MA103-FinalExam-Answers

A boat heads north at 40 miles per hour, and the wind blows east at 30 miles per hour (mph). What is the speed of the boat?

Choose one answer.

	a. 10 mph
	b. 30 mph
	c. 40 mph
	d. 50 mph

Question 2

A boat heads north at 40 miles per hour, and the wind blows east at 30 miles per hour. What is the direction of the boat?

Choose one answer.

	a. 53°
	b. 45°
	c. 90°
	d. None of the above

Question 3

Choose the best answer to fill in the blanks. If

, then

and

are _____________ vectors, and if

, then

is _____________ to both

and

Choose one answer.

	a. Non-zero, equal
	b. Orthogonal, parallel
	c. Parallel, orthogonal
	d. Zero, orthogonal

Question 4

Choose the best answer to fill in the blanks. The __________ of the unit tangent vector remains constant; therefore, as a particle moves, the only thing that changes is the ___________ of T.

Choose one answer.

	a. length, magnitude
	b. direction, magnitude
	c. length, direction
	d. magnitude, length

Question 5

Convert the vector

to component form.

Choose one answer.

	a. <-2,2>
	b. <2,-3,1>
	c. <-2,3,-1>
	d. <2,3,-1>

Question 6

Evaluate L' (t), if L(t)= r(t)×v(t), where r(t) and v(t) are both vector-valued functions and r' (t)=v(t) and r'' (t)=a(t).

Choose one answer.

	a. L'(t) = v(t) ⋅ v(t) + r(t) × v(t)
	b. L'(t) = v(t) ⋅ v(t)
	c. L'(t) = r(t) × a(t)
	d. L'(t) = v(t) ⋅ v(t) + r(t) × a(t)

Question 7

Evaluate the length of the curve r(t)= < t, sin(2t), cos(2t) > on the interval 0 ≤ t ≤ 2π.

Choose one answer.

	a. √2
	b. 2√5
	c. 5π√2
	d. 2π√5

Question 8

Fill in the blank. Suppose v(t) is a vector such that ‖v(t)‖=c for all t, then v'(t) is ___________ to v(t).

Choose one answer.

	a. equal
	b. orthogonal
	c. anti-derivative
	d. parallel

Question 9

Fill in the blank. The speed of an object is the ____________ of the velocity vector.

Choose one answer.

	a. distance
	b. magnitude
	c. derivative
	d. anti-derivative

Question 10

Find

Choose one answer.

	a. 0
	b. –
	c.
	d. None of the above

Question 11

Find the acceleration of the following function x(t) = <t³,t²,t²>.

Choose one answer.

	a. <6t, 2, 2>
	b. <3t, 2t, 2t>
	c. <6t, 0, 0>
	d. None of the above

Question 12

Find the angle in degrees between vectors

and

Choose one answer.

	a. 0°
	b. 90°
	c. 180°
	d. 45°

Question 13

Find the arc length function for the curve given by r(t)= <sin(3t),cos⁡(3t)>.

Choose one answer.

	a. 3t
	b. 9t
	c. 3
	d. All of the above

Question 14

Find the curvature k for the curve given by

Choose one answer.

	a.
	b.
	c.
	d.

Question 15

Find the equation of the plane through the point ( 0,0,0), ( 1,2,0), and ( 2,1,0).

Choose one answer.

	a.
	b.
	c.
	d.

Question 16

Find the linear acceleration for the curve given by r(t)= < cost, sint >.

Choose one answer.

	a. 0
	b. 1
	c. k
	d. None of the above

Question 17

Find the unit normal N(t) for the curve given by r(t)= <-2t,-t²>.

Choose one answer.

	a.
	b.
	c.
	d.

Question 18

Find the unit tangent vector to the curve

Choose one answer.

	a.
	b.
	c.
	d.

