Deck 13: Partial Differentiation

Question 1

Write the equation for the surface obtained by revolving the curve z = y² (in the yz-plane) about the z-axis.

A) z = - x² - y²
B) z = - x² + y²
C) z = x² - y²
D) z = x² + y²
E) z = x + y²

Answer 1

z = x² + y²

Question 2

Find the domain of the function f(x, y) = ln(9 - x² - 9y²).

A)
B)
C)
D)
E)

Answer 2

Question 3

Find the domain of the function f(x) =

A) {(x, y) : x + y > 0}
B) {(x, y) : y $\neq$ -x and |x| $\neq$ |y|}
C) {(x, y) : y $\neq$ -x and y $\neq$ x}
D) {(x, y) : x + y > 0 and x - y > 0}
E) {(x, y) : x > 0 and y > 0}

Answer 3

{(x, y) : y $\neq$ -x and |x| $\neq$ |y|}

Question 4

The set of all points (x,y) in the plane satisfying + $\le$ 1 is the domain of which function?

A) f(x,y) =
B) f(x,y) =
C) f(x,y) =
D) f(x,y) =
E) f(x,y) =

Question 5

Describe the graph of the function f(x,y) = , where a > 0.

A) the sphere of radius a centred at the origin
B) the set of points inside or on the sphere of radius a centred at the origin
C) the points on the sphere of radius a centred at the origin that lie on or above the xy-plane
D) the circle centred at the origin in the xy-plane
E) the points inside or on the sphere of radius a centred at the origin that lie on or above the xy-plane

Question 6

Describe the level curves of the function f(x,y) = , where a > 0.

A) all circles in the xy-plane
B) all circles in the xy-plane having centres at the origin.
C) all circles in the xy-plane having centres at the origin and radii less than a
D) the origin together with all circles in the xy-plane having centres at the origin and radii in the interval (0, a]
E) a circle in the xy -plane having its centre at the origin and radius = a

Question 7

Let f(x, y, z) = x² - y² + z² -1. The level surfaces f(x, y, z) = C, whereC < -1, are all hyperboloids of one sheet with centre at the origin.

Question 8

Describe the level curves f(x, y) = .

A) all parabolas in the xy-plane tangent to the x-axis at the origin
B) all parabolas in the xy-plane tangent to the y-axis at the origin
C) all parabolas in the xy-plane tangent to the y-axis
D) all parabolas in the xy-plane tangent to the x-axis
E) all parabolas in the xy-plane

Question 9

Which function f(x,y) has the level curves corresponding to c = -1, 0, and 1 shown in the figure below?

A) f(x,y) =
B) f(x,y) =
C) f(x,y) =
D) f(x,y) =
E) f(x,y) =

Question 10

The graph below shows level curves f(x,y) = C for a function f and equally spaced values of C. Which of the following functions f is consistent with the graph?
(a) f(x,y) = + , (b) f(x,y) = (c) f(x,y) =

A) function (a)
B) function (b)
C) function (c)
D) all three functions
E) none of the functions

Question 11

Identify the function f(x,y) whose domain is the shaded region shown in the figure below.

A) f(x,y) =
B) f(x,y) =
C) f(x,y) =
D) f(x,y) = )
E) f(x,y) =

Question 12

Describe the level surfaces of f(x, y, z) = .

A) elliptic paraboloids symmetric about the z-axis and having horizontal cross-sections that are ellipses with major axis parallel to the y-axis and eccentricity 1/
B) elliptic paraboloids symmetric about the z-axis and having horizontal cross-sections that are ellipses with major axis parallel to the x-axis and eccentricity 1/ https://d2lvgg3v3hfg70.cloudfront.net/TB9661/.
C) elliptic paraboloids symmetric about the z-axis and having horizontal cross-sections that are ellipses with major axis parallel to the y-axis and eccentricity 1/2
D) elliptic paraboloids symmetric about the z-axis and having horizontal cross-sections that are ellipses with major axis parallel to the y-axis and eccentricity 1/2.
E) none of the above

Question 13

Evaluate the limit .

