Question 1

Let f(x, y,) = xy + 5. Compute f(1, 2 + k) - f(1, 2).
Enter a polynomial in k in standard form.

Accepted Answer

k

Question 2

Let f(x, y) = $\frac { x } { y }$ - 2xy. Compute f $\left( 1 + h , \frac { 1 } { 2 } \right)$ .
Enter your answer as a polynomial in h in standard form.

Accepted Answer

h + 1

Question 3

A petroleum company has a Cobb-Douglas production function $f ( x , y ) = 70 x ^ { 2 / 5 } y ^ { 3 / 5 }$ where x is the utilization of labor and y is the utilization of capital. Determine the number of units of petroleum produced when 1200 units of labor and 2100 units of capital are used.
A) 175,174 units
B) 150,000 units
C) 105,074 units
D) 117,517 units

Accepted Answer

Plugging the values of $x = 1200$ and $y = 2100$ into the Cobb-Douglas production function $f(x, y) = 70x^{2/5}y^{3/5}$, we get $f(1200, 2100) = 70(1200)^{2/5}(2100)^{3/5}$, which calculates to approximately 117,517 units.

Question 4

Let f(x, y) = 3 $x ^ { 1 / 4 }$ $y ^ { 3 / 4 }$ . Compute f(4a, 4b).
Enter your answer as c $a ^ { n }$ $\mathrm { b^m }$ .

Accepted Answer

The answer of Let f(x, y) = 3 \(x ^...

Question 5

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute f(3, 4).
Enter just an integer.

Accepted Answer

The answer of Let f(x, y) = \(\sqrt { x...

Question 6

Let f (x, y, z) = $\frac { x y } { x + z }$ . Compute f(1, -1, -2).
Enter just an integer.

Accepted Answer

The answer of Let f (x, y, z) = \(\frac...

Question 7

A company has a Cobb-Douglas production function $f(x, y)=20 x ^{0.33} y^{0.67}$  where x is the utilization of labor and y is the utilization of capital. Determine the number of units of product produced when 1728 units of labor and 27,000 units of capital are used.
A) 29,292 units
B) 46,605 units
C) 217,988 units
D) 85,612 units

Accepted Answer

The answer of A company has a Cobb-Douglas production function...

Question 8

A rectangular garden is to be surrounded on three sides by a fence costing $5 per foot and on one side by a stone wall costing $15 per foot. Let x be the length of the side with the stone wall, and let y be the length of each of the other three sides. Express the cost of enclosing the garden as a function of two variables.
A) C(x, y) = (15x)(5y)
B) C(x, y) = 15x + 5y
C) C(x, y) = (20x)(10y)
D) C(x, y) = 15x + 15y
E) none of these

Accepted Answer

The length of the fence required for three sides (which cost $5 per foot) is 3y. Therefore, the cost of the fence is 3y * 5 = 15y.

The length of the wall required for one side (which costs $15 per foot) is x. Therefore, the cost of the wall is x * 15 = 15x.

The total cost of enclosing the garden is the sum of the cost of the fence and the wall, which is 15y + 15x. Therefore, the best choice for the function is D.

Question 9

Let f(x, y, z) = xyz(1 + $\text { eyz }$
 ). Find $\frac { \partial f } { \partial z }$ .
A) xy(1 + $\text { eyz }$ ) + $x y ^ { 2 }$ $\text { zeyz }$
B) xy(1 + $e ^ { y }$ )
C) xy(1 + $\text { eyz }$ ) + xy(1 + y $\text { eyz }$ )
D) xy(1 +  )
E) none of these

Accepted Answer

The answer of Let f(x, y, z) = xyz(1 +...

Question 10

Let f(x, y) = 8x + 9y - 6. Compute f(-6, -5).
A) 12
B) -93
C) -108
D) -107

Accepted Answer

The answer of Let f(x, y) = 8x + 9y...

Question 11

Are these the level curves of heights 0, 1, 2, and 3 for $f ( x , y ) = 3 x - 5 y + 1$ ?

Accepted Answer

The answer of Are these the level curves of heights...

Question 12

Suppose the number of leather bags produced by a certain firm, utilizing x units of raw materials and y units of labor, is given by the formula f(x, y) = 10 $x ^ { 1 / 2 }$ $y ^ { 1 / 2 }$ . What happens to the production of bags if supplies of both raw materials and labor are tripled?
A) Production is increased by a factor of 6.
B) Production is increased by a factor of 9.
C) Production is increased by a factor of 3.
D) Production is increased by a factor of $\sqrt { 3 }$ .
E) none of these

Accepted Answer

The answer of Suppose the number of leather bags produced...