Question 19

Find the velocity of the following function x(t) = <t³,t²,t²>

Choose one answer.

	a. <3t²,1,2t>
	b. <3t, 0, 2>
	c. <3t²,t,t>
	d. <3t²,2t,2t>

Question 20

Let

and

be vectors, then what is

Choose one answer.

	a.
	b.
	c.
	d. None of the above

Question 21

Suppose a car brakes and decelerates at a constant rate of 10 ft/s². Suppose it takes the car 100 ft before coming to a stop. How fast was the car traveling when the driver applied the brakes?

Choose one answer.

	a. 10 ft/s
	b. 44.72 ft/s²
	c. 44.72 ft/s
	d. 10.25 ft/s

Question 22

The curvature of a circle is equal to which of the following?

Choose one answer.

	a. 0
	b. The reciprocal of its radius
	c. Its radius
	d. None of the above

Question 23

What is the standard parameterization of a circle with radius 1 and center (0, 2)?

Choose one answer.

	a.
	b.
	c.
	d.

Question 24

What is the velocity v(2) at the point of tangency to r(t)= <2t²,t³>?

Choose one answer.

	a. <8, 8>
	b. <8, 12>
	c. <12, 8>
	d. <2, 2>

Question 25

Find the vector

with initial point P₁ (-1,-3) and final point P₂ (0,2).

Choose one answer.

	a.
	b.
	c.
	d.

Question 26

Determine where the function

is continuous?

Choose one answer.

	a. Continuous for x≠-y
	b. Continuous for x≠y
	c. Not a continuous function
	d. None of the above

Question 27

Evaluate ^df⁄_dt given that f(x,y)= x²+y² and x(t)= t⁵,y(t)=2 cos⁡(t).

Choose one answer.

	a. 10t⁹-8cos⁡(t)sin(t)
	b. 10t⁴-4ysin(t)
	c. 2t⁵+4cos⁡(t)
	d. None of the above

Question 28

Evaluate

Choose one answer.

	a. ¾
	b. -¾
	c. -1
	d. Undefined

Question 29

Fill in the blank. A function

is _________ at the point

, if

Choose one answer.

	a. defined
	b. undefined
	c. continuous
	d. differentiable

Question 30

Fill in the blank. The domain of the function

is __________.

Choose one answer.

	a. not connected
	b. undefined
	c. dom(f)={(x,y)\|x≠4}
	d. connected

Question 31

Fill in the blank. The domain of the function

is ____________.

Choose one answer.

	a. closed and unbounded
	b. connected and bounded
	c. open, connected, and unbounded
	d. closed, connected, and bounded

Question 32

Find

Choose one answer.

	a.
	b.
	c.
	d.

Question 33

Find each of the directional derivatives of f(x,y)= x cos⁡(y) in the direction of v = <1,2>?

Choose one answer.

	a.
	b.
	c.
	d.

Question 34

Find f_x (x,y), if f(x,y) = y sin⁡(x).

Choose one answer.

	a. f_x(x,y) = cos⁡(x)
	b. f_x(x,y) = -cos⁡(x)
	c. f_x(x,y) = y cos⁡(x)
	d. f_x(x,y) = y cos⁡(-x)

Question 35

Find f_xx (x,y), if f(x,y) = 2x³+ x²y⁵.

Choose one answer.

	a. f_xx(x,y) = 6x+y⁵
	b. f_xx(x,y) = 6+10y³
	c. f_xx(x,y) = 12x+2y⁵
	d. None of the above

Question 36

Find

, if

Choose one answer.

	a.
	b.
	c.
	d.

Question 37

Find f_yx(x,y), if f(x,y) = x sin(y).

Choose one answer.

	a. f_yx(x,y) = cos⁡(y)
	b. f_yx(x,y) = xcos⁡(y)
	c. f_yx(x,y) = -cos⁡(y)
	d. f_yx(x,y) = -xcos⁡(y)

Question 38

Find the domain of

Choose one answer.

	a.
	b.
	c.
	d.

Question 39

Suppose f and g are continuous at point (a, b), then which of the following functions is also continuous?

