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Calculus 1 : Calculus

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Example Questions

Example Question #311 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=cos(\sqrt[5]{t})$ p(0)=0 . Approximate p(9) using Euler's Method and three steps.

Possible Answers:

0.958

1.376

3.958

4.376

Correct answer:

4.376

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 $v(t)=cos(\sqrt[5]{t})$ p(0)=0

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

$\Delta t = \frac{9-0}{3}=3$

p_0=0;t_0=0

$p_1=0+(3)cos(\sqrt[5]{0})=3$

$p_2=3+(3)cos(\sqrt[5]{3})=3.958$

$p_3=3.958+(3)cos(\sqrt[5]{6})=4.376$

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Example Question #312 : Spatial Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=csc^3(t^2) p(1)=2 . Approximate p(1.6) using Euler's Method and three steps.

Possible Answers:

3.459

2.794

2.541

2.336

3.021

Correct answer:

2.794

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)=csc^3(t^2) p(1)=2

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

$\Delta t = \frac{1.6-1}{3}=0.2$

p_0=2;t_0=1

p_1=2+(0.2)csc^3(1^2)=2.336

p_2=2.336+(0.2)csc^3(1.2^2)=2.541

p_3=2.541+(0.2)csc^3(1.4^2)=2.794

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Example Question #311 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by v(t)=sin(4t^4) p(1.1)=0 . Approximate p(1.4) using Euler's Method and three steps.

Possible Answers:

-0.065

0.049

-0.091

0.083

-0.042

Correct answer:

-0.042

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)=sin(4t^4) p(1.1)=0

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

$\Delta t = \frac{1.4-1.1}{3}=0.1$

p_0=0;t_0=1.1

p_1=0+(0.1)sin(4(1.1)^4)=-0.041

p_2=-0.041+(0.1)sin(4(1.2)^4)=0.049

p_3=0.049+(0.1)sin(4(1.3)^4)=-0.042

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Example Question #312 : Velocity

The velocity function of a particle and a position of this particle at a known time are given by v(t)=5sin(t^3-t^2) p(1)=0 . Approximate p(7) using Euler's Method and three steps.

Possible Answers:

-9.543

-6.904

-15.251

-7.625

-13.808

Correct answer:

-15.251

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)=5sin(t^3-t^2) p(1)=0

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

$\Delta t = \frac{7-1}{3}=2$

p_0=0;t_0=1

p_1=0+(2)(5sin(1^3-1^2))=0

p_2=0+(2)(5sin(2^3-2^2))=-7.568

p_3=-7.568+(2)(5sin(4^3-4^2))=-15.251

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Example Question #313 : Velocity

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

Possible Answers:

1.137

0.137

4.137

3.137

2.137

Correct answer:

2.137

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(t^4-t) p(0)=0

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

$\Delta t = \frac{3-0}{3}=1$

p_0=0;t_0=0

p_1=0+(1)cos(0^4-0)=1

p_2=1+(1)cos(1^4-1)=2

p_3=2+(1)cos(2^4-2)=2.137

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Example Question #314 : Velocity

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=1.2^{t^3-t^2}$ p(0)=4 . Approximate p(3) using Euler's Method and three steps.

Possible Answers:

7.702

8.073

9.702

6.073

Correct answer:

8.073

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 $v(t)=1.2^{t^3-t^2}$ p(0)=4

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

$\Delta t = \frac{3-0}{3}=1$

p_0=4;t_0=0

$p_1=4+(1)1.2^{0^3-0^2}=5$

$p_2=5+(1)1.2^{1^3-1^2}=6$

$p_3=6+(1)1.2^{2^3-2^2}=8.073$

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Example Question #315 : Velocity

The velocity function of a particle and a position of this particle at a known time are given by v(t)=2t^3 p(0)=0 . Approximate p(0.9) using Euler's Method and three steps.

Possible Answers:

0.2358

0.1458

0.1296

0.0162

0.3393

Correct answer:

0.1458

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)=2t^3 p(0)=0

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

$\Delta t = \frac{0.9-0}{3}=0.3$

p_0=0;t_0=0

p_1=0+(0.3)2(0)^3=0

p_2=0+(0.3)2(0.3)^3=0.0162

p_3=0.0162+(0.3)2(0.6)=0.1458

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Example Question #316 : Velocity

The velocity function of a particle and a position of this particle at a known time are given by v(t)=t! p(1)=8 . Approximate p(4) using Euler's Method and three steps.

Possible Answers:

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)=t! p(1)=8

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

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

p_0=8;t_0=1

p_1=8+(1)1!=9

p_2=9+(1)2!=11

p_3=11+(1)3!=17

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Example Question #319 : Calculus

The velocity function of a particle and a position of this particle at a known time are given by $v(t)=cos(\pi t!)$ p(1)=0 . Approximate p(10) using Euler's Method and three steps.

Possible Answers:

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 $v(t)=cos(\pi t!)$ p(1)=0

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

$\Delta t = \frac{10-1}{3}=3$

p_0=0;t_0=1

$p_1=0+(3)cos(\pi (1)!)=-3$

$p_2=-3+(3)cos(\pi (3)!)=0$

$p_3=0+(3)cos(\pi (6)!)=3$

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Example Question #317 : Velocity

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

Possible Answers:

$\frac{11}{7}$

$\frac{11}{4}$

$\frac{7}{2}$

$\frac{11}{2}$

$\frac{7}{4}$

Correct answer:

$\frac{7}{2}$

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 $v(t)=sin^2(\frac{\pi}{3}t!)$ p(1)=2

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

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

p_0=2;t_0=1

$p_1=2+(1)sin^2(\frac{\pi}{3}(1)!)=\frac{11}{4}$

$p_2=\frac{11}{4}+(1)sin^2(\frac{\pi}{3}(2)!)=\frac{7}{2}$

$p_3=\frac{7}{2}+(1)sin^2(\frac{\pi}{3}(3)!)=\frac{7}{2}$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #311 : Spatial Calculus

Example Question #312 : Spatial Calculus

Example Question #311 : Calculus

Example Question #312 : Velocity

Example Question #313 : Velocity

Example Question #314 : Velocity

Example Question #315 : Velocity

Example Question #316 : Velocity

Example Question #319 : Calculus

Example Question #317 : Velocity

All Calculus 1 Resources

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