Calculus homework

Question 1

1.

Find fx, fy, and fz. f(x,y,z) = xe(x2 + y2 + z2)

fx = (1 + 2x2) e(x2 + y2 + z2); fy = xy2e(x2 + y2 + z2); fz = xz2e(x2 + y2 + z2)

fx = (1 + 2x2) e(x2 + y2 + z2); fy = 2xye(x2 + y2 + z2); fz = 2xze(x2 + y2 + z2)

fx = 2x2e(x2 + y2 + z2); fy = xye(x2 + y2 + z2); fz = 2xze(x2 + y2 + z2)

fx = (1 + 2x2) e(x2 + y2 + z2); fy = xe(x2 + y2 + z2); fz = xe(x2 + y2 + z2)

5 points

Question 2

1.

Find fx, fy, and fz.

f(x, y, z) =

fx = ; fy = ; fz =

fx = - ; fy = - ; fz = -

fx = - ; fy = - ; fz = -

fx = ; fy = ; fz =

5 points

Question 3

1.

Find and . f(x, y) = ln yx

= ln y; = - xln y

= ln y; =

= 0; = -

= xln y; = -

5 points

Question 4

1.

Find all the second order partial derivatives of the given function. f(x, y) = ex/y

= ; = ; = =

= ; = ex/y ; = = -ex/y

= ; = -ex/y ; = = ex/y

= ; = ex/y ; = = -ex/y

5 points

Question 5

1.

Find and .

f(x, y) =

= ; =

= - ; = -

= ; = -

= ; = -

5 points

Question 6

1.

Find fx, fy, and fz.

f(x, y, z) =

fx = ; fy = ; fz =

fx = - ; fy = ; fz = -

fx = ; fy = - ; fz =

fx = - ; fy = - ; fz =

5 points

Question 7

1.

Find and . f(x, y) = xye-y

= ye-y; = xe-y(y - 1)

= ye-y; = xe-y(1 - y)

= ye-y; = - xye-y

= ye-y; = xe-y

5 points

Question 8

1.

Find the specific function value. Find f(3, 0, 9) when f(x, y, z) = 3x2 + 3y2 - z2.

-48

-72

108

-54

5 points

Question 9

1.

Find the specific function value. Find f(0, 1, -1) when f(x, y, z) = 3x - 4yz + 9x.

-3

5

4

-4

5 points

Question 10

1.

Find and .

f(x, y) =

= - ; = -

= ; =

= - ; = -

= - ; = -

5 points

Question 11

1.

Find fx, fy, and fz. f(x, y, z) = e(yz + sin x)

fx = (cos x)e(yz + sin x); fy = ze(yz + sin x); fz = ye(yz + sin x)

fx = (cos x)e(yz + sin x); fy = e(yz + sin x); fz = e(yz + sin x)

fx = e(yz + sin x); fy = ze(yz + sin x); fz = ye(yz + sin x)

fx = (cos x + yz)e(yz + sin x); fy = ze(yz + sin x); fz = ye(yz + sin x)

5 points

Question 12

1.

Find all the second order partial derivatives of the given function. f(x, y) = ln (x2y - x)

= ; = - ; = = -

= ; = - ; = = -

= ; = ; = =

= ; = - ; = = -

5 points

Question 13

1.

Find the specific function value.

Find f(3, 4) when f(x, y) = .

-

-

-

-

5 points

Question 14

1.

Find fx, fy, and fz. f(x, y, z) = cos x sin2 (yz)

fx = sin x sin2 (yz); fy = -2z cos x sin (yz) cos (yz); fz = - 2y cos x sin (yz) cos (yz)

fx = sin x sin2 (yz); fy = - cos x sin (yz) cos (yz); fz = -cos x sin (yz) cos (yz)

fx = -sin x sin2 (yz); fy = z cos x sin (yz) cos (yz); fz = y cos x sin (yz) cos (yz)

fx = -sin x sin2 (yz); fy = 2z cos x sin (yz) cos (yz); fz = 2y cos x sin (yz) cos (yz)

5 points

Question 15

1.

Find all the second order partial derivatives of the given function.

f(x, y) =

= - ; = ; = =

= - ; = ; = =

= - ; = ; = =

= ; = - ; = =

5 points

Question 16

1.

Find and .

f(x, y) =

= - ; = -

= ; =

= - ; = -

= - ; = -

5 points

Question 17

1.

Find and .

f(x, y) =

= - ; = -

= ; = -

= ; =

= ; = -

5 points

Question 18

1.

Find all the second order partial derivatives of the given function. f(x, y) = xye-y2

= ye-y2; = 2xye-y2(y2 - 1); = = e-y2(1 - y2)

= 0 ; = 2xye-y2(2y2 - 3); = = e-y2(1 - y2)

= 0; = 2xye-y2(2y2 - 3); = = e-y2(1 - 2y2)

= ye-y2; = 2xye-y2(2y2 - 6); = = e-y2(1 - 2y2)

5 points

Question 19

1.

Find the specific function value. Find f(2, 3) when f(x, y) = (x + y)3.

25

15

125

35

5 points

Question 20

1.

Find and . f(x, y) = x3 + 9x2y + 2xy3

= x2 + 9xy + 2y3; = 9x2 + 2xy2

= 3x2 + 2xy + 2y3; = 9x2 + 3xy2

= 3x2; = 9x2 + 6xy2

= 3x2 + 18xy + 2y3; = 9x2 + 6xy2

5 points