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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. 
video by The Organic Chemistry Tutor 

Practice
Limits
Evaluate \(\displaystyle{ \lim_{x \to 3}{ \frac{x^2 + 2x + 15}{x^29} } }\)
Problem Statement 

Evaluate \(\displaystyle{ \lim_{x \to 3}{ \frac{x^2 + 2x + 15}{x^29} } }\)
Final Answer 

\(4/3\)
Problem Statement
Evaluate \(\displaystyle{ \lim_{x \to 3}{ \frac{x^2 + 2x + 15}{x^29} } }\)
Solution
Calculus Topics In This Problem 

Finite Limits 
Limits  Substitution Technique 
Limits  Factoring Technique 
video by The Organic Chemistry Tutor 

Final Answer
\(4/3\)
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Find the value of \(c\) that makes \(f(x)\) continuous using limits.
\(\displaystyle{f(x) = \left\{\begin{array}{ll} 2cx6 & x \lt 3 \\ x^2 + cx & x \geq 3 \end{array} \right. }\)
Problem Statement 

Find the value of \(c\) that makes \(f(x)\) continuous using limits.
\(\displaystyle{f(x) = \left\{\begin{array}{ll} 2cx6 & x \lt 3 \\ x^2 + cx & x \geq 3 \end{array} \right. }\)
Final Answer 

\( c = 5 \)
Problem Statement
Find the value of \(c\) that makes \(f(x)\) continuous using limits.
\(\displaystyle{f(x) = \left\{\begin{array}{ll} 2cx6 & x \lt 3 \\ x^2 + cx & x \geq 3 \end{array} \right. }\)
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  OneSided Limits 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

Final Answer
\( \cos(x) \)
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Evaluate \(\displaystyle{ \lim_{x \to 4} { \frac{1/x  1/4}{x4} } }\)
Problem Statement 

Evaluate \(\displaystyle{ \lim_{x \to 4} { \frac{1/x  1/4}{x4} } }\)
Final Answer 

\(1/16\)
Problem Statement
Evaluate \(\displaystyle{ \lim_{x \to 4} { \frac{1/x  1/4}{x4} } }\)
Solution
Calculus Topics In This Problem 

Finite Limits 
Limits  Factoring 
video by The Organic Chemistry Tutor 

Final Answer
\(1/16\)
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Evaluate \(\displaystyle{ \lim_{x \to 9} { \frac{x^281}{\sqrt{x}3} } }\)
Problem Statement 

Evaluate \(\displaystyle{ \lim_{x \to 9} { \frac{x^281}{\sqrt{x}3} } }\)
Final Answer 

\( 108 \)
Problem Statement
Evaluate \(\displaystyle{ \lim_{x \to 9} { \frac{x^281}{\sqrt{x}3} } }\)
Solution
Calculus Topics In This Problem 

Limits  Rationalizing 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

Final Answer
\( 3/5 \)
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Evaluate \(\displaystyle{ \lim_{x \to 0}{ [12x]^{1/x} } }\)
Problem Statement 

Evaluate \(\displaystyle{ \lim_{x \to 0}{ [12x]^{1/x} } }\)
Final Answer 

\( e^{2} \)
Problem Statement
Evaluate \(\displaystyle{ \lim_{x \to 0}{ [12x]^{1/x} } }\)
Solution
Calculus Topics In This Problem 

Finite Limits 
L'Hopitals Rule 
video by The Organic Chemistry Tutor 

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 
OneSided Limits 
Difference Between Limits at Infinity and Limits That Do Not Exist 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 

\( 16x(3x7)[2x^37x^2]^7 \)
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
\( 16x^{15}(3x7)[2x7]^7 \)
Calculus Topics In This Problem 

Derivatives  Chain Rule 
video by The Organic Chemistry Tutor 

Final Answer
\( 16x(3x7)[2x^37x^2]^7 \)
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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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

Final Answer
\( e^{8x}[ 8\ln x \sin x + (\sin x)/x + \ln x \cos x ] \)
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Evaluate \(\displaystyle{ \frac{d}{dx} \left[ \frac{x^2+3}{x^34} \right] }\) giving your answer in completely factored form.
Problem Statement 

Evaluate \(\displaystyle{ \frac{d}{dx} \left[ \frac{x^2+3}{x^34} \right] }\) giving your answer in completely factored form.
Final Answer 

\(\displaystyle{ \frac{x(x^3+9x+8)}{(x^34)^2} }\)
Problem Statement
Evaluate \(\displaystyle{ \frac{d}{dx} \left[ \frac{x^2+3}{x^34} \right] }\) giving your answer in completely factored form.
Solution
Calculus Topics In This Problem 

Derivatives  Power Rule 
Derivatives  Quotient Rule 
video by The Organic Chemistry Tutor 

Final Answer
\(\displaystyle{ \frac{x(x^3+9x+8)}{(x^34)^2} }\)
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Integrals
Evaluate \(\displaystyle{ \int{ \frac{4x^5+x^43x^2}{x^2} ~ dx } }\)
Problem Statement 

Evaluate \(\displaystyle{ \int{ \frac{4x^5+x^43x^2}{x^2} ~ dx } }\)
Final Answer 

\(\displaystyle{ x^4 + \frac{1}{3}x^3  3x + C }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{4x^5+x^43x^2}{x^2} ~ dx } }\)
Solution
Calculus Topics In This Problem 

Integration 
Integration  Power Rule 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

Final Answer
\(5\)
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\(\displaystyle{ \int_0^5{ \sqrt{25x^2} ~ dx } }\)
Problem Statement 

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

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

\( 25 \pi /4 \)
Problem Statement
\(\displaystyle{ \int_0^5{ \sqrt{25x^2} ~ dx } }\)
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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

Final Answer
\( 67\pi / 6 \)
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Find the value of \(c\) guaranteed by Rolleâ€™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 Rolleâ€™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 Rolleâ€™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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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The acceleration of a particle is given by \( a(t) = 2t6 \). The initial velocity of the particle is 8ft/sec and is located 5ft east of the origin along the xaxis 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 \( a(t) = 2t6 \). The initial velocity of the particle is 8ft/sec and is located 5ft east of the origin along the xaxis 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 
video by The Organic Chemistry Tutor 

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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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

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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 
video by The Organic Chemistry Tutor 

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 
video by The Organic Chemistry Tutor 

Final Answer
\( 59 \)
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Really UNDERSTAND Calculus
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all single variable calculus topics 
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