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ \frac{4x^5+x^4-3x^2}{x^2} ~ dx } }\)

Solution

Calculus Topics In This Problem

Integration

Integration - Power Rule

The Organic Chemistry Tutor - 3114 video solution

Final Answer

\(\displaystyle{ x^4 + \frac{1}{3}x^3 - 3x + C }\)

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Evaluate \(\displaystyle{ \int{ 2x \sqrt{3x^2+5} ~ dx } }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ 2x \sqrt{3x^2+5} ~ dx } }\)

Final Answer

\(\displaystyle{ \frac{2}{9}[3x^2+5]^{3/2} + C }\)

Problem Statement

Evaluate \(\displaystyle{ \int{ 2x \sqrt{3x^2+5} ~ dx } }\)

Solution

Calculus Topics In This Problem

Integration by Substitution

The Organic Chemistry Tutor - 3117 video solution

Final Answer

\(\displaystyle{ \frac{2}{9}[3x^2+5]^{3/2} + C }\)

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Evaluate \(\displaystyle{ \frac{d}{dx} \int_{x^2}^{4} { \sqrt{5+t^4} ~ dt } }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} \int_{x^2}^{4} { \sqrt{5+t^4} ~ dt } }\)

Final Answer

\( -2x \sqrt{5+x^8} \)

Problem Statement

Evaluate \(\displaystyle{ \frac{d}{dx} \int_{x^2}^{4} { \sqrt{5+t^4} ~ dt } }\)

Solution

Calculus Topics In This Problem

First Fundamental Theorem of Calculus

The Organic Chemistry Tutor - 3126 video solution

Final Answer

\( -2x \sqrt{5+x^8} \)

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If \(\displaystyle{ \int_1^8{ f(x)~dx } = -7 }\) and \(\displaystyle{ \int_1^5{ f(x)~dx } = -12 }\), what is \(\displaystyle{ \int_8^5{ f(x)~dx } }\)?

Problem Statement

If \(\displaystyle{ \int_1^8{ f(x)~dx } = -7 }\) and \(\displaystyle{ \int_1^5{ f(x)~dx } = -12 }\), what is \(\displaystyle{ \int_8^5{ f(x)~dx } }\)?

Final Answer

\(-5\)

Problem Statement

If \(\displaystyle{ \int_1^8{ f(x)~dx } = -7 }\) and \(\displaystyle{ \int_1^5{ f(x)~dx } = -12 }\), what is \(\displaystyle{ \int_8^5{ f(x)~dx } }\)?

Solution

Calculus Topics In This Problem

Basic Integration Rules

The Organic Chemistry Tutor - 3140 video solution

Final Answer

\(-5\)

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\(\displaystyle{ \int_0^5{ \sqrt{25-x^2} ~ dx } }\)

Problem Statement

\(\displaystyle{ \int_0^5{ \sqrt{25-x^2} ~ dx } }\)

Hint

When you get to calculus 2, you will be able to evaluate this integral directly using trig substitution. Since you probably have not had that topic yet, think about what this integral represents and what the integrand looks like.

Problem Statement

\(\displaystyle{ \int_0^5{ \sqrt{25-x^2} ~ dx } }\)

Final Answer

\( 25 \pi /4 \)

Problem Statement

\(\displaystyle{ \int_0^5{ \sqrt{25-x^2} ~ dx } }\)

Hint

When you get to calculus 2, you will be able to evaluate this integral directly using trig substitution. Since you probably have not had that topic yet, think about what this integral represents and what the integrand looks like.

Solution

Calculus Topics In This Problem

Basic Integration and The Meaning of the Integral

The Organic Chemistry Tutor - 3141 video solution

Final Answer

\( 25 \pi /4 \)

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Graphing

Identify all intervals where \(f(x)\) is increasing given \(f(x) = x^3 + (3/2)x^2 - 36x - 9 \). Also locate any local maximums and minimums.

Problem Statement

Identify all intervals where \(f(x)\) is increasing given \(f(x) = x^3 + (3/2)x^2 - 36x - 9 \). Also locate any local maximums and minimums.

