Question 1

Find the domain and range and describe the level curves for the function f(x,y).&#10;f(x, y) = -6x + 8y

Accepted Answer

The function $f(x, y) = -6x + 8y$ can take any real number values for $x$ and $y$, so its domain is all points in the $xy$-plane. The function can produce any real number as output, so its range is all real numbers. The level curves are defined by setting $f(x, y)$ equal to a constant $c$, resulting in lines of the form $-6x + 8y = c$, where $c$ can be any real number.

Question 2

Find the specific function value.&#10; &#10;A) 108&#10;B) -72&#10;C) -49&#10;D) -54

Accepted Answer

Plug in the values given in the function:
$f(3,0,9) = 3(3)^2 + 3(0)^2 - (9)^2 = 27 + 0 - 81 = -54$
Therefore, the answer is D) -54.

Question 3

Find the domain and range and describe the level curves for the function f(x,y).&#10;

Accepted Answer

The domain of $f(x, y) = \sqrt{25 - x^2 - y^2}$ includes all points $(x, y)$ for which the expression under the square root is non-negative, i.e., $25 - x^2 - y^2 \geq 0$, which simplifies to $x^2 + y^2 \leq 25$. This represents a disk of radius 5 centered at the origin. The range is the set of all possible output values, which are non-negative (since it's a square root) and at most 5 (when $x = 0$ and $y = 0$), hence $0 \leq z \leq 5$. The level curves, which are the sets of points $(x, y)$ where $f(x, y)$ is constant, correspond to circles centered at the origin, with radii less than or equal to 5, since for any constant value $c$, $f(x, y) = c$ implies $25 - x^2 - y^2 = c^2$, which rearranges to $x^2 + y^2 = 25 - c^2$, a circle with radius $\sqrt{25 - c^2}$.

Question 4

Match the surface show below to the graph of its level curves.&#10;

Accepted Answer

The surface appears to be a saddle shape, which is represented by the level curves in choice C. The curves have a symmetrical pattern around the central point, which is characteristic of saddle surfaces. The other choices do not match the shape of the surface.

Question 5

Match the surface show below to the graph of its level curves.&#10;

Accepted Answer

The level curves on this graph are evenly spaced and circular, indicating a function of the form f(x,y) = k, which is consistent with choice C. The other graphs have level curves that are not evenly spaced, have sharp corners, or have elliptical shapes, which are not consistent with f(x,y) = k.

Question 6

Find the specific function value.&#10; &#10;A) -302&#10;B) -310&#10;C) -232&#10;D) -237

Accepted Answer

The answer of Find the specific function value.&#10; &#10;A) -302&#10;B)...

Question 7

Find the specific function value.&#10;Find f(-3, 6) when f(x, y) = 5x + 2y - 3.&#10;A) -3&#10;B) -6&#10;C) -11&#10;D) -8

Accepted Answer

The answer of Find the specific function value.&#10;Find f(-3, 6)...

Question 8

Find the domain and range and describe the level curves for the function f(x,y).&#10;

Accepted Answer

The answer of Find the domain and range and describe...

Question 9

Find the domain and range and describe the level curves for the function f(x,y).&#10; &#10;A) Domain: all points in the first quadrant of the xy-plane; range: all real numbers; level curves: lines x + y = c&#10;B) Domain: all points in the xy-plane; range: real numbers z > 0; level curves: lines x + y = c&#10;C) Domain: all points in the first quadrant of the xy-plane; range: real numbers z > 0; level curves: lines x + y = c&#10;D) Domain: all points in the xy-plane; range: all real numbers; level curves: lines x + y = c

Accepted Answer

The function $f(x, y) = e^{x+y}$ is defined for all real numbers $x$ and $y$, so its domain is all points in the xy-plane. The exponential function always yields positive values, so the range is $z > 0$. The level curves are given by setting $f(x, y) = e^{x+y} = c$, where $c$ is a constant, leading to $x + y = \ln(c)$, which are lines in the xy-plane.

Question 10

Find the specific function value.&#10;

Accepted Answer

Substituting in $x=3$ and $y=6$ into the function $f(x,y)=\sqrt{3x+y^2}$ results in 
$$f(3,6)=\sqrt{3(3)+(6)^{2}}=\sqrt{9+36}=\sqrt{45}=3\sqrt{5}$$
Therefore, the answer is choice D.

Question 11

Find the specific function value.&#10;

Accepted Answer

The answer of Find the specific function value.&#10; ...

Question 12

Find the specific function value.&#10;Find f(100, 3) when f(x, y) = y log x.&#10;A) 3&#10;B) 60&#10;C) 6&#10;D) 30

Accepted Answer

The answer of Find the specific function value.&#10;Find f(100, 3)...

Question 13

Find the domain and range and describe the level curves for the function f(x,y).&#10;

Accepted Answer

The answer of Find the domain and range and describe...

Question 14

Find the specific function value.&#10; &#10;A) -7&#10;B) 8&#10;C) -6&#10;D) 7

Accepted Answer

The answer of Find the specific function value.&#10; &#10;A) -7&#10;B)...