A) 1
B) 2
C) 4
D) The limit does not exist.
E) None of the above

Question 14

Evaluate the limit .

A) 0
B) 1
C) -1
D) $\infty$
E) The limit does not exist.

Question 15

Evaluate the limit.

A) -2
B) 0
C) -1
D) $\infty$
E) The limit does not exist.

Question 16

Show that the function g(x,y) = is continuous at (x, y) = (0, 0).

Question 17

Let f(x,y) = where k is a constant real number. Find all values of k so that the function f is continuousat (0, 0).

A) - or
B) k R
C) -1 or 2
D) 1 or -2
E) or -

Question 18

Evaluate the limit .

A) 1
B) 0
C)
D) -1
E) The limit does not exist.

Question 19

Evaluate the limit .

A) 0
B) 1
C) 2
D) -1
E) The limit does not exist.

Question 20

Evaluate the limit .

A) 1
B)
C) 0
D) 2
E) The limit does not exist.

Question 21

Evaluate the limit.

A) 0
B) 1
C) -1
D) $\infty$
E) The limit does not exist.

Question 22

Evaluate the limit .

A) 0
B) 1
C)
D) -
E) The limit does not exist.

Question 23

Given z = f(x, y) = y - , find .

A) 3 y -
B) 3 y - y
C) y - x
D)
E)

Question 24

Given z = f(x, y) = y - , find .

A) y - x
B) - y
C) - x
D) y -
E) 3 y - y

Question 25

Find and if z = f(x, y) = cos (ln( + xy + )).

A) = - ; = -
B) = - ; = -
C) = - ; = -
D) = ; =
E) = - ; = -

Question 26

Find and if z = f(x, y) = sin(x - 2y).

A)
B)
C)
D)
E)

Question 27

Find (3, 4) and (3, 4) if f(x, y) = ln (x + ).

A) (3, 4) = , (3, 4) =
B) (3, 4) = , (3, 4) =
C) (3, 4) = , (3, 4) =
D) (3, 4) = , (3, 4) =
E) (3, 4) = , (3, 4) =

Question 28

Evaluate ( $\pi$ , - ln 6) if f(x, y) = cos (x ).

A) -
B) -
C)
D)
E)

Question 29

z = (y/x) + cos (x/y) satisfies the partial differential equationx + y = 0 provided (x, y) $\neq$ (0, 0).

Question 30

Find the equation of the plane tangent to the surface 5x² - 2y² + 2z = - 9 and parallel to the plane 5x -4y + z = 2.

A) , t
B) ^{5x - 4y + z = - 6}
C) 10x -4y + 2 = 0
D) 5x - 4y + z = 9
E) , t 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 11ee7b4e_7d8e_093a_ae82_6552af45ee55_TB9661_11

Question 31

Find the slope of the tangent line to the curve that is the intersection of the surface z = x² - y² with the plane x = 2 at the point (2, 1, 3).

A) -2
B) 2
C) 0
D) -4
E) 4

Question 32

Find an equation of the plane tangent to the surface z = x² - y² at the point (2, 1, 3).

A) 4x - 2y + z = 9
B) 4x - 2y - z = 3
C) 4x + 2y + z = 13
D) 4x + 2y - z = 7
E) 4x - 2y + z = -9

Question 33

Find all points where the surface z = xy has a horizontal tangent plane.

A) , , , , (0, 0)
B) , , ,
C) , , (0, 0)
D) ,
E) The tangent plane is never horizontal.

Question 34

Find all horizontal planes that are tangent to the surface z = x³ - 3xy² + 6y² + 1.

A) z = 1 and z = 3
B) z = 1
C) z = 3
D) z = -15
E) z = 9

Question 35

Find the equation of the tangent plane to the surface z = at the point(2, -2, 1).

