Question 1

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute $\frac { \partial f } { \partial x }$ at (3, 4).
Enter just a reduced fraction $\frac { a } { b }$ .

Accepted Answer

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

Question 2

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.

Accepted Answer

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

Question 3

Use the method of your choice to obtain the formula for the least-squares line to fit the data (0, 6), (1, 3), $( 2,3 ) .$ Enter your answer in standard point-intercept form with any fractions reduced of form $\frac { a } { b }$ .

Accepted Answer

The answer of Use the method of your choice to...

Question 4

Let f(x, y) = $x ^ { 2 }$ - xy + $y ^ { 2 }$ + 2y - 4. The point $- \frac { 2 } { 3 } , - \frac { 4 } { 3 }$ is a

Accepted Answer

A)  relative maximum 
B)  not a relative extreme point 
C)  absolute maximum 
D)  relative minimum

Question 5

Maximize the function f(x, y) = $x ^ { 2 }$ - $y ^ { 2 }$ subject to the constraint $y - x ^ { 2 } = - \frac { 1 } { 2 } , x , y > 0$ 
Enter your answer as just a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

The answer of Maximize the function f(x, y) = \(x...

Question 6

Minimize the function f(x) = x + y, subject to the constraint xy = 100, x > 0, y > 0. Use the method of Lagrange multipliers.
Enter your answer exactly as just (a, b), c where (a, b) gives the minimum and c is the Lagrange multiplier as a reduced fraction of form $\frac { d } { e }$ (no words or labels).

Accepted Answer

The answer of Minimize the function f(x) = x +...

Question 7

Let f(x, y) = 5 $x ^ { 2 }$ - 5 $y ^ { 2 }$ + 2xy + 34x + 38y + 12. At which point does f(x, y) have a possible maximum or minimum value?

Accepted Answer

A)  (4, 3) 
B)  (-4, -3) 
C)  (-4, 3) 
D)  (4, -3)

Question 8

Design a cylindrical can of volume 100 cubic units that requires a minimum amount of aluminum; that is, the can is to have a minimum surface area.
Enter your answer exactly as just r, h where r is exactly of form $\sqrt [ 3 ] { \frac { a } { b } }$ representing radius, and h is exactly of form $c \sqrt [ 3 ] { \frac { \mathrm { d } } { \mathrm { e } } }$ representing height (no labels, words, or units).

Accepted Answer

The answer of Design a cylindrical can of volume 100...

Question 9

Let f(x, y) = $\frac { x } { y + 1 }$ . Find $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ .
Enter just ± $( \mathrm { P } ( \mathrm { y } ) ) ^ { \mathrm { a } }$ where P is a polynomial in standard form (do not label).

Accepted Answer

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

Question 10

Let f(x, y) = $\frac { \ln x y } { y }$ . Find $\frac { \partial f } { \partial x }$ . Is $( x y ) ^ { - 1 }$ the correct answer?

Accepted Answer

A) True 
 B)False

Question 11

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

Accepted Answer

The answer of Let f(x, y,) = xy + 5....

Question 12

Find the greatest possible volume of a rectangular box that has length plus girth equal to 60 inches. Enter your answer as a single integer (no units).

Accepted Answer

The answer of Find the greatest possible volume of a...

Question 13

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

Accepted Answer

A) True 
 B)False

Question 14

Find all points (x, y) where $f ( x , y ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } \right) }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Accepted Answer

A)  (0, 1) gives a relative minimum point 
B)  (0, 0) gives a relative minimum point 
C)  (0, 1) gives a relative maximum point 
D)  (0, 0) gives a relative maximum point

Question 15

Calculate the iterated integral $\int _ { - 1 } ^ { 2 } \left( \int _ { 0 } ^ { 1 / x ^ { 2 } } x ^ { 3 } e ^ { x ^ { 3 } y } d y \right)$ dx.
Enter your answer exactly in the form $e ^ { a }$ ± b - $\mathrm { e } ^ { \mathrm { C } }$ .

Accepted Answer

The answer of Calculate the iterated integral \(\int _ {...

Question 16

Let E = $( 2 A + B - 3 ) ^ { 2 }$ + (A + B + 2) + (4A + B - 1 $)^2$ . What is $\frac { \partial \mathrm { E } } { \partial \mathrm { A } }$ ?

Accepted Answer

A)  42A + 24B - 9 
B)  14A + 6B - 4 
C)  42A + 14B - 9 
D)  24A + 6B - 4 
E)  none of these

Question 17

Find all points (x, y) where $f ( x , y ) = \frac { 1 } { x } + x y - \frac { 1 } { y }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Accepted Answer

A)  (1, 1) gives a relative minimum point 
B)  (-1, 1) gives a relative maximum point 
C)  (-1, 1) gives a relative minimum point 
D)  Neither a relative maximum nor a relative minimum at (-1, 1)

Question 18

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 19

Maximize f(x, y, z) = x + y + z subject to the constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 , x , y , z > 0$ Enter your answer as just $3 ^a$ where a is a reduced fraction of form $\frac { b } { c }$ .