Question 15

Find the specific function value.&#10; &#10;A) 100&#10;B) 1000&#10;C) 30&#10;D) 280

Accepted Answer

The answer of Find the specific function value.&#10; &#10;A) 100&#10;B)...

Question 16

Find the domain and range and describe the level curves for the function f(x,y).&#10;

Accepted Answer

The answer of Find the domain and range and describe...

Question 17

Find the domain and range and describe the level curves for the function f(x,y).&#10;

Accepted Answer

The answer of Find the domain and range and describe...

Question 18

Find the domain and range and describe the level curves for the function f(x,y).&#10;f(x, y) = ln (5x + 4y)

Accepted Answer

The answer of Find the domain and range and describe...

Question 19

Find the domain and range and describe the level curves for the function f(x,y).&#10;

Accepted Answer

The answer of Find the domain and range and describe...

Question 20

Find the domain and range and describe the level curves for the function f(x,y).&#10;

Accepted Answer

The answer of Find the domain and range and describe...

Question 21

Match the surface show below to the graph of its level curves.&#10;

Accepted Answer

The answer of Match the surface show below to the...

Question 22

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 23

Match the surface show below to the graph of its level curves.&#10;

Accepted Answer

The answer of Match the surface show below to the...

Question 24

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 25

Match the surface show below to the graph of its level curves.&#10;

Accepted Answer

The answer of Match the surface show below to the...

Question 26

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 27

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 28

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 29

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 30

Match the surface show below to the graph of its level curves.&#10;

Accepted Answer

The answer of Match the surface show below to the...

Question 31

Match the surface show below to the graph of its level curves.&#10;

Accepted Answer

The answer of Match the surface show below to the...

Question 32

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 33

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 34

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 35

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 36

Match the surface show below to the graph of its level curves.&#10;

Accepted Answer

The answer of Match the surface show below to the...

Question 37

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 38

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 39

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 40

Sketch the surface z = f(x,y).&#10;

Accepted Answer

The answer of Sketch the surface z = f(x,y).&#10; ...

Question 41

Find the equation for the level surface of the function through the given point.&#10;

Accepted Answer

The answer of Find the equation for the level surface...

Question 42

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 43

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 44

Sketch a typical level surface for the function.&#10;

Accepted Answer

The answer of Sketch a typical level surface for the...

Question 45

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 46

Sketch a typical level surface for the function.&#10;

Accepted Answer

The answer of Sketch a typical level surface for the...

Question 47

Sketch a typical level surface for the function.&#10;

Accepted Answer

The answer of Sketch a typical level surface for the...

Question 48

Find the equation for the level surface of the function through the given point.&#10;

Accepted Answer

The answer of Find the equation for the level surface...

Question 49

Find the equation for the level surface of the function through the given point.&#10;

Accepted Answer

The answer of Find the equation for the level surface...

Question 50

Find the equation for the level surface of the function through the given point.&#10;

Accepted Answer

The answer of Find the equation for the level surface...

Question 51

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 52

Sketch a typical level surface for the function.&#10;

Accepted Answer

The answer of Sketch a typical level surface for the...

Question 53

Sketch a typical level surface for the function.&#10;

Accepted Answer

The answer of Sketch a typical level surface for the...

Question 54

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 55

Provide an appropriate response.&#10;Find the extrema of f(x, y, z) = x + yz on the line defined by x = 8(6 + t), y = t - 8, and z = t + 6. Classify each extremum as a minimum or maximum. (Hint: w = f(x, y, z) is a differentiable function of t.)&#10;A) (24, -11, 3), minimum&#10;B) (72, -5, 9), minimum; (0, -8, 0), minimum&#10;C) (24, -11, 3), minimum; (72, -5, 9), maximum&#10;D) (72, -5, 9), minimum

Accepted Answer

The answer of Provide an appropriate response.&#10;Find the extrema of...

Question 56

Sketch a typical level surface for the function.&#10;

Accepted Answer

The answer of Sketch a typical level surface for the...

Question 57

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 58

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 59

Sketch a typical level surface for the function.&#10;

Accepted Answer

The answer of Sketch a typical level surface for the...

Question 60

Solve the problem.&#10;

Accepted Answer

The answer of Solve the problem.&#10; ...

Question 61

Find the limit.&#10;

Accepted Answer

The answer of Find the limit.&#10; ...

Question 62

Find the limit.&#10;

Accepted Answer

The answer of Find the limit.&#10; ...

Question 63

Find the limit.&#10; &#10;A) 8&#10;B) 4&#10;C) 0&#10;D) No limit

Accepted Answer

The answer of Find the limit.&#10; &#10;A) 8&#10;B) 4&#10;C) 0&#10;D)...

Question 64

Find the limit.&#10; &#10;A) 16&#10;B) 4&#10;C) - 4&#10;D) No limit

Accepted Answer

The answer of Find the limit.&#10; &#10;A) 16&#10;B) 4&#10;C) -...

Deck 15: Partial Derivatives