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AP Calculus AB : Derivatives

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Example Questions

Example Question #91 : Derivative At A Point

$\begin{align*}&\text{Approximate the acceleration of a particle using}\\&\text{a central difference approach at }t=13\\&\text{Using the following table of positions:}\\&t&p(t)\\&1&33\\&4&9\\&7&5\\&10&42\\&13&-22\\&16&26\end{align*}$

Possible Answers:

$-\frac{16}{9}$

$-\frac{16}{3}$

$-\frac{64}{3}$

$\frac{112}{9}$

Correct answer:

$\frac{112}{9}$

Explanation:

$\begin{align*}&\text{Acceleration is the 2nd time derivative of position. In other}\\&\text{words, you take the derivative of a position function twice}\\&\text{to define it. In this problem, we’re not given a function, but}\\&\text{ a table of times and positions, requiring an approximation,}\\&\text{a 2nd order central difference. This has the form:}\\&f''(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{This approximation is actually born from two types of first}\\&\text{order approximations: the forward and backward differences,}\\&\text{which have the forms:}\\&f'(x)_{forward}=\frac{f(x+h)-f(x)}{h}\text{ and }f'(x)_{backward}=\frac{f(x)-f(x-h)}{h}\\&\text{The difference of these two, divided by our increment, gives}\\&\text{a derivative of a derivative.}\\&\text{Applying the formula above, with }h=3\text{, we find:}\\&a(13)\approx\frac{26+(42)-2(-22)}{3^2}\\&a(13)\approx\frac{112}{9}\end{align*}$

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Example Question #92 : Derivative At A Point

$\begin{align*}&t&p(t)\\&1&32\\&6&-26\\&11&43\\&16&-15\\&\text{The table above lists the positions of a particle}\\&\text{At various times. Approximate the acceleration at }t=6\\&\text{Using a central difference approach.}\end{align*}$

Possible Answers:

$\frac{11}{25}$

$\frac{11}{5}$

$\frac{127}{25}$

$-\frac{58}{5}$

Correct answer:

$\frac{127}{25}$

Explanation:

$\begin{align*}&\text{Acceleration is the 2nd time derivative of position. In other}\\&\text{words, you take the derivative of a position function twice}\\&\text{to define it. In this problem, we’re not given a function, but}\\&\text{ a table of times and positions, requiring an approximation,}\\&\text{a 2nd order central difference. This has the form:}\\&f''(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{This approximation is actually born from two types of first}\\&\text{order approximations: the forward and backward differences,}\\&\text{which have the forms:}\\&f'(x)_{forward}=\frac{f(x+h)-f(x)}{h}\text{ and }f'(x)_{backward}=\frac{f(x)-f(x-h)}{h}\\&\text{The difference of these two, divided by our increment, gives}\\&\text{a derivative of a derivative.}\\&\text{Applying the formula above, with }h=5\text{, we find:}\\&a(6)\approx\frac{43+(32)-2(-26)}{5^2}\\&a(6)\approx\frac{127}{25}\end{align*}$

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Example Question #93 : Derivative At A Point

$\begin{align*}&t&p(t)\\&2&-28\\&4&25\\&6&-25\\&8&1\\&10&20\\&12&39\\&14&46\\&16&5\\&\text{The table above lists the positions of a particle}\\&\text{At various times. Approximate the acceleration at }t=4\\&\text{Using a central difference approach.}\end{align*}$

Possible Answers:

$\frac{3}{2}$

$\frac{53}{2}$

$-\frac{103}{4}$

$\frac{3}{4}$

Correct answer:

$-\frac{103}{4}$

Explanation:

$\begin{align*}&\text{Acceleration is the 2nd time derivative of position. In other}\\&\text{words, you take the derivative of a position function twice}\\&\text{to define it. In this problem, we’re not given a function, but}\\&\text{ a table of times and positions, requiring an approximation,}\\&\text{a 2nd order central difference. This has the form:}\\&f''(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{This approximation is actually born from two types of first}\\&\text{order approximations: the forward and backward differences,}\\&\text{which have the forms:}\\&f'(x)_{forward}=\frac{f(x+h)-f(x)}{h}\text{ and }f'(x)_{backward}=\frac{f(x)-f(x-h)}{h}\\&\text{The difference of these two, divided by our increment, gives}\\&\text{a derivative of a derivative.}\\&\text{Applying the formula above, with }h=2\text{, we find:}\\&a(4)\approx\frac{-25+(-28)-2(25)}{2^2}\\&a(4)\approx-\frac{103}{4}\end{align*}$

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Example Question #31 : Approximate Rate Of Change From Graphs And Tables Of Values

$\begin{align*}&\text{Approximate the acceleration of a particle using}\\&\text{a central difference approach at }t=6\\&\text{Using the following table of positions:}\\&t&p(t)\\&2&26\\&6&-23\\&10&18\\&14&16\\&18&-34\end{align*}$