Final Answer

The function \( f(x) \) is increasing in the intervals \( (-\infty, -4) \) and \( (3, \infty) \). The critical value \( x = -4 \) is a relative maximum, \( x= 3 \) is a relative minimum.

Problem Statement

Identify all intervals where \(f(x)\) is increasing given \(f(x) = x^3 + (3/2)x^2 - 36x - 9 \). Also locate any local maximums and minimums.

Solution

Calculus Topics In This Problem

Graphing - Increasing and Decreasing Intervals

Critical Values

First Derivative Test

The Organic Chemistry Tutor - 3119 video solution

Final Answer

The function \( f(x) \) is increasing in the intervals \( (-\infty, -4) \) and \( (3, \infty) \). The critical value \( x = -4 \) is a relative maximum, \( x= 3 \) is a relative minimum.

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Identify the location and maximum value of the function \( f(x) = 16x - x^2 + 5 \) using calculus.

Problem Statement

Identify the location and maximum value of the function \( f(x) = 16x - x^2 + 5 \) using calculus.

Final Answer

The location is \(x=8\) and the maximum value is \(f(8)=69\).

Problem Statement

Identify the location and maximum value of the function \( f(x) = 16x - x^2 + 5 \) using calculus.

Solution

Calculus Topics In This Problem

Critical Points

First Derivative Test

The Organic Chemistry Tutor - 3120 video solution

Final Answer

The location is \(x=8\) and the maximum value is \(f(8)=69\).

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Identify all intervals where the function \( f(x) = x^3 - 6x^2 + 5x +1 \) is concave downward.

Problem Statement

Identify all intervals where the function \( f(x) = x^3 - 6x^2 + 5x +1 \) is concave downward.

Final Answer

\( (-\infty, 2) \)

Problem Statement

Identify all intervals where the function \( f(x) = x^3 - 6x^2 + 5x +1 \) is concave downward.

Solution

Calculus Topics In This Problem

Inflection Points

Graphing - Concavity

The Organic Chemistry Tutor - 3123 video solution

Final Answer

\( (-\infty, 2) \)

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Applications

Water is flowing into a cylinder with a diameter of 6ft and a height of 10ft. If the height of the water in the cylinder is increasing at 3ft/min, at what rate is the volume of the water in the cylinder changing?

Problem Statement

Water is flowing into a cylinder with a diameter of 6ft and a height of 10ft. If the height of the water in the cylinder is increasing at 3ft/min, at what rate is the volume of the water in the cylinder changing?

Final Answer

\( dV/dt = 27\pi \) cubic ft per min

Problem Statement

Water is flowing into a cylinder with a diameter of 6ft and a height of 10ft. If the height of the water in the cylinder is increasing at 3ft/min, at what rate is the volume of the water in the cylinder changing?

Solution

Calculus Topics In This Problem

Related Rates - Volume

The Organic Chemistry Tutor - 3118 video solution

Final Answer

\( dV/dt = 27\pi \) cubic ft per min

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Find the area of the region bounded by \( y=x/2 \) and \( y=x^{1/2} \).

Problem Statement

Find the area of the region bounded by \( y=x/2 \) and \( y=x^{1/2} \).

Final Answer

\( 4/3 \)

Problem Statement

Find the area of the region bounded by \( y=x/2 \) and \( y=x^{1/2} \).

Solution

Calculus Topics In This Problem

Application - Area Between Curves

The Organic Chemistry Tutor - 3127 video solution

Final Answer

\( 4/3 \)

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Calculate the volume of the solid formed by revolving the region bounded by \( y =x^{1/2} \), \( y = 0 \) and \( x = 3 \) about the line \( x = 6 \). Give your answer in exact form.

Problem Statement

Calculate the volume of the solid formed by revolving the region bounded by \( y =x^{1/2} \), \( y = 0 \) and \( x = 3 \) about the line \( x = 6 \). Give your answer in exact form.