A) 4x - 8y - z = 23
B) 4x + 8y - z = -9
C) 4x - 8y + z = 25
D) 4x + 2y + z = 5
E) 4x - 8y - z = 9

Question 36

Find the equation of the normal line to the surface z = at the point(2, -2, 1).

A) = =
B) = =
C) = =
D) = =
E) = =

Question 37

Find the distance from the point (0, 0, 1) to the surface z = x² + 2y².

A) units
B) units
C) 2 units
D) unit
E) 1 unit

Question 38

Find f₁₁(x, y) and f₂₂(x, y) if f(x, y) = e^x sin y + 2xy + y.

A) f₁₁(x, y) = sin y + 2, (x, y) = sin y
B) f₁₁(x, y) = cos y, f₂₂(x, y) = sin y
C) f₁₁(x, y) = sin y, f₂₂(x, y) = cos y
D) f₁₁(x, y) = sin y, f₂₂(x, y) = sin y
E) f₁₁(x, y) = cos y, f₂₂(x, y) = cos y

Question 39

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

A) (x, y) = - , (x, y) = -
B) (x, y) = - , (x, y) = -
C) (x, y) = - , (x, y) = -
D) (x, y) = - , (x, y) = -
E) (x, y) = - , (x, y) = -

Question 40

Find (x, y) and (x, y) if f(x, y) = .

A) (x, y) = ; (x, y) =
B) (x, y) = ; (x, y) =
C) (x, y) = ; (x, y) =
D) (x, y) = ; (x, y) =
E) (x, y) = ; (x, y) =

Question 41

Find if z = .

A) x(x - 1)
B) x ln(x)
C) x ln(y)
D) (1 + xln(y))
E) (1 + xln(x - 1))

Question 42

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

A) (x, y) = ; (x, y) =
B) (x, y) = ; (x, y) =
C) (x, y) = 1 - ; (x, y) = 1 +
D) (x, y) = ; (x, y) =
E) (x, y) = (xy) - ; (x, y) = (xy) +

Question 43

Find the value of the positive constant real number k such that the function is harmonic for all points in 3-space.

A)
B) 2 -
C) 2
D) 2 +
E) 3

Question 44

Compute and if f(x, y) = xy + sin xy + x² + 5xln y.

A) = sin xy + 2, = - sin xy -
B) = - sin xy + 2, = - sin xy -
C) = - sin xy + 2, = sin xy -
D) = - sin xy + 2, = - sin xy +
E) = - sin xy + 2, = - sin xy -

Question 45

Calculate and simplify + if z = .

A) -
B)
C)
D) 0
E)

Question 46

Let F(x,y) = (a) Use the definitions of partial derivatives to calculate (0, 0) and (0, 0).
(b) Calculate and for (x, y) ≠ (0, 0).
(c) Use part (b) and the definitions of partial derivatives to calculate the mixed partial .

Question 47

Evaluate (3, 2, 1) if f(x, y, z) = x (yz).

A)
B) -
C) -
D)
E) 0

Question 48

Find if w = .

A) 2x(1 - yz)
B) 2x(1 + yz)
C) x(1 - yz)
D) 2(1 + yz)
E) 2(1 - yz)

Question 49

Calculate and simplify u_t - u_xx - u_yy for the function u = .

A) -
B)
C)
D) 0
E) -

Question 50

For what value(s) of the constant k is the function f(x, y, z) = sin(kz) a harmonic function in 3-space? Note that a harmonic function f in 3-space satisfies + + = 0.

A) k = 7
B) k = ± 1
C) k = ± 5
D) k = 0
E) k = ±4

Question 51

Find if z = x sin y, where x = t² + 2t + 1 and y = ln t.

A) = 2(t + 1) sin(ln t) + cos(ln t)
B) = 2(t + 1) sin(t) + cos(t)
C) = 2(t + 1) sin(ln t) - sin(ln t)
D) = 2(t + 1) cos(ln t) + sin(ln t)
E) = 2(t + 1) sin(ln t) + cos(ln t)

Question 52

Find if z = , where x = ln t and y = ln t².

