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Calculus 1 : Other Differential Functions

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Example Questions

Example Question #711 : Differential Functions

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

Possible Answers:

Correct answer:

Explanation:

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

$\Delta x =\frac{4-1}{3}=1$

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=2

y_1=y_0+1(1^2)=2+1=3

y_2=3+1(2^2)=7

y_3=7+1(3^2)=16

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Example Question #712 : Differential Functions

Given that $\frac{dy}{dx}=tan(x^3+5)$ and y(4)=7 , use Euler's method to approximate y(6) using three steps.

Possible Answers:

6.54

6.11

6.80

6.92

Correct answer:

6.54

Explanation:

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

$\Delta x =\frac{6-4}{3}=\frac{2}{3}$

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=7

$y_1=7+\frac{2}{3}tan(4^3+5)=6.92$

$y_2=6.92+\frac{2}{3}tan((\frac{14}{3})^3+5)=6.80$

$y_3=6.80+\frac{2}{3}tan((\frac{16}{3})^3+5)=6.54$

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Example Question #713 : Differential Functions

Given that $\frac{dy}{dx}=e^{\frac{5}{x}}$ and y(3)=12 , use Euler's method to approximate y(4) using three steps.

Possible Answers:

13.76

16.55

14.88 17.97

15.25

Correct answer:

16.55

Explanation:

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

$\Delta x =\frac{4-3}{3}=\frac{1}{3}$

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=12

$y_1=12+\frac{1}{3}e^{\frac{5}{3}}=13.76$

$y_2=13.76+\frac{1}{3}e^{\frac{5}{\frac{10}{3}}}=15.25$

$y_3=15.25+\frac{1}{3}e^{\frac{5}{\frac{11}{3}}}=16.55$

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Example Question #1750 : Calculus

Given that $\frac{dy}{dx}=e^{x^3-1}$ and y(0)=1.8 , use Euler's method to approximate y(0.3) using three steps.

Possible Answers:

2.032

1.8

2.259

1.874

1.911

Correct answer:

1.911

Explanation:

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

$\Delta x =\frac{0.3-0}{3}=0.1$

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=1.8

$y_1=1.8+0.1(e^{0^3-1})=1.837$

$y_2=1.837+0.1(e^{0.1^3-1})=1.874$

$y_3=1.874+0.1(e^{0.2^3-1})=1.911$

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Example Question #721 : 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:

5.13

4.72

4.61

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 #723 : 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:

7.08

6.92

7.17

6.68

7.56

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 #721 : 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

7.119

13.24

0.308

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 #726 : 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:

3.112

2.665

2.833

2.500

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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Calculus 1 : Other Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #711 : Differential Functions

Example Question #712 : Differential Functions

Example Question #713 : Differential Functions

Example Question #1750 : Calculus

Example Question #721 : Functions

Example Question #722 : Functions

Example Question #723 : Functions

Example Question #721 : Differential Functions

Example Question #725 : Functions

Example Question #726 : Functions

All Calculus 1 Resources

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