Final Answer

\(\displaystyle{ \frac{84\pi\sqrt{3}}{5} }\)

Problem Statement

Calculate the volume of the solid formed by revolving the region bounded by \( y =x^{1/2} \), \( y = 0 \) and \( x = 3 \) about the line \( x = 6 \). Give your answer in exact form.

Solution

Calculus Topics In This Problem

Integrals - Volume of Revolution

The Organic Chemistry Tutor - 3128 video solution

Final Answer

\(\displaystyle{ \frac{84\pi\sqrt{3}}{5} }\)

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Calculate the volume generated by rotating the region bounded by \( y = x^2 \), \( y = 0 \), \( x = 1 \) and \( x = 2 \) about the line \( x = 4 \). Give your answer in exact form.

Problem Statement

Calculate the volume generated by rotating the region bounded by \( y = x^2 \), \( y = 0 \), \( x = 1 \) and \( x = 2 \) about the line \( x = 4 \). Give your answer in exact form.

Final Answer

\( 67\pi / 6 \)

Problem Statement

Calculate the volume generated by rotating the region bounded by \( y = x^2 \), \( y = 0 \), \( x = 1 \) and \( x = 2 \) about the line \( x = 4 \). Give your answer in exact form.

Solution

Calculus Topics In This Problem

Integrals - Volume of Revolution

The Organic Chemistry Tutor - 3129 video solution

Final Answer

\( 67\pi / 6 \)

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Find the value of \(c\) guaranteed by Rolle’s Theorem in the function \( f(x) = x^2 - 8x + 12 \) on the interval \([2,6]\).

Problem Statement

Find the value of \(c\) guaranteed by Rolle’s Theorem in the function \( f(x) = x^2 - 8x + 12 \) on the interval \([2,6]\).

Final Answer

\( c = 4 \)

Problem Statement

Find the value of \(c\) guaranteed by Rolle’s Theorem in the function \( f(x) = x^2 - 8x + 12 \) on the interval \([2,6]\).

Solution

Calculus Topics In This Problem

Rolle's Theorem

The Organic Chemistry Tutor - 3134 video solution

Final Answer

\( c = 4 \)

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Find the value of \(c\) guaranteed by the Mean Value Theorem in the function \(f(x) = x^3 -4x\) on the interval \([-2,4]\).

Problem Statement

Find the value of \(c\) guaranteed by the Mean Value Theorem in the function \(f(x) = x^3 -4x\) on the interval \([-2,4]\).

Final Answer

\( c = 2 \)

Problem Statement

Find the value of \(c\) guaranteed by the Mean Value Theorem in the function \(f(x) = x^3 -4x\) on the interval \([-2,4]\).

Solution

This video solution has a great explanation of the Mean Value Theorem including graphs to help you get a feel for what the MVT means.

Calculus Topics In This Problem

Mean Value Theorem

The Organic Chemistry Tutor - 3135 video solution

Final Answer

\( c = 2 \)

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A ball is thrown upward at 96ft/sec from a height of 256ft.
a. How long will it take the ball to hit the ground?
b. What will the velocity of the ball be 4 seconds after it is thrown?
c. Calculate the velocity of the ball just before it hits the ground.
d. Calculate the maximum height of the ball.

Problem Statement

A ball is thrown upward at 96ft/sec from a height of 256ft.
a. How long will it take the ball to hit the ground?
b. What will the velocity of the ball be 4 seconds after it is thrown?
c. Calculate the velocity of the ball just before it hits the ground.
d. Calculate the maximum height of the ball.

Hint

The height of the ball with respect to time is given by the equation \( h(t) = -16t^2 + 96t + 256 \).

Problem Statement

A ball is thrown upward at 96ft/sec from a height of 256ft.
a. How long will it take the ball to hit the ground?
b. What will the velocity of the ball be 4 seconds after it is thrown?
c. Calculate the velocity of the ball just before it hits the ground.
d. Calculate the maximum height of the ball.

Hint

The height of the ball with respect to time is given by the equation \( h(t) = -16t^2 + 96t + 256 \).