A) = -
B) =
C) = -
D) =
E) = -

Question 53

Find if z = sin(uv)cos(uv), where u = e^t and y = e^2t.

A) 3
B) 3
C) 3
D)
E)

Question 54

A) - ln ( )
B) + ln ( )
C) - + ln ( )
D) - - ln ( )
E) + ln (2)

Question 55

Let h(t) = f(x, y), where f(x, y) = + , x = t, y = t². Find (2).

A) - 2
B) + 4
C) - 4
D) + 2
E) - 4

Question 56

Let z = f(x, y), where x = u , and y = . Use the chain rule to find at (u, v) = (2,- 1) given that (2,- 1) = - 7, (2,- 1) = 5, (2,- 2) = 3, and (2,- 2) = -2.

A) -5
B) 4
C) -12
D) 5
E) 2

Question 57

If W = , where x = t + sin(t) and y = 2t -1, then the value of at t = 0 is equal to:

A)
B)
C)
D)
E) 0

Question 58

Let g be a function of a single variable having continuous second order derivatives, and letu(x, y) = g(y + mx) , for some constant real number m. Determine all values of m such that u(x, y) satisfies the partial differential equation - 10 + 24 = 0.

A) - 2 and 12
B) - 4 and - 6
C) 2 and - 12
D) 4 and 6
E) 3 and 8

Question 59

Use the chain rule to find the values of and . at (u, v) = (0, 1), where z = x³y⁵,x = u - v, and y = u + v.

A) = -2; = -8
B) = 8; = 2
C) = -2; = 8
D) = -8; = -2
E) = 2; = 8

Question 60

Assuming that the function f has continuous first partial derivatives and , calculate and simplify f(x²y, x ).

A)
B)
C)
D)
E)

Question 61

Assuming that the function f has continuous second partial derivatives and that a and b are constants, calculate and simplify + f(ax + by, bx - ay).

A)
B)
C)
D)
E)

Question 62

Assuming that the function f has continuous partial derivatives of orders 1 and 2, calculate and simplify f( y, x ).

A)
B)
C)
D)
E)

Question 63

Let z = f(x,y) = + y where x = + -4 and y = uv -1. Find the value of at (u , v) = (2 , -1).

A) -2
B) -8
C) -10
D) 0
E) -13

Question 64

Find the linearization L(x,y) of f(x,y) = x (y) about the point .

A)
B)
C)
D)
E)

Question 65

The linearization of f(x,y) =ln (x² + y² + xy) at the point (1,-1) is given by

A) L(x,y) = ln(x - y - 2)
B) L(x, y) = x - y - 2
C) L(x, y) = x - y + 2
D) L(x, y) = x - y
E) L(x, y) = 3x - y - 4

Question 66

Use the linearization of f(x,y) = x³ e^-y about the point (2, 0) to find an approximate value for f(2.1, 0.1).

A) 8.4
B) 10.0
C) 9.2
D) 7.6
E) 6.3

Question 67

Find the differential of the function f(x, y) = .

A) (y dx + x dy)
B) (y dx - x dy)
C) - (y dx + x dy)
D) - (y dx - x dy)
E) - (y dy - x dx)

Question 68

The pressure P, the volume V, and the temperature T (in Kelvin) of a confined gas are related by the ideal gas law P V = kT , where k is a constant. If P = 0.5 pascal when V = 50 cm³ and T = 360 K, determine by approximately what percentage P changes if V and T change to52 cm³ and 351 K, respectively.

A) 6.5 %
B) 1.5 %
C) -4.5 %
D) -6.5 %
E) -1.5 %

Question 69

The area of a triangle is given by the formula A = ab sin $\theta$ , where $\theta$ is the angle between the sides having lengths a and b. If measurements indicate that a = 4 m with error ± 1 cm,b = 3 m with error ± 1 cm, and $\theta$ = 60° with error ± 2°, use differentials to determine the approximate maximum error in the calculated area of the triangle.