Question 13

$\text { Let } f ( x , y ) = x ^ { 2 } y ^ { 2 } - 1$Compute $\frac { f ( - 1,1 + k ) - f ( - 1,1 ) } { k }$ for k ≠ 0.
Enter your answer as a polynomial in k in standard form.

Accepted Answer

The answer of \(\text { Let } f ( x...

Question 14

Let f(x, y) = ln(y + 2) + $e ^ { 2 x y }$ . Compute f(0, -1).
A) e
B) 1
C) 1 + e
D) 0

Accepted Answer

Substituting x = 0 and y = -1 into the function f(x, y) = ln(y + 2) + $e ^ { 2 x y }$ gives f(0, -1) = ln(-1 + 2) + $e ^ { 2 \cdot 0 \cdot -1 }$ = ln(1) + $e^0$ = 0 + 1 = 1.

Question 15

Let $f ( x , y , z ) = ( \sqrt { x y z } + \sqrt { - 2 z } + \sqrt { x - z } ) .$ Compute f(1, -2, -1).
Enter your answer as a $\sqrt { b }$ .

Accepted Answer

The answer of Let \(f ( x , y ,...

Question 16

A company makes cylindrical cans of radius r and height h at a cost of a cents per unit area for the top and bottom and b cents per unit area for the side. Express the cost of producing a can as a function of the two variables r, and h.
Enter your answer in the form: $2 \pi r ( \mathrm { cr } \pm \mathrm { dh } )$

Accepted Answer

The answer of A company makes cylindrical cans of radius...

Question 17

Let f(x, y) = x $y ^ { 2 }$ + $x ^ { 2 }$ . Simplify $\frac { f ( x + h , y ) - f ( x , y ) } { h }$ for h ≠ 0.
Enter your answer exactly in the form: $a x + y ^ { n } + b$

Accepted Answer

The answer of Let f(x, y) = x \(y ^...

Question 18

Let g(x, y) = 9 $y ^ { 2 }$ - 2xy. Compute g(-7, -3).
A) 46
B) 399
C) 39
D) 396

Accepted Answer

To compute $g(-7, -3)$, substitute $x = -7$ and $y = -3$ into the given function: $g(x, y) = 9y^2 - 2xy$. Thus, $g(-7, -3) = 9(-3)^2 - 2(-7)(-3) = 9(9) - 2(7)(3) = 81 - 42 = 39$.

Question 19

Let g(x, y) = $\frac { x - 6 y } { x ^ { 2 } + y ^ { 2 } }$ . Compute g(3, 4).
A) - $\frac { 25 } { 21 }$
B) - $\frac { 21 } { 25 }$
C) - $\frac { 21 } { 5 }$
D) - $\frac { 5 } { 21 }$

Accepted Answer

To compute $g(3, 4)$, substitute $x = 3$ and $y = 4$ into the given function: $g(3, 4) = \frac{3 - 6(4)}{3^2 + 4^2} = \frac{3 - 24}{9 + 16} = \frac{-21}{25}$.

Question 20

Let f(x, y) = $\frac { 1 } { 2 }$ $x ^ { 2 }$ ∙ $e ^ { x / y }$ . Find $\frac { \partial f } { \partial y }$ .
A) $\frac { x ^ { 2 } } { y }$ $e ^ { x / y }$
B) $\frac { x ^ { 3 } } { y }$ $e ^ { x / y }$ - 1
C) - $\frac { x ^ { 3 } } { 2 y ^ { 2 } }$ $e ^ { x / y }$
D) $\frac { 1 } { 2 }$ $x ^ { 2 }$ $e ^ { x / y }$
E) none of these

Accepted Answer

To find $\frac{\partial f}{\partial y}$, we apply the chain rule to the exponential function $e^{x/y}$. Differentiating $f(x, y) = \frac{1}{2}x^2e^{x/y}$ with respect to $y$, we treat $x$ as a constant. The derivative of $e^{x/y}$ with respect to $y$ involves the chain rule, giving $-\frac{x}{y^2}e^{x/y}$. Multiplying this by $\frac{1}{2}x^2$ gives $-\frac{x^3}{2y^2}e^{x/y}$, which matches option C.

Quiz 7: Functions of Several Variables

Let f(x, y,) = xy + 5. Compute f(1, 2 + k) - f(1, 2). Enter a polynomial in k in standard form.

