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Calculus 1 : Calculus

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Example Questions

Example Question #371 : How To Find Velocity

The velocity function of a particle and a position of this particle at a known time are given by v(t)=t^4+sin(t) p(0)=3 . Approximate $p(\pi)$ using Euler's Method and three steps.

Possible Answers:

2.934

5.166

26.222

33.288

41.950

Correct answer:

26.222

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=t^4+sin(t) p(0)=3

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{\pi-0}{3}=\frac{\pi}{3}$

p_0=3;t_0=0

$p_1=3+(\frac{\pi}{3})(0^4+sin(0))=3$

$p_2=3+(\frac{\pi}{3})((\frac{\pi}{3})^4+sin(\frac{\pi}{3}))=5.166$

$p_3=5.166+(\frac{\pi}{3})((\frac{2\pi}{3})^4+sin(\frac{2\pi}{3}))=26.222$

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Example Question #372 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=ln(t^3) p(10)=0 . Approximate p(1000) using Euler's Method and three steps.

Possible Answers:

12340.9

8020.7

23017.1

5508.2

14492.5

Correct answer:

14492.5

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=ln(t^3) p(10)=0

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{1000-10}{3}=330$

p_0=0;t_0=10

p_1=0+(330)ln(10^3)=2279.6

p_2=2279.6+(330)ln(340^3)=8050.3

p_3=8050.3+(330)ln(670^3)=14492.5

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Example Question #372 : How To Find Velocity

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=te^{t}cos(t)$ p(0)=0 . Approximate p(1) using Euler's Method and two steps

Possible Answers:

1.096

0.183

0.734

0.362

Correct answer:

0.362

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told $v(t)=te^{t}cos(t)$ p(0)=0

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{1-0}{2}=0.5$

p_0=0;t_0=0

$p_1=0+(0.5)(0)e^{0}cos(0)=0$

$p_2=0+(0.5)(0.5)e^{0.5}cos(0.5)=0.362$

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Example Question #371 : How To Find Velocity

The velocity function of a particle and a position of this particle at a known time are given by v(t)=(t-7)^3 p(6)=0 . Approximate p(7) using Euler's Method and two steps

Possible Answers:

-0.063

-0.563

-0.5

0.5

Correct answer:

-0.563

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=(t-7)^3 p(6)=0

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{7-6}{2}=0.5$

p_0=0;t_0=6

p_1=0+(0.5)(6-7)^3=-0.5

p_2=-0.5+(0.5)(6.5-7)^3=-0.563

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Example Question #375 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=4t^2+3t+1 p(0)=0 . Approximate p(2) using Euler's Method and two steps.

Possible Answers:

$\frac{56}{3}$

Correct answer:

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=4t^2+3t+1 p(0)=0

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{2-0}{2}=1$

p_0=0;t_0=0

p_1=0+(1)(4(0)^2+3(0)+1)=1

p_2=1+(1)(4(1)^2+3(1)+1)=9

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Example Question #376 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=4sin^2(t) and $p(\frac{\pi}{2})=1$ . Approximate $p(\pi)$ using Euler's Method and two steps.

Possible Answers:

4.142

5.713

2.089

6.552

1.726

Correct answer:

5.713

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=4sin^2(t) and $p(\frac{\pi}{2})=1$

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{\pi-\frac{\pi}{2}}{2}=\frac{\pi}{4}$

$p_0=1;t_0=\frac{\pi}{2}$

$p_1=1+(\frac{\pi}{4})(4sin^2(\frac{\pi}{2}))=4.142$

$p_2=4.142+(\frac{\pi}{4})(4sin^2(\frac{3\pi}{4}))=5.713$

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Example Question #377 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=tan^2(t^2) and p(0)=4 . Approximate $p(0.5\pi)$ using Euler's Method and two steps.

Possible Answers:

4.126

1.395

4.395

5.261

1.261

Correct answer:

4.395

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=tan^2(t^2) and p(0)=4

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{0.5\pi-0}{2}=0.25\pi$

p_0=4;t_0=0

$p_1=4+(0.25\pi)(tan^2(0^2))=4$

$p_2=4+(0.25\pi)(tan^2((0.25\pi)^2))=4.395$

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Example Question #378 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=cos^3(t) and p(0)=7 . Approximate $p(\pi)$ using Euler's Method and two steps.

Possible Answers:

2.316

1.571

8.571

9.316

Correct answer:

8.571

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=cos^3(t) and p(0)=7

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{\pi-0}{2}=\frac{\pi}{2}$

p_0=7;t_0=0

$p_1=7+(\frac{\pi}{2})cos^3(0)=8.571$

$p_2=8.571+(\frac{\pi}{2})cos^3(\frac{\pi}{2})=8.571$

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Example Question #379 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=ln(cos(t^3)) and p(0)=1 . Approximate p(8) using Euler's Method and two steps.

Possible Answers:

-0.75

4.75

3.75

-2.75

Correct answer:

-2.75

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told v(t)=ln(cos(t^3)) and p(0)=1

Knowing this, we may take the steps to estimate our function value at our desired value:

$\Delta t = \frac{8-0}{2}=4$

p_0=1;t_0=0

p_1=1+(4)ln(cos(0^3))=1

p_2=1+(4)ln(cos(4^3))=-2.75

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Example Question #380 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=e^{\frac{t^2}{4}}$ and p(1)=3 . Approximate p(3) using Euler's Method and two steps.

Possible Answers:

8.432

7.002

4.284

5.718

Correct answer:

7.002

Explanation:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

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

In the case of this problem, this can be rewritten as:

$p(t_n)=p(t_{n+1}) +\Delta t v(t_n)$

To calculate the step size find the difference between the final and initial value of and divide by the number of steps to be used:

$\Delta t = \frac{t_f-t_i}{Steps}$

For this problem, we are told $v(t)=e^{\frac{t^2}{4}}$ and p(1)=3

Knowing this, we may take the steps to estimate our function value at our desired value:

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

p_0=3;t_0=1

$p_1=3+(1)e^{\frac{1^2}{4}}=4.284$

$p_2=4.284+(1)e^{\frac{2^2}{4}}=7.002$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #371 : How To Find Velocity

Example Question #372 : Calculus

Example Question #372 : How To Find Velocity

Example Question #371 : How To Find Velocity

Example Question #375 : Calculus

Example Question #376 : Calculus

Example Question #377 : Calculus

Example Question #378 : Calculus

Example Question #379 : Calculus

Example Question #380 : Calculus

All Calculus 1 Resources

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