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Calculus 1 : Calculus

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Example Questions

Example Question #721 : Differential Functions

Given that $\frac{dy}{dx}=\sin(x^2)$ and y(1)=4 , use Euler's method to approximate y(2) using three steps.

Possible Answers:

4.72

4.61

5.13

4.44

4.28

Correct answer:

4.72

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{2-1}{3}=\frac{1}{3}$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

y_0=4

$y_1=4+\frac{1}{3}\sin(1^2)=4.28$

$y_2=4.28+\frac{1}{3}\sin((\frac{4}{3})^2)=4.61$

$y_3=4.61+\frac{1}{3}\sin((\frac{5}{3})^2)=4.72$

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Example Question #722 : Functions

Given that $\frac{dy}{dx}=4\sin(2\pi x)$ and y(1)=3 , use Euler's method to approximate y(2.2) using three steps.

Possible Answers:

2.42

1.68

3.94

5.46

Correct answer:

2.42

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{2.2-1}{3}=0.4$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

y_0=3

$y_1=3 +0.4(4sin(2\pi (1)))=3$

$y_2=3+0.4(4sin(2\pi (1.4)))=3.94$

$y_3=3.94+0.4(4sin(2\pi (1.8)))=2.42$

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Example Question #721 : Functions

Given that $\frac{dy}{dx}=3^{x^2}$ and y(1)=6 , use Euler's method to approximate y(1.3) using three steps.

Possible Answers:

6.92

7.56

6.68

7.17

7.08

Correct answer:

7.17

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{1.3-1}{3}=0.1$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

y_0=6

$y_1=6+0.1(3^{1^2})=6.3$

$y_2=6.3+0.1(3^{1.1^2})=6.68$

$y_3=6.68+0.1(3^{1.2^2})=7.17$

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Example Question #531 : Other Differential Functions

Given that $\frac{dy}{dx}=e^{e^{e^{x}}}$ and y(0)=0 , use Euler's method to approximate y(0.02) using two steps.

Possible Answers:

30.776

0.308

7.119

13.24

Correct answer:

0.308

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{0.02-0}{2}=0.01$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

$\frac{dy}{dx}=e^{e^{e^{x}}}$

y_0=0;x_0=0

$y_1=0+0.01e^{e^{e^{0}}}=0.152$

$y_2=0.152+0.01e^{e^{e^{0.01}}}=0.308$

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Example Question #725 : Functions

Given that $\frac{dy}{dx}=4^{x^5}$ and y(1)=0 , use Euler's method to approximate y(1.2) using four steps.

Possible Answers:

2.088

1.772

3.191

0.812

Correct answer:

1.772

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{1.2-1}{4}=0.05$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

$\frac{dy}{dx}=4^{x^5}$

y_0=0;x_0=1

$y_1=0+0.05(4^{1^5})=0.2$

$y_2=0.2+0.05(4^{1.05^5})=0.493$

$y_3=0.493+0.05(4^{1.1^5})=0.959$

$y_4=0.959+0.05(4^{1.15^5})=1.772$

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Example Question #531 : How To Find Differential Functions

Given that $\frac{dy}{dx}=\sin(\cos(x))$ and y(3)=3 , use Euler's method to approximate y(3.6) using three steps.

Possible Answers:

2.665

2.500

3.112

2.833

3.254

Correct answer:

2.500

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{3.6-3}{3}=0.2$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

y_0=3

$y_1=3+0.2\sin(\cos(3))=2.833$

$y_2=2.833+0.2\sin(\cos(3.2))=2.665$

$y_3=2.665+0.2\sin(\cos(3.4))=2.500$

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Example Question #541 : Other Differential Functions

Given that $\frac{dy}{dx}=\ln(x^3)$ and y(1)=13 , use Euler's method to approximate y(1.9) using three steps.

Possible Answers:

15.44

13.66

14.12

14.89

Correct answer:

13.66

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{1.9-1}{3}=0.3$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

y_0=13,x_0=1

$y_1=13+0.3(\ln(1^3))=13$

$y_2=13+0.3(\ln(1.3^3))=13.24$

$y_3=13.24+0.3(\ln(1.6^3))=13.66$

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Example Question #541 : Other Differential Functions

Given that $\frac{dy}{dx}=2^{\ln(x^4)}$ and y(1)=3 , use Euler's method to approximate y(1.8) using four steps.

Possible Answers:

4.46

4.04

5.22

4.93

4.78

Correct answer:

4.78

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{1.8-1}{4}=0.2$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

y_0=3;x_0=1

$y_1=3+0.2(2^{\ln(1^4)})=3.2$

$y_2=3.2+0.2(2^{\ln(1.2^4)})=3.53$

$y_3=3.53+0.2(2^{\ln(1.4^4)})=4.04$

$y_4=4.04+0.2(2^{\ln(1.6^4)})=4.78$

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Example Question #722 : Functions

Given that $\frac{dy}{dx}=\ln(2^{x})$ and y(0)=6 , use Euler's method to approximate y(1.5) using three steps.

Possible Answers:

6.347

6.782

6.520

6.173

6.999

Correct answer:

6.520

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{1.5-0}{3}=0.5$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

$\frac{dy}{dx}=\ln(2^{x}),y_0=6,x_0=0$

$y_1=6+0.5(\ln(2^{0}))=6$

$y_2=6+0.5(\ln(2^{0.5}))=6.173$

$y_3=6.173+0.5(\ln(2^{1}))=6.520$

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Example Question #541 : Other Differential Functions

Given that $\frac{dy}{dx}=\sin(x^2)$ and y(0)=0 , use Euler's method to approximate y(2) using five steps.

Possible Answers:

0.219

1.424

0.700

0.763

0.920

Correct answer:

0.920

Explanation:

When using Euler's method, the first step is to calculate step size:

$\Delta x =\frac{2-0}{5}=0.4$

Now, to approximate function values using Euler's method, utilize the following formula:

$y_{n+1}=y_n + \Delta x \cdot f'(x_n,y_n)$

After that, it's merely a matter of taking the steps:

$\frac{dy}{dx}=\sin(x^2);y_0=0;x_0=0$

$y_1=0+0.4\sin(0^2)=0$

$y_2=0+0.4\sin(0.4^2)=0.064$

$y_3=0.064+0.4\sin(0.8^2)=0.303$

$y_4=0.303+0.4\sin(1.2^2)=0.700$

$y_5=0.700+0.4\sin(1.6^2)=0.920$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #721 : Differential Functions

Example Question #722 : Functions

Example Question #721 : Functions

Example Question #531 : Other Differential Functions

Example Question #725 : Functions

Example Question #531 : How To Find Differential Functions

Example Question #541 : Other Differential Functions

Example Question #541 : Other Differential Functions

Example Question #722 : Functions

Example Question #541 : Other Differential Functions

All Calculus 1 Resources

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