Let f(x, y) = $\frac { x } { y }$ - 2xy. Compute f $\left( 1 + h , \frac { 1 } { 2 } \right)$ . Enter your answer as a polynomial in h in standard form.

A petroleum company has a Cobb-Douglas production function $f ( x , y ) = 70 x ^ { 2 / 5 } y ^ { 3 / 5 }$ where x is the utilization of labor and y is the utilization of capital. Determine the number of units of petroleum produced when 1200 units of labor and 2100 units of capital are used.

Let f(x, y) = 3 $x ^ { 1 / 4 }$ $y ^ { 3 / 4 }$ . Compute f(4a, 4b). Enter your answer as c $a ^ { n }$ $\mathrm { b^m }$ .

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute f(3, 4). Enter just an integer.

Let f (x, y, z) = $\frac { x y } { x + z }$ . Compute f(1, -1, -2). Enter just an integer.

A company has a Cobb-Douglas production function $f(x, y) =20 x ^{0.33} y^{0.67}$ where x is the utilization of labor and y is the utilization of capital. Determine the number of units of product produced when 1728 units of labor and 27,000 units of capital are used.

Let f(x, y, z) = xyz(1 + $\text { eyz }$ ) . Find $\frac { \partial f } { \partial z }$ .

Let f(x, y) = 8x + 9y - 6. Compute f(-6, -5) .

Are these the level curves of heights 0, 1, 2, and 3 for $f ( x , y ) = 3 x - 5 y + 1$ ?

Suppose the number of leather bags produced by a certain firm, utilizing x units of raw materials and y units of labor, is given by the formula f(x, y) = 10 $x ^ { 1 / 2 }$ $y ^ { 1 / 2 }$ . What happens to the production of bags if supplies of both raw materials and labor are tripled?

$\text { Let } f ( x , y ) = x ^ { 2 } y ^ { 2 } - 1$ Compute $\frac { f ( - 1,1 + k ) - f ( - 1,1 ) } { k }$ for k ≠ 0. Enter your answer as a polynomial in k in standard form.

Let f(x, y) = ln(y + 2) + $e ^ { 2 x y }$ . Compute f(0, -1) .

Let $f ( x , y , z ) = ( \sqrt { x y z } + \sqrt { - 2 z } + \sqrt { x - z } ) .$ Compute f(1, -2, -1). Enter your answer as a $\sqrt { b }$ .

Let f(x, y) = x $y ^ { 2 }$ + $x ^ { 2 }$ . Simplify $\frac { f ( x + h , y ) - f ( x , y ) } { h }$ for h ≠ 0. Enter your answer exactly in the form: $a x + y ^ { n } + b$

Let g(x, y) = 9 $y ^ { 2 }$ - 2xy. Compute g(-7, -3) .

Let g(x, y) = $\frac { x - 6 y } { x ^ { 2 } + y ^ { 2 } }$ . Compute g(3, 4) .

Let f(x, y) = $\frac { 1 } { 2 }$ $x ^ { 2 }$ ∙ $e ^ { x / y }$ . Find $\frac { \partial f } { \partial y }$ .

Quiz 7: Functions of Several Variables

Let f(x, y,) = xy + 5. Compute f(1, 2 + k) - f(1, 2). Enter a polynomial in k in standard form.

Let f(x, y) = xy\frac { x } { y }yx​ - 2xy. Compute f (1+h,12)\left( 1 + h , \frac { 1 } { 2 } \right)(1+h,21​) . Enter your answer as a polynomial in h in standard form.

Let f(x, y) = 3 x1/4x ^ { 1 / 4 }x1/4 y3/4y ^ { 3 / 4 }y3/4 . Compute f(4a, 4b). Enter your answer as c ana ^ { n }an bm\mathrm { b^m }bm .

Let f(x, y) = x2+y2\sqrt { x ^ { 2 } + y ^ { 2 } }x2+y2​ . Compute f(3, 4). Enter just an integer.

Let f (x, y, z) = xyx+z\frac { x y } { x + z }x+zxy​ . Compute f(1, -1, -2). Enter just an integer.

Let f(x, y, z) = xyz(1 + eyz \text { eyz } eyz ) . Find ∂f∂z\frac { \partial f } { \partial z }∂z∂f​ .

Let f(x, y) = 8x + 9y - 6. Compute f(-6, -5) .

Are these the level curves of heights 0, 1, 2, and 3 for f(x,y)=3x−5y+1f ( x , y ) = 3 x - 5 y + 1f(x,y)=3x−5y+1 ?