Solution

Calculus Topics In This Problem

Derivatives - Linear Motion

The Organic Chemistry Tutor - 3137 video solution

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The acceleration of a particle is given by \( a(t) = 2t-6 \). The initial velocity of the particle is 8ft/sec and is located 5ft east of the origin along the x-axis at \( t = 1 \).
a. Write a function for the velocity, \( v(t) \), of the particle.
b. When is the particle moving to the right?
c. What is the position of the particle at \( t = 5 \)?
d. Calculate the displacement and total distance traveled by the particle in the first 6 seconds.

Problem Statement

The acceleration of a particle is given by \( a(t) = 2t-6 \). The initial velocity of the particle is 8ft/sec and is located 5ft east of the origin along the x-axis at \( t = 1 \).
a. Write a function for the velocity, \( v(t) \), of the particle.
b. When is the particle moving to the right?
c. What is the position of the particle at \( t = 5 \)?
d. Calculate the displacement and total distance traveled by the particle in the first 6 seconds.

Solution

This video contains an excellent explanation of acceleration, velocity and position.
Following the video solution, we have included the video that he talks about at the end of this solution.

Calculus Topics In This Problem

Integrals - Linear Motion

The Organic Chemistry Tutor - 3138 video solution

3138 video solution

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The rate of water flowing into an initially empty tank can be modeled by the equation \( R(t) = 0.1t^2 + 0.4t + 12 \) gal/min for \( t \geq 0 \). Calculate the total volume of water accumulated in the tank after 10 minutes.

Problem Statement

The rate of water flowing into an initially empty tank can be modeled by the equation \( R(t) = 0.1t^2 + 0.4t + 12 \) gal/min for \( t \geq 0 \). Calculate the total volume of water accumulated in the tank after 10 minutes.

Final Answer

\( 520/3 \) gals

Problem Statement

The rate of water flowing into an initially empty tank can be modeled by the equation \( R(t) = 0.1t^2 + 0.4t + 12 \) gal/min for \( t \geq 0 \). Calculate the total volume of water accumulated in the tank after 10 minutes.

Solution

Calculus Topics In This Problem

Related Rates - Volume

The Organic Chemistry Tutor - 3139 video solution

Final Answer

\( 520/3 \) gals

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A farmer wants to set up a rectangular fence adjacent to a river. The area of the field is 16200 square feet. What dimensions will require the least amount of fencing if no fencing is needed along the river?

Problem Statement

A farmer wants to set up a rectangular fence adjacent to a river. The area of the field is 16200 square feet. What dimensions will require the least amount of fencing if no fencing is needed along the river?

Solution

Calculus Topics In This Problem

Optimization

The Organic Chemistry Tutor - 3142 video solution

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Calculate the average rate of change of the function \( f(x) = x^2 -5x + 2 \) over the interval \( [1,5] \).

Problem Statement

Calculate the average rate of change of the function \( f(x) = x^2 -5x + 2 \) over the interval \( [1,5] \).

Final Answer

\(1\)

Problem Statement

Calculate the average rate of change of the function \( f(x) = x^2 -5x + 2 \) over the interval \( [1,5] \).

Solution

Calculus Topics In This Problem

Average Rate of Change

The Organic Chemistry Tutor - 3143 video solution

Final Answer

\(1\)

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Calculate the average value of the function \( f(x) = x^3 + 8x - 4 \) over the interval \( [1, 5] \).

Problem Statement

Calculate the average value of the function \( f(x) = x^3 + 8x - 4 \) over the interval \( [1, 5] \).

Final Answer

\( 59 \)

Problem Statement

Calculate the average value of the function \( f(x) = x^3 + 8x - 4 \) over the interval \( [1, 5] \).

Solution

Calculus Topics In This Problem

Integration Application - Average Value of a Function

The Organic Chemistry Tutor - 3144 video solution

Final Answer

\( 59 \)

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Really UNDERSTAND Calculus

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Topics You Need To Understand For This Page

all single variable calculus topics

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100 calculus 2 problems (in ONE take)

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