Possible Answers:

$-\frac{49}{4}$

$\frac{45}{8}$

$-\frac{1}{2}$

Correct answer:

$\frac{45}{8}$

Explanation:

$\begin{align*}&\text{Acceleration is the 2nd time derivative of position. In other}\\&\text{words, you take the derivative of a position function twice}\\&\text{to define it. In this problem, we’re not given a function, but}\\&\text{ a table of times and positions, requiring an approximation,}\\&\text{a 2nd order central difference. This has the form:}\\&f''(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{This approximation is actually born from two types of first}\\&\text{order approximations: the forward and backward differences,}\\&\text{which have the forms:}\\&f'(x)_{forward}=\frac{f(x+h)-f(x)}{h}\text{ and }f'(x)_{backward}=\frac{f(x)-f(x-h)}{h}\\&\text{The difference of these two, divided by our increment, gives}\\&\text{a derivative of a derivative.}\\&\text{Applying the formula above, with }h=4\text{, we find:}\\&a(6)\approx\frac{18+(26)-2(-23)}{4^2}\\&a(6)\approx\frac{45}{8}\end{align*}$

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Example Question #101 : Derivative At A Point

$\begin{align*}&\text{Approximate the acceleration of a particle using}\\&\text{a central difference approach at }t=5\\&\text{Using the following table of positions:}\\&t&p(t)\\&0&28\\&5&32\\&10&37\end{align*}$

Possible Answers:

$\frac{9}{5}$

$\frac{4}{5}$

$\frac{9}{25}$

$\frac{1}{25}$

Correct answer:

$\frac{1}{25}$

Explanation:

$\begin{align*}&\text{Acceleration is the 2nd time derivative of position. In other}\\&\text{words, you take the derivative of a position function twice}\\&\text{to define it. In this problem, we’re not given a function, but}\\&\text{ a table of times and positions, requiring an approximation,}\\&\text{a 2nd order central difference. This has the form:}\\&f''(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{This approximation is actually born from two types of first}\\&\text{order approximations: the forward and backward differences,}\\&\text{which have the forms:}\\&f'(x)_{forward}=\frac{f(x+h)-f(x)}{h}\text{ and }f'(x)_{backward}=\frac{f(x)-f(x-h)}{h}\\&\text{The difference of these two, divided by our increment, gives}\\&\text{a derivative of a derivative.}\\&\text{Applying the formula above, with }h=5\text{, we find:}\\&a''(5)\approx\frac{37+(28)-2(32)}{5^2}\\&a'(5)\approx\frac{1}{25}\end{align*}$

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Example Question #102 : Derivative At A Point

$\begin{align*}&t&p(t)\\&1&41\\&6&-32\\&11&-24\\&16&-36\\&\text{Using the above table of a particle's positions}\\&\text{ make a central difference approximation of its acceleration at }t=6\end{align*}$

Possible Answers:

$-\frac{13}{5}$

$-\frac{73}{5}$

$\frac{81}{25}$

Correct answer:

$\frac{81}{25}$

Explanation:

$\begin{align*}&\text{Acceleration is the 2nd time derivative of position. In other}\\&\text{words, you take the derivative of a position function twice}\\&\text{to define it. In this problem, we’re not given a function, but}\\&\text{ a table of times and positions, requiring an approximation,}\\&\text{a 2nd order central difference. This has the form:}\\&f''(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{This approximation is actually born from two types of first}\\&\text{order approximations: the forward and backward differences,}\\&\text{which have the forms:}\\&f'(x)_{forward}=\frac{f(x+h)-f(x)}{h}\text{ and }f'(x)_{backward}=\frac{f(x)-f(x-h)}{h}\\&\text{The difference of these two, divided by our increment, gives}\\&\text{a derivative of a derivative.}\\&\text{Applying the formula above, with }h=5\text{, we find:}\\&a(6)\approx\frac{-24+(41)-2(-32)}{5^2}\\&a(6)\approx\frac{81}{25}\end{align*}$

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Example Question #103 : Derivative At A Point

$\begin{align*}&t&p(t)\\&0&36\\&3&12\\&6&-15\\&9&1\\&12&-10\\&15&-43\\&\text{The table above lists the positions of a particle}\\&\text{At various times. Approximate the acceleration at }t=3\\&\text{Using a central difference approach.}\end{align*}$

Possible Answers:

$-\frac{1}{3}$

$-\frac{17}{3}$

Correct answer:

$-\frac{1}{3}$

Explanation:

$\begin{align*}&\text{Acceleration is the 2nd time derivative of position. In other}\\&\text{words, you take the derivative of a position function twice}\\&\text{to define it. In this problem, we’re not given a function, but}\\&\text{ a table of times and positions, requiring an approximation,}\\&\text{a 2nd order central difference. This has the form:}\\&f''(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{This approximation is actually born from two types of first}\\&\text{order approximations: the forward and backward differences,}\\&\text{which have the forms:}\\&f'(x)_{forward}=\frac{f(x+h)-f(x)}{h}\text{ and }f'(x)_{backward}=\frac{f(x)-f(x-h)}{h}\\&\text{The difference of these two, divided by our increment, gives}\\&\text{a derivative of a derivative.}\\&\text{Applying the formula above, with }h=3\text{, we find:}\\&a(3)\approx\frac{-15+(36)-2(12)}{3^2}\\&a(3)\approx-\frac{1}{3}\end{align*}$

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Example Question #104 : Derivative At A Point

$\begin{align*}&\text{Approximate the acceleration of a particle using}\\&\text{a central difference approach at }t=3\\&\text{Using the following table of positions:}\\&t&p(t)\\&1&-45\\&3&41\\&5&45\\&7&-1\end{align*}$

Possible Answers:

$\frac{45}{2}$

$-\frac{41}{2}$

Correct answer:

$-\frac{41}{2}$

Explanation:

$\begin{align*}&\text{Acceleration is the 2nd time derivative of position. In other}\\&\text{words, you take the derivative of a position function twice}\\&\text{to define it. In this problem, we’re not given a function, but}\\&\text{ a table of times and positions, requiring an approximation,}\\&\text{a 2nd order central difference. This has the form:}\\&f''(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{This approximation is actually born from two types of first}\\&\text{order approximations: the forward and backward differences,}\\&\text{which have the forms:}\\&f'(x)_{forward}=\frac{f(x+h)-f(x)}{h}\text{ and }f'(x)_{backward}=\frac{f(x)-f(x-h)}{h}\\&\text{The difference of these two, divided by our increment, gives}\\&\text{a derivative of a derivative.}\\&\text{Applying the formula above, with }h=2\text{, we find:}\\&a(3)\approx\frac{45+(-45)-2(41)}{2^2}\\&a(3)\approx-\frac{41}{2}\end{align*}$

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Example Question #105 : Derivative At A Point

$\begin{align*}&t&p(t)\\&0&28\\&2&-11\\&4&-26\\&6&-10\\&8&-41\\&10&-37\\&12&45\\&14&46\\&\text{The table above lists the positions of a particle}\\&\text{At various times. Approximate the acceleration at }t=8\\&\text{Using a central difference approach.}\end{align*}$

Possible Answers:

$-\frac{31}{2}$

$-\frac{27}{2}$

$\frac{35}{4}$

$-\frac{27}{4}$

Correct answer:

$\frac{35}{4}$

Explanation:

$\begin{align*}&\text{Acceleration is the 2nd time derivative of position. In other}\\&\text{words, you take the derivative of a position function twice}\\&\text{to define it. In this problem, we’re not given a function, but}\\&\text{ a table of times and positions, requiring an approximation,}\\&\text{a 2nd order central difference. This has the form:}\\&f''(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^2}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{This approximation is actually born from two types of first}\\&\text{order approximations: the forward and backward differences,}\\&\text{which have the forms:}\\&f'(x)_{forward}=\frac{f(x+h)-f(x)}{h}\text{ and }f'(x)_{backward}=\frac{f(x)-f(x-h)}{h}\\&\text{The difference of these two, divided by our increment, gives}\\&\text{a derivative of a derivative.}\\&\text{Applying the formula above, with }h=2\text{, we find:}\\&a(8)\approx\frac{-37+(-10)-2(-41)}{2^2}\\&a(8)\approx\frac{35}{4}\end{align*}$

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Example Question #106 : Derivative At A Point

$\begin{align*}&t&p(t)\\&2&24\\&3&6\\&4&-32\\&5&10\\&6&-20\\&7&-37\\&8&-29\\&9&40\\&\text{The above table lists a particle's postion at different times.}\\&\text{Approximate its velocity with a forward difference at }t=2\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(2)\text{ and }f(3)\\&\text{Applying the formula above, with }h=1\text{, we find:}\\&v(2)\approx\frac{6-(24)}{1}\\&v(2)\approx-18\end{align*}$

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AP Calculus AB : Derivatives

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #91 : Derivative At A Point

Example Question #92 : Derivative At A Point

Example Question #93 : Derivative At A Point

Example Question #31 : Approximate Rate Of Change From Graphs And Tables Of Values

Example Question #101 : Derivative At A Point

Example Question #102 : Derivative At A Point

Example Question #103 : Derivative At A Point

Example Question #104 : Derivative At A Point

Example Question #105 : Derivative At A Point

Example Question #106 : Derivative At A Point

All AP Calculus AB Resources

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