Accepted Answer

The answer of Maximize f(x, y, z) = x +...

Question 20

Let R be the rectangle consisting of all points (x, y) such that 1 ≤ x ≤ 25, 1 ≤ y ≤ 16. 
Calculate $\iint _ { R } 5 \sqrt { x y } d y d x$

Accepted Answer

A)  795,600 
B)  156,240 
C)  17,360 
D)  198,900

Exam 7: Functions of Several Variables

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute $\frac { \partial f } { \partial x }$ at (3, 4). Enter just a reduced fraction $\frac { a } { b }$ .

Use the method of your choice to obtain the formula for the least-squares line to fit the data (0, 6), (1, 3), $( 2,3 ) .$ Enter your answer in standard point-intercept form with any fractions reduced of form $\frac { a } { b }$ .

Let f(x, y) = $x ^ { 2 }$ - xy + $y ^ { 2 }$ + 2y - 4. The point $- \frac { 2 } { 3 } , - \frac { 4 } { 3 }$ is a

Maximize the function f(x, y) = $x ^ { 2 }$ - $y ^ { 2 }$ subject to the constraint $y - x ^ { 2 } = - \frac { 1 } { 2 } , x , y > 0$ Enter your answer as just a reduced fraction of form $\frac { a } { b }$ .

Let f(x, y) = 5 $x ^ { 2 }$ - 5 $y ^ { 2 }$ + 2xy + 34x + 38y + 12. At which point does f(x, y) have a possible maximum or minimum value?

Let f(x, y) = $\frac { x } { y + 1 }$ . Find $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ . Enter just ± $( \mathrm { P } ( \mathrm { y } ) ) ^ { \mathrm { a } }$ where P is a polynomial in standard form (do not label).

Let f(x, y) = $\frac { \ln x y } { y }$ . Find $\frac { \partial f } { \partial x }$ . Is $( x y ) ^ { - 1 }$ the correct answer?

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

Find the greatest possible volume of a rectangular box that has length plus girth equal to 60 inches. Enter your answer as a single integer (no units).

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

Find all points (x, y) where $f ( x , y ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } \right) }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Calculate the iterated integral $\int _ { - 1 } ^ { 2 } \left( \int _ { 0 } ^ { 1 / x ^ { 2 } } x ^ { 3 } e ^ { x ^ { 3 } y } d y \right)$ dx. Enter your answer exactly in the form $e ^ { a }$ ± b - $\mathrm { e } ^ { \mathrm { C } }$ .

Let E = $( 2 A + B - 3 ) ^ { 2 }$ + (A + B + 2) + (4A + B - 1 $) ^2$ . What is $\frac { \partial \mathrm { E } } { \partial \mathrm { A } }$ ?

Find all points (x, y) where $f ( x , y ) = \frac { 1 } { x } + x y - \frac { 1 } { y }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

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: $a x + y ^ { n } + b$

Maximize f(x, y, z) = x + y + z subject to the constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 , x , y , z > 0$ Enter your answer as just $3 ^a$ where a is a reduced fraction of form $\frac { b } { c }$ .

Let R be the rectangle consisting of all points (x, y) such that 1 ≤ x ≤ 25, 1 ≤ y ≤ 16. Calculate $\iint _ { R } 5 \sqrt { x y } d y d x$

Exam 7: Functions of Several Variables

Let f(x, y) = x2+y2\sqrt { x ^ { 2 } + y ^ { 2 } }x2+y2​ . Compute ∂f∂x\frac { \partial f } { \partial x }∂x∂f​ at (3, 4). Enter just a reduced fraction ab\frac { a } { b }ba​ .

Use the method of your choice to obtain the formula for the least-squares line to fit the data (0, 6), (1, 3), (2,3).( 2,3 ) .(2,3). Enter your answer in standard point-intercept form with any fractions reduced of form ab\frac { a } { b }ba​ .

Let f(x, y) = x2x ^ { 2 }x2 - xy + y2y ^ { 2 }y2 + 2y - 4. The point −23,−43- \frac { 2 } { 3 } , - \frac { 4 } { 3 }−32​,−34​ is a

Maximize the function f(x, y) = x2x ^ { 2 }x2 - y2y ^ { 2 }y2 subject to the constraint y−x2=−12,x,y>0y - x ^ { 2 } = - \frac { 1 } { 2 } , x , y > 0y−x2=−21​,x,y>0 Enter your answer as just a reduced fraction of form ab\frac { a } { b }ba​ .

