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Calculus 1 : Differential Functions

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Example Questions

Example Question #481 : Other Differential Functions

Find the divergence of the function $F(x,y)=xcos(y)\widehat{i}+xcos(y)\widehat{j}$ at $(\pi,\pi)$

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

$1+\pi$

$\pi$

$1-\pi$

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y)=xcos(y)\widehat{i}+xcos(y)\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

d(cos(u))=-sin(u)du

Then sum the results together:

divF(x,y)=cos(y)-xsin(y)

At the point $(\pi,\pi)$

$divF(\pi,\pi)=cos(\pi)-\pi sin(\pi)$

$divF(\pi,\pi)=1$

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Example Question #671 : Differential Functions

Find the divergence of the function $F(x,y)=x^2cos(y)\widehat{i}+x^2sin(y)\widehat{j}$ at $(3,2\pi)$

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y)=x^2cos(y)\widehat{i}+x^2sin(y)\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

d(sin(u))=cos(u)du

divF(x,y)=2xcos(y)+x^2cos(y)

At the point $(3,2\pi)$

$divF(3,2\pi)=2(3)cos(2\pi)+3^2cos(2\pi)=6+9=15$

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Example Question #672 : Differential Functions

Find the divergence of the function $F(x,y)=e^xln(y)\widehat{i}+cos(x)sin(y)\widehat{j}$ at (2,0.2)

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

-6.6

-12.3

5.9

-1.2

11.8

Correct answer:

-12.3

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y)=e^xln(y)\widehat{i}+cos(x)sin(y)\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

d(e^u)=e^udu

d(sin(u))=cos(u)du

div F(x,y)=e^xln(y)+cos(x)cos(y)

At the point (2,0.2)

div F(2,0.2)=e^2ln(0.2)+cos(2)cos(0.2)=-12.3

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Example Question #673 : Differential Functions

Find the divergence of the function $F(x,y)=18xy^2\widehat{i}+21x^2y\widehat{j}$ at (0.2,0.1)

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

10.2

20.4

1.02

3.12

Correct answer:

1.02

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y)=18xy^2\widehat{i}+21x^2y\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

divF(x,y)=18y^2+21x^2

At the point (0.2,0.1)

divF(0.2,0.1)=18(0.1)^2+21(0.2)^2=1.02

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Example Question #488 : Other Differential Functions

Find the divergence of the function $F(x,y)=sin(xy^2)\widehat{i}+e^ycos(2x)\widehat{j}$ at (3,1)

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

2.62

1.62

1.18

4.44

3.95

Correct answer:

1.62

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y)=sin(xy^2)\widehat{i}+e^ycos(2x)\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

Derivative of an exponential:

d[e^u]=e^udu

Trigonometric derivative:

d[sin(u)]=cos(u)du

Note that u may represent large functions, and not just individual variables!

divF(x,y)=y^2cos(xy^2)+e^ycos(2x)

At the point (3,1)

divF(3,1)=1^2cos(3(1)^2)+e^1cos(2(3))=1.62

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Example Question #1701 : Calculus

Find the divergence of the function $F(x,y)=x^y\widehat{i}+y^x\widehat{j}$ at (2,3)

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

16.2

3.7

14.2

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y)=x^y\widehat{i}+y^x\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

$divF(x,y)=yx^{y-1}+xy^{x-1}$

At the point (2,3)

$divF(2,3)=(3)(2^{3-1})+(2)(3^{2-1})=18$

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Example Question #1702 : Calculus

Find the divergence of the function $F(x,y)=y^x\widehat{i}+x^y\widehat{j}$ at (2,3)

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

7.9

12.2

15.4

Correct answer:

15.4

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y)=y^x\widehat{i}+x^y\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

Derivative of an exponential:

d[a^u]=a^uduln(a)

Note that u may represent large functions, and not just individual variables!

divF(x,y)=y^xln(y)+x^yln(x)

At the point (2,3)

divF(2,3)=3^2ln(3)+2^3ln(2)=15.4

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Example Question #676 : Differential Functions

Find the divergence of the function $F(x,y,z)=4xz^2\widehat{i}+3xyz\widehat{j}+2yz^3\widehat{k}$ at (4,1,1)

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y,z)=4xz^2\widehat{i}+3xyz\widehat{j}+2yz^3\widehat{k}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

divF(x,y,z)=4z^2+3xz+6yz^2

At the point (4,1,1)

divF(4,1,1)=4(1)^2+3(4)(1)+6(1)(1)^2=22

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Example Question #677 : Differential Functions

Find the divergence of the function $F(x,y,z)=y^2z^3\widehat{i}+3x^3y^2\widehat{j}+x^2z\widehat{k}$ at (3,2,4)

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

216

297

333

153

Correct answer:

333

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y,z)=y^2z^3\widehat{i}+3x^3y^2\widehat{j}+x^2z\widehat{k}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

divF(x,y,z)=0+6x^3y+x^2

At the point (3,2,4)

divF(3,2,4)=6(3)^3(2)+3^2=333

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Example Question #678 : Differential Functions

Find the divergence of the function $F(x,y)=ln(x^2y)\widehat{i}+3^{xy^2}\widehat{j}$ at (1.3,1.7)

Hint: $divF(x_1,x_2,...x_n)=\frac{\delta F_{x_1}}{\delta x_1}+\frac{\delta F_{x_2}}{\delta x_2}+...+\frac{\delta F_{x_n}}{\delta x_n}$

Possible Answers:

617.89

302.71

12.13

441.23

189.22

Correct answer:

302.71

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

$F(x,y)=ln(x^2y)\widehat{i}+3^{xy^2}\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $(\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

Derivative of an exponential:

d[a^u]=a^uduln(a)

Derivative of a natural log:

$d[ln(u)]=\frac{du}{u}$

Note that u may represent large functions, and not just individual variables!

$divF(x,y)=\frac{2xy}{x^2y}+2xy(3^{xy^2})ln(3)$

$divF(x,y)=\frac{2}{x}+2xy(3^{xy^2})ln(3)$

At the point (1.3,1.7)

$divF(1.3,1.7)=\frac{2}{1.3}+2(1.3)(1.7)(3^{(1.3)(1.7)^2})ln(3)$

divF(1.3,1.7)=302.71

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Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #481 : Other Differential Functions

Example Question #671 : Differential Functions

Example Question #672 : Differential Functions

Example Question #673 : Differential Functions

Example Question #488 : Other Differential Functions

Example Question #1701 : Calculus

Example Question #1702 : Calculus

Example Question #676 : Differential Functions

Example Question #677 : Differential Functions

Example Question #678 : Differential Functions

All Calculus 1 Resources

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