Exam 7: Functions of Several Variables

Let f(x, y) = $2 y ^ { 3 }$ + 4 $x ^ { 4 }$ + 2xy. Find $\frac { \partial f } { \partial y }$ . Enter just a polynomial in y plus or minus a polynomial in x both in standard form (no label, no parentheses).

Let f(x, y) = ( $x ^ { 2 }$ $y ^ { 3 }$ + $x ^ { 3 }$ $5 ^ { 5 }$ . Find $\frac { \partial f } { \partial y }$ . Is $5 \left( x ^ { 3 } + y ^ { 3 } x ^ { 2 } \right) ^ { 4 } \left( 3 y ^ { 2 } x ^ { 2 } \right)$ the correct answer?

Determine the minimum value of f(x, y) = $x ^ { 2 }$ - xy + 2 $y ^ { 2 }$ + 4 subject to the constraint $x - y - 1 = 0$ Enter your answer exactly as just a, b where a is the minimum and b is the Lagrange multiplier, using integers or reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ (no words or units).

Which of the following pairs of values (x, y) maximizes the function $f ( x , y ) = x + 3 y$ subject to the constraint $x ^ { 2 } + 9 y ^ { 2 } = 72$ assuming x and y are positive? (I) (-6, -2) (II) $\left( 8 , \sqrt { \frac { 8 } { 9 } } \right)$ III) (2, -6) (IV) (6, 2)

Find all points (x, y) where f(x, y) = $x ^ { 2 }$ + xy + $y ^ { 2 }$ - x - y + 2 has a possible relative maximum or minimum. Enter your answer exactly as just (a, b) where a, b are reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ or integers.

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

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

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

Find all points (x, y) where $f ( x , y ) = x ^ { 3 } - 12 y + y ^ { 2 }$ 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 R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 1, 0 ≤ y≤ 2. Calculate $\iint _ { R } \left( e ^ { x - y } \right) d x d y$ Enter your answer exactly in the form (e - a)(1 - $\mathrm { e } ^ { \mathrm { b } }$ ).

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 R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 3, 0 ≤ y ≤ 1. Calculate $\iint _ { R } 4 x ^ { 2 } y ^ { 2 }$ dy dx.

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

Let f(x, y) = ( $x ^ { 2 }$ + x) $e y - x$ . Find $\frac { \partial f } { \partial x }$ . Enter your answer as just (P(x)) $e y - x$ where P is a polynomial in x in standard form.

Calculate the iterated integral $\int _ { 0 } ^ { 1 } \left( \int _ { 0 } ^ { \sqrt { x } } ( x - 2 y - 7 ) d y \right)$ dx. Enter just a reduced fraction of form $\frac { a } { b }$ .

Find all points (x, y) where $f ( x , y ) = x ^ { 3 } - y ^ { 2 } - 3 x + y + 5$ has a possible relative maximum or minimum. Enter your answer exactly as just (a, b), (c, d) with a > c and where a, b, c, d are either integers or reduced fractions of form $\frac { e } { f }$ .

Let f(x, y) = 3 $x ^ { 2 }$ + 2xy. Find $\frac { \partial f } { \partial x }$ . Enter your answer exactly as just a polynomial in x plus or minus a polynomial in y both in standard form (do not label, no parentheses).

Let f(x, y) = 4 $y ^ { 2 }$ - 2 $x ^ { 3 }$ + 5 $x y ^ { 2 }$ . Find $\frac { \partial f } { \partial x }$ . Enter just a polynomial in x plus or minus a polynomial in y both in standard form (do not label, no parentheses).

Let R be the rectangle consisting of all points (x, y) such that 0 ≤ x ≤ 3, 0 ≤ y ≤ 4. Calculate $\iint _ { R } x y d y d x$ Enter just an integer.

Let f(x, y) = $e ^ { x - y }$ . Find $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ . Is $e ^ { x - y }$ the correct answer?