Let f(x, y) = 5 x2x ^ { 2 }x2 - 5 y2y ^ { 2 }y2 + 2xy + 34x + 38y + 12. At which point does f(x, y) have a possible maximum or minimum value?

Let f(x, y) = ln⁡xyy\frac { \ln x y } { y }ylnxy​ . Find ∂f∂x\frac { \partial f } { \partial x }∂x∂f​ . Is (xy)−1( x y ) ^ { - 1 }(xy)−1 the correct answer?

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

Find the greatest possible volume of a rectangular box that has length plus girth equal to 60 inches. Enter your answer as a single integer (no units).

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 ?

Find all points (x, y) where f(x,y) =e(x2+y2) f ( x , y ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } \right) }f(x,y) =e(x2+y2) has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Calculate the iterated integral ∫−12(∫01/x2x3ex3ydy)\int _ { - 1 } ^ { 2 } \left( \int _ { 0 } ^ { 1 / x ^ { 2 } } x ^ { 3 } e ^ { x ^ { 3 } y } d y \right)∫−12​(∫01/x2​x3ex3ydy) dx. Enter your answer exactly in the form eae ^ { a }ea ± b - eC\mathrm { e } ^ { \mathrm { C } }eC .

Let E = (2A+B−3) 2( 2 A + B - 3 ) ^ { 2 }(2A+B−3) 2 + (A + B + 2) + (4A + B - 1 ) 2) ^2) 2 . What is ∂E∂A\frac { \partial \mathrm { E } } { \partial \mathrm { A } }∂A∂E​ ?

Find all points (x, y) where f(x,y) =1x+xy−1yf ( x , y ) = \frac { 1 } { x } + x y - \frac { 1 } { y }f(x,y) =x1​+xy−y1​ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

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

Maximize f(x, y, z) = x + y + z subject to the constraint x2+y2+z2=1,x,y,z>0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 , x , y , z > 0x2+y2+z2=1,x,y,z>0 Enter your answer as just 3a3 ^a3a where a is a reduced fraction of form bc\frac { b } { c }cb​ .

Let R be the rectangle consisting of all points (x, y) such that 1 ≤ x ≤ 25, 1 ≤ y ≤ 16. Calculate ∬R5xydydx\iint _ { R } 5 \sqrt { x y } d y d x∬R​5xy​dydx

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute $\frac { \partial f } { \partial x }$ at (3, 4). Enter just a reduced fraction $\frac { a } { b }$ .

Use the method of your choice to obtain the formula for the least-squares line to fit the data (0, 6), (1, 3), $( 2,3 ) .$ Enter your answer in standard point-intercept form with any fractions reduced of form $\frac { a } { b }$ .

Let f(x, y) = $x ^ { 2 }$ - xy + $y ^ { 2 }$ + 2y - 4. The point $- \frac { 2 } { 3 } , - \frac { 4 } { 3 }$ is a

Maximize the function f(x, y) = $x ^ { 2 }$ - $y ^ { 2 }$ subject to the constraint $y - x ^ { 2 } = - \frac { 1 } { 2 } , x , y > 0$ Enter your answer as just a reduced fraction of form $\frac { a } { b }$ .

Let f(x, y) = 5 $x ^ { 2 }$ - 5 $y ^ { 2 }$ + 2xy + 34x + 38y + 12. At which point does f(x, y) have a possible maximum or minimum value?

Let f(x, y) = $\frac { \ln x y } { y }$ . Find $\frac { \partial f } { \partial x }$ . Is $( x y ) ^ { - 1 }$ the correct answer?

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

Find all points (x, y) where $f ( x , y ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } \right) }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Calculate the iterated integral $\int _ { - 1 } ^ { 2 } \left( \int _ { 0 } ^ { 1 / x ^ { 2 } } x ^ { 3 } e ^ { x ^ { 3 } y } d y \right)$ dx. Enter your answer exactly in the form $e ^ { a }$ ± b - $\mathrm { e } ^ { \mathrm { C } }$ .

Let E = $( 2 A + B - 3 ) ^ { 2 }$ + (A + B + 2) + (4A + B - 1 $) ^2$ . What is $\frac { \partial \mathrm { E } } { \partial \mathrm { A } }$ ?

Find all points (x, y) where $f ( x , y ) = \frac { 1 } { x } + x y - \frac { 1 } { y }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

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: $a x + y ^ { n } + b$

Maximize f(x, y, z) = x + y + z subject to the constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 , x , y , z > 0$ Enter your answer as just $3 ^a$ where a is a reduced fraction of form $\frac { b } { c }$ .

Let R be the rectangle consisting of all points (x, y) such that 1 ≤ x ≤ 25, 1 ≤ y ≤ 16. Calculate $\iint _ { R } 5 \sqrt { x y } d y d x$