Suppose the number of leather bags produced by a certain firm, utilizing x units of raw materials and y units of labor, is given by the formula f(x, y) = 10 x1/2x ^ { 1 / 2 }x1/2 y1/2y ^ { 1 / 2 }y1/2 . What happens to the production of bags if supplies of both raw materials and labor are tripled?

Let f(x,y)=x2y2−1\text { Let } f ( x , y ) = x ^ { 2 } y ^ { 2 } - 1 Let f(x,y)=x2y2−1 Compute f(−1,1+k)−f(−1,1)k\frac { f ( - 1,1 + k ) - f ( - 1,1 ) } { k }kf(−1,1+k)−f(−1,1)​ for k ≠ 0. Enter your answer as a polynomial in k in standard form.

Let f(x, y) = ln(y + 2) + e2xye ^ { 2 x y }e2xy . Compute f(0, -1) .

Let f(x,y,z)=(xyz+−2z+x−z).f ( x , y , z ) = ( \sqrt { x y z } + \sqrt { - 2 z } + \sqrt { x - z } ) .f(x,y,z)=(xyz​+−2z​+x−z​). Compute f(1, -2, -1). Enter your answer as a b\sqrt { b }b​ .

Let f(x, y) = x y2y ^ { 2 }y2 + x2x ^ { 2 }x2 . Simplify f(x+h,y)−f(x,y)h\frac { f ( x + h , y ) - f ( x , y ) } { h }hf(x+h,y)−f(x,y)​ for h ≠ 0. Enter your answer exactly in the form: ax+yn+ba x + y ^ { n } + bax+yn+b

Let g(x, y) = 9 y2y ^ { 2 }y2 - 2xy. Compute g(-7, -3) .

Let g(x, y) = x−6yx2+y2\frac { x - 6 y } { x ^ { 2 } + y ^ { 2 } }x2+y2x−6y​ . Compute g(3, 4) .

Let f(x, y) = 12\frac { 1 } { 2 }21​ x2x ^ { 2 }x2 ∙ ex/ye ^ { x / y }ex/y . Find ∂f∂y\frac { \partial f } { \partial y }∂y∂f​ .

Let f(x, y) = $\frac { x } { y }$ - 2xy. Compute f $\left( 1 + h , \frac { 1 } { 2 } \right)$ . Enter your answer as a polynomial in h in standard form.

Let f(x, y) = 3 $x ^ { 1 / 4 }$ $y ^ { 3 / 4 }$ . Compute f(4a, 4b). Enter your answer as c $a ^ { n }$ $\mathrm { b^m }$ .

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute f(3, 4). Enter just an integer.

Let f (x, y, z) = $\frac { x y } { x + z }$ . Compute f(1, -1, -2). Enter just an integer.

Let f(x, y, z) = xyz(1 + $\text { eyz }$ ) . Find $\frac { \partial f } { \partial z }$ .

Are these the level curves of heights 0, 1, 2, and 3 for $f ( x , y ) = 3 x - 5 y + 1$ ?

Suppose the number of leather bags produced by a certain firm, utilizing x units of raw materials and y units of labor, is given by the formula f(x, y) = 10 $x ^ { 1 / 2 }$ $y ^ { 1 / 2 }$ . What happens to the production of bags if supplies of both raw materials and labor are tripled?

$\text { Let } f ( x , y ) = x ^ { 2 } y ^ { 2 } - 1$ Compute $\frac { f ( - 1,1 + k ) - f ( - 1,1 ) } { k }$ for k ≠ 0. Enter your answer as a polynomial in k in standard form.

Let f(x, y) = ln(y + 2) + $e ^ { 2 x y }$ . Compute f(0, -1) .

Let $f ( x , y , z ) = ( \sqrt { x y z } + \sqrt { - 2 z } + \sqrt { x - z } ) .$ Compute f(1, -2, -1). Enter your answer as a $\sqrt { b }$ .

Let f(x, y) = x $y ^ { 2 }$ + $x ^ { 2 }$ . Simplify $\frac { f ( x + h , y ) - f ( x , y ) } { h }$ for h ≠ 0. Enter your answer exactly in the form: $a x + y ^ { n } + b$

Let g(x, y) = 9 $y ^ { 2 }$ - 2xy. Compute g(-7, -3) .

Let g(x, y) = $\frac { x - 6 y } { x ^ { 2 } + y ^ { 2 } }$ . Compute g(3, 4) .

Let f(x, y) = $\frac { 1 } { 2 }$ $x ^ { 2 }$ ∙ $e ^ { x / y }$ . Find $\frac { \partial f } { \partial y }$ .