A) about 0.135 m²
B) about 6.015 m²
C) about 0.603 m²
D) about 0.014 m²
E) about 0.862 m²

Question 70

When the ellipse b²x² + a²y² = a²b² is rotated about the x-axis, the volume V of the spheroid is 4 $\pi$ ab²/3. If a and b are each increased by 1%, use differentials to find the approximate percentage change in V.

A) increase of 1%
B) increase of 2%
C) increase of 3%
D) increase of 4%
E) increase of 5%

Question 71

In an electric circuit the current measured in amperes is related to the voltage and the resistance by Ohm's Law V = IR. If the voltage V drops from 24 to 23 volts and the resistance R drops from 100 to 80 Ohms, use differentials to determine whether the current I will increase or decrease and by approximately how much?

A) decrease of 0.067 amps
B) increase of 0.094 amps
C) increase of 0.038 amps
D) decrease of 0.028 amps
E) decrease of 0.062 amps

Question 72

Find the Jacobian matrix Df(x, y, z) of the transformation f(x, y, z) = (x² + xy, y² - ln(z)).

A)
B)
C)
D)
E)

Question 73

Use the Jacobian matrix for the transformation f(x, y, z) = to find an approximate value for f(1.98, 0.03, -0.01).

A) (2.00, -0.02, 0.03)
B) (2.04, -0.02, 0.03)
C) (2.02, 0.02, 0.97)
D) (1.98, 0.02, -0.03)
E) (2, 0, 0)

Question 74

Find a unit vector in the direction of which the function f(x, y) = 6 sin (a) increases most rapidly, and (b) decreases most rapidly at the point (2, $\pi$ ).

A) (a) (3 i - j) (b) (-3 i + j)
B) (a) (3 i + j) (b) - (3 i + j)
C) (a) - (3 i + j) (b) (3 i + j)
D) (a) (-3 i + j) (b) (3 i - j)
E) (a) (3 i - j) (b) (-3 i + j)

Question 75

Find the directional derivative of f(x, y) = 3x² + 2xy - 7y² at the point (-1, 3) in the direction making an angle of 60° with the positive x- and y-axes.

A) -2 - 30
B) -
C) 1 - 15
D) -1 - 15
E) 1 -

Question 76

Calculate the directional derivative of the function at (1, 1, 1) in the direction from (1, 1, 1) toward the point (-1, -2, 3).

A) -
B)
C)
D) -
E) -

Question 77

Let f(x, y, z) = -2 - 3y + 4 and let P be the point (1, -1, 3). Which of the following statements is false?

A)
B) The maximum rate of change of f at P is 9.
C) There is no direction in which the rate of change of f at P is equal to 10.
D) There is a direction in which the rate of change of f at P is equal to -12.
E) The function f increases most rapidly at P in the direction of the vector -6 i - 3 j + 6 k.

Question 78

At the point (1, 2) the function f(x, y) has a directional derivative of 2 in the direction toward (2, 2) and a directional derivative of -2 in the direction toward (1, 1). Find the directional derivative of f at (1, 2) in the direction toward the point (4, 6).

A) -
B) -
C)
D)
E) 4

Question 79

The temperature in the xy-plane is a function T(x, y). At the point (a,b) the directional derivative of T in the direction of the vector 3 i + 4 j is 4 and the directional derivative in the direction of 5 i - 12 j is -2. Find T(a,b).

A) (68 i + 89 j)
B) (43 i - 17 j)
C) (-53 i + 117 j)
D) (28 i + 37 j)
E) - (172 i + 89 j)

Question 80

Find an equation of the tangent plane to the surface x² + 2y² + 3z² = 6 at the point(1, 1, 1).

A) x + y + z = 3
B) x + 4y + 9z = 14
C) x + 2y + 3z = 6
D) 3x + 2y + z = 6
E) x + 2y - 3z = 6