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AP Calculus BC : Euler's Method and L'Hopital's Rule

Study concepts, example questions & explanations for AP Calculus BC

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2 Diagnostic Tests 86 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #45 : Applications Of Derivatives

$\begin{align*}&\text{The expression of a particular function is unknown;}\\&\text{however, we have an expression for its derivative.}\\&\text{Knowing that }f'(x)=tan(x)\text{ and }f(3)=12\\&\text{Approximate } f(21)\text{ using Euler's Method and }4\text{ steps.}\end{align*}$

Possible Answers:

25.23

100.94

10.47

8.41

11.36

Correct answer:

25.23

Explanation:

$\begin{align*}&\text{The general form of Euler's method, when a derivative function,}\\&\text{initial value, and step size are known, is:}\\&y_n=y_{n+1} +\Delta x f'(x_n,y_n)\\&\text{To calculate the step size find the distance between the final}\\&\text{and initial value and divide by the number of steps to be used:}\\&\Delta x = \frac{x_f-x_i}{Steps}\\&\text{For this problem, we are told: }f'(x)=tan(x);f(3)=12\\&\text{Knowing this, we may take the steps to estimate our function}\\&\text{value at our final x value:}\\&\Delta x = \frac{21-(3)}{4}=4.5\\&y_0=12;x_0=3\\&y_{1}=12+(4.5)(tan{(3)})=11.36\\&y_{2}=11.36+(4.5)(tan{(7.5)})=23.54\\&y_{3}=23.54+(4.5)(tan{(12)})=20.67\\&y_{4}=20.67+(4.5)(tan{(16.5)})=25.23\\&\\&\end{align*}$

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Example Question #41 : Applications Of Derivatives

$\begin{align*}&\text{The expression of a particular function is unknown;}\\&\text{however, we have an expression for its derivative.}\\&\text{Knowing that }f'(x)=13sin(3x)\text{ and }f(2)=-15\\&\text{Approximate } f(11.4)\text{ using Euler's Method and }2\text{ steps.}\end{align*}$

Possible Answers:

-10.45

25.92

12.96

-32.07

103.68

Correct answer:

25.92

Explanation:

$\begin{align*}&\text{The general form of Euler's method, when a derivative function,}\\&\text{initial value, and step size are known, is:}\\&y_n=y_{n+1} +\Delta x f'(x_n,y_n)\\&\text{To calculate the step size find the distance between the final}\\&\text{and initial value and divide by the number of steps to be used:}\\&\Delta x = \frac{x_f-x_i}{Steps}\\&\text{For this problem, we are told: }f'(x)=13sin(3x);f(2)=-15\\&\text{Knowing this, we may take the steps to estimate our function}\\&\text{value at our final x value:}\\&\Delta x = \frac{11.4-(2)}{2}=4.7\\&y_0=-15;x_0=2\\&y_{1}=-15+(4.7)(13sin(3{(2)}))=-32.07\\&y_{2}=-32.07+(4.7)(13sin(3{(6.7)}))=25.92\\&\\&\end{align*}$

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Example Question #47 : Applications Of Derivatives

$\begin{align*}&\text{The expression of a particular function is unknown;}\\&\text{however, we have an expression for its derivative.}\\&\text{Knowing that }f'(x)=-14cos(4x)\text{ and }f(-1)=17\\&\text{Approximate } f(7.6)\text{ using Euler's Method and }2\text{ steps.}\end{align*}$

Possible Answers:

7.84

2.61

56.35

9.62

15.67

Correct answer:

7.84

Explanation:

$\begin{align*}&\text{The general form of Euler's method, when a derivative function,}\\&\text{initial value, and step size are known, is:}\\&y_n=y_{n+1} +\Delta x f'(x_n,y_n)\\&\text{To calculate the step size find the distance between the final}\\&\text{and initial value and divide by the number of steps to be used:}\\&\Delta x = \frac{x_f-x_i}{Steps}\\&\text{For this problem, we are told: }f'(x)=-14cos(4x);f(-1)=17\\&\text{Knowing this, we may take the steps to estimate our function}\\&\text{value at our final x value:}\\&\Delta x = \frac{7.6-(-1)}{2}=4.3\\&y_0=17;x_0=-1\\&y_{1}=17+(4.3)(-14cos(4{(-1)}))=56.35\\&y_{2}=56.35+(4.3)(-14cos(4{(3.3)}))=7.84\\&\\&\end{align*}$

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Example Question #221 : Derivatives

$\begin{align*}&\text{The expression of a particular function is unknown;}\\&\text{however, we have an expression for its derivative.}\\&\text{Knowing that }f'(x)=-6x^{2}\text{ and }f(-1)=-20\\&\text{Approximate } f(11.9)\text{ using Euler's Method and }3\text{ steps.}\end{align*}$

Possible Answers:

-1816.97

-363.39

-869.66

-45.8

-9084.85

Correct answer:

-1816.97

Explanation:

$\begin{align*}&\text{The general form of Euler's method, when a derivative function,}\\&\text{initial value, and step size are known, is:}\\&y_n=y_{n+1} +\Delta x f'(x_n,y_n)\\&\text{To calculate the step size find the distance between the final}\\&\text{and initial value and divide by the number of steps to be used:}\\&\Delta x = \frac{x_f-x_i}{Steps}\\&\text{For this problem, we are told: }f'(x)=-6x^{2};f(-1)=-20\\&\text{Knowing this, we may take the steps to estimate our function}\\&\text{value at our final x value:}\\&\Delta x = \frac{11.9-(-1)}{3}=4.3\\&y_0=-20;x_0=-1\\&y_{1}=-20+(4.3)(-6{(-1)}^{2})=-45.8\\&y_{2}=-45.8+(4.3)(-6{(3.3)}^{2})=-326.76\\&y_{3}=-326.76+(4.3)(-6{(7.6)}^{2})=-1816.97\\&\\&\end{align*}$

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Example Question #42 : Applications Of Derivatives

$\begin{align*}&\text{The expression of a particular function is unknown;}\\&\text{however, we have an expression for its derivative.}\\&\text{Knowing that }f'(x)=-2cos(6x)\text{ and }f(-1)=-11\\&\text{Approximate } f(6.2)\text{ using Euler's Method and }4\text{ steps.}\end{align*}$

Possible Answers:

-36.81

-6.13

-12.27

-14.46

-12.76

Correct answer:

-12.27

Explanation:

$\begin{align*}&\text{The general form of Euler's method, when a derivative function,}\\&\text{initial value, and step size are known, is:}\\&y_n=y_{n+1} +\Delta x f'(x_n,y_n)\\&\text{To calculate the step size find the distance between the final}\\&\text{and initial value and divide by the number of steps to be used:}\\&\Delta x = \frac{x_f-x_i}{Steps}\\&\text{For this problem, we are told: }f'(x)=-2cos(6x);f(-1)=-11\\&\text{Knowing this, we may take the steps to estimate our function}\\&\text{value at our final x value:}\\&\Delta x = \frac{6.2-(-1)}{4}=1.8\\&y_0=-11;x_0=-1\\&y_{1}=-11+(1.8)(-2cos(6{(-1)}))=-14.46\\&y_{2}=-14.46+(1.8)(-2cos(6{(0.8)}))=-14.77\\&y_{3}=-14.77+(1.8)(-2cos(6{(2.6)}))=-11.19\\&y_{4}=-11.19+(1.8)(-2cos(6{(4.4)}))=-12.27\\&\\&\end{align*}$

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AP Calculus BC : Euler's Method and L'Hopital's Rule

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #45 : Applications Of Derivatives

Example Question #41 : Applications Of Derivatives

Example Question #47 : Applications Of Derivatives

Example Question #221 : Derivatives

Example Question #42 : Applications Of Derivatives

All AP Calculus BC Resources

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