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AP Calculus AB : Derivatives

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Example Questions

Example Question #107 : Derivative At A Point

$\begin{align*}&t&p(t)\\&3&-31\\&4&-41\\&5&-19\\&6&-4\\&7&-40\\&\text{The above table lists a particle's postion at different times.}\\&\text{Approximate its velocity with a forward difference at }t=6\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(6)\text{ and }f(7)\\&\text{Applying the formula above, with }h=1\text{, we find:}\\&v(6)\approx\frac{-40-(-4)}{1}\\&v(6)\approx-36\end{align*}$

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Example Question #108 : Derivative At A Point

$\begin{align*}&t&p(t)\\&0&33\\&3&-16\\&6&-21\\&9&25\\&12&-49\\&15&-46\\&18&17\\&21&10\\&\text{Using the above table of positions,}\\&\text{make a forward difference velocity approximation at }t=9\end{align*}$

Possible Answers:

$\frac{25}{3}$

$-\frac{74}{3}$

$-\frac{37}{3}$

$\frac{46}{3}$

Correct answer:

$-\frac{74}{3}$

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(9)\text{ and }f(12)\\&\text{Applying the formula above, with }h=3\text{, we find:}\\&v(9)\approx\frac{-49-(25)}{3}\\&v(9)\approx-\frac{74}{3}\end{align*}$

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Example Question #109 : Derivative At A Point

$\begin{align*}&\text{Approximate a particle's velocity with a forward difference at }t=4\\&\text{Using the following table of positions:}\\&t&p(t)\\&4&6\\&6&-10\\&8&-44\\&10&28\\&12&-16\\&14&11\\&16&24\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(4)\text{ and }f(6)\\&\text{Applying the formula above, with }h=2\text{, we find:}\\&v(4)\approx\frac{-10-(6)}{2}\\&v(4)\approx-8\end{align*}$

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Example Question #110 : Derivative At A Point

$\begin{align*}&t&p(t)\\&5&24\\&10&-45\\&15&-43\\&20&-42\\&25&30\\&\text{The above table lists a particle's postion at different times.}\\&\text{Approximate its velocity with a forward difference at }t=20\end{align*}$

Possible Answers:

$\frac{36}{5}$

$-\frac{42}{5}$

$\frac{1}{5}$

$\frac{72}{5}$

Correct answer:

$\frac{72}{5}$

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(20)\text{ and }f(25)\\&\text{Applying the formula above, with }h=5\text{, we find:}\\&v(20)\approx\frac{30-(-42)}{5}\\&v(20)\approx\frac{72}{5}\end{align*}$

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Example Question #511 : Derivatives

$\begin{align*}&t&p(t)\\&0&14\\&1&-17\\&2&16\\&3&25\\&4&8\\&5&24\\&6&-27\\&\text{Using the above table of positions,}\\&\text{make a forward difference velocity approximation at }t=4\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(4)\text{ and }f(5)\\&\text{Applying the formula above, with }h=1\text{, we find:}\\&v(4)\approx\frac{24-(8)}{1}\\&v(4)\approx16\end{align*}$

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Example Question #512 : Derivatives

$\begin{align*}&t&p(t)\\&2&19\\&4&10\\&6&29\\&\text{The above table lists a particle's postion at different times.}\\&\text{Approximate its velocity with a forward difference at }t=2\end{align*}$

Possible Answers:

$-\frac{9}{4}$

$\frac{9}{2}$

$\frac{9}{4}$

$-\frac{9}{2}$

Correct answer:

$-\frac{9}{2}$

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(2)\text{ and }f(4)\\&\text{Applying the formula above, with }h=2\text{, we find:}\\&v(2)\approx\frac{10-(19)}{2}\\&v(2)\approx-\frac{9}{2}\end{align*}$

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Example Question #513 : Derivatives

$\begin{align*}&\text{Approximate a particle's velocity with a forward difference at }t=4\\&\text{Using the following table of positions:}\\&t&p(t)\\&2&5\\&4&-27\\&6&14\\&8&-2\\&10&-35\\&12&28\\&14&-40\end{align*}$

Possible Answers:

$\frac{41}{2}$

$\frac{41}{4}$

$-\frac{27}{2}$

Correct answer:

$\frac{41}{2}$

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(4)\text{ and }f(6)\\&\text{Applying the formula above, with }h=2\text{, we find:}\\&v(4)\approx\frac{14-(-27)}{2}\\&v(4)\approx\frac{41}{2}\end{align*}$

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Example Question #514 : Derivatives

$\begin{align*}&\text{Approximate a particle's velocity with a forward difference at }t=3\\&\text{Using the following table of positions:}\\&t&p(t)\\&0&-39\\&3&29\\&6&-21\end{align*}$

Possible Answers:

$-\frac{25}{3}$

$\frac{68}{3}$

$-\frac{50}{3}$

$\frac{29}{3}$

Correct answer:

$-\frac{50}{3}$

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(3)\text{ and }f(6)\\&\text{Applying the formula above, with }h=3\text{, we find:}\\&v(3)\approx\frac{-21-(29)}{3}\\&v(3)\approx-\frac{50}{3}\end{align*}$

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Example Question #515 : Derivatives

$\begin{align*}&t&p(t)\\&3&16\\&7&-39\\&11&44\\&15&-32\\&19&-24\\&23&30\\&27&-1\\&\text{The above table lists a particle's postion at different times.}\\&\text{Approximate its velocity with a forward difference at }t=19\end{align*}$

Possible Answers:

$\frac{27}{2}$

$\frac{27}{4}$

Correct answer:

$\frac{27}{2}$

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(19)\text{ and }f(23)\\&\text{Applying the formula above, with }h=4\text{, we find:}\\&v(19)\approx\frac{30-(-24)}{4}\\&v(19)\approx\frac{27}{2}\end{align*}$

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Example Question #516 : Derivatives

$\begin{align*}&t&p(t)\\&3&-5\\&7&11\\&11&-45\\&\text{Using the above table of positions,}\\&\text{make a forward difference velocity approximation at }t=3\end{align*}$

Possible Answers:

$-\frac{5}{4}$

Correct answer:

Explanation:

$\begin{align*}&\text{Velocity is the time derivative of position. However, since}\\&\text{we do not have a position function, we cannot take the derivative;}\\&\text{we must use an approximation. To find a forward derivative approximation,}\\&\text{we consider function two values: one at the point of interest}\\&\text{and another succeeding it, and finally that distance between}\\&\text{the two. Mathematically, this follows the form:}\\&f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{Where h is the size of the increment between points on the x-axis.}\\&\text{In approximating a derivative, it is generally prudent to minimize}\\&\text{the distance between function values. In other words, it’s a}\\&\text{good idea to look at two adjacent points. For our problem, that’d}\\&\text{be:}\\&f(3)\text{ and }f(7)\\&\text{Applying the formula above, with }h=4\text{, we find:}\\&v(3)\approx\frac{11-(-5)}{4}\\&v(3)\approx4\end{align*}$

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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 #107 : Derivative At A Point

Example Question #108 : Derivative At A Point

Example Question #109 : Derivative At A Point

Example Question #110 : Derivative At A Point

Example Question #511 : Derivatives

Example Question #512 : Derivatives

Example Question #513 : Derivatives

Example Question #514 : Derivatives

Example Question #515 : Derivatives

Example Question #516 : Derivatives

All AP Calculus AB Resources

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