Choose one answer.

	a. f/g
	b. f + g
	c. f – g
	d. All of the above

Question 40

Suppose that A(x,t) and B(x,t) are both solutions to the heat equation, then which of the following is also a solution?

Choose one answer.

	a. cA(x,t), for a constant c
	b. cB(x,t), for a constant c
	c. αA(x,t)+ βB(x,t), for constants α,β
	d. All of the above

Question 41

The directional derivative, given by

, provides which of the following?

Choose one answer.

	a. The rate of change of the unit vector u=<a,b> in a given direction
	b. The rate of change of ‖f(x,y)‖ in the direction of a vector
	c. The rate of change of f(x,y) in the direction of the unit vector u=<a,b>
	d. None of the above

Question 42

Use Lagrange Multipliers to find the maximum of f(x,y) = 4xy subject to the constraint x²+y²=1.

Choose one answer.

	a. Maximum value of 0 occurs at (0,0).
	b. Maximum value of 2 occurs at ±.
	c. Maximum value of 2 occurs at and
	d. None of the above

Question 43

What are boundary conditions?

Choose one answer.

	a. Constraints on the solutions at different points in space
	b. Constraints indicating that the rate of change of a function must be along a normal vector
	c. Conditions that hold for surfaces without boundaries
	d. None of the above

Question 44

What do the partial derivatives represent?

Choose one answer.

	a. The rates of change of the functions as the variables change
	b. The change in the magnitude of the function
	c. The points at which the value of the function is zero
	d. None of the above

Question 45

What is

Choose one answer.

	a. Does not exist
	b. 0
	c. -1
	d. x = 0

Question 46

What is the definition of the partial derivative of f(x,y) with respect to x?

Choose one answer.

	a.
	b.
	c.
	d.

Question 47

What is the domain of the function

Choose one answer.

	a. Connected
	b. ℝ
	c. ℝ - {0, 0}
	d. None of the above

Question 48

What is the general solution to the differential equationy^''+k² y=0?

Choose one answer.

	a. y(t)= A cos⁡(kt)+ B sin(kt)
	b. y(t)=Ae^kt
	c. y(t)=A cosh⁡(kt)+ B sinh⁡(kt)
	d. None of the above

Question 49

Which of the following statements is true of f(x,y) = x cos(y)?

Choose one answer.

	a. f_xx = f_yy
	b. f_xy = f_yx
	c. f_xy = -f_yx
	d. All of the above

Question 50

Which of the following statements is true of f(x,y)= x+y?

Choose one answer.

	a. f_xx = f_yy
	b. f_xy = f_yx
	c. f_xy = -f_yx
	d. All of the above

Question 51

Find the domain of

Choose one answer.

	a.
	b.
	c.
	d.

Question 52

Calculate the divergence of

Choose one answer.

	a. 2x + 2y + 2z
	b. 2x + yz
	c. xyz
	d. 2x

Question 53

Choose the best answers to fill in the blanks. If

is the gradient of some function

, then

is called a _____________, and function

is said to be a ______________ for

Choose one answer.

	a. tangent vector field, conservative vector field
	b. conservative vector field, potential
	c. potential, conservative vector field
	d. conservative vector field, tangent vector field

Question 54

Compute the following integral:

Choose one answer.

	a. 0
	b. 1
	c. π
	d. None of the above

Question 55

Evaluate

Choose one answer.

	a. (-5)/4
	b. 4/5
	c. 0
	d. -1

Question 56

Evaluate

Choose one answer.

	a. 1/2π
	b. (-1)/2π
	c. 1
	d. 0

Question 57

Evaluate

, where C is the right half of the circle parameterized by x=3 cos⁡(t),y=3 sin(t) for t in

Choose one answer.

	a. 27
	b. 54
	c. 50
	d. 81

Question 58

Fill in the blank. The vector field

is ___________.

Choose one answer.

	a. conservative
	b. constant
	c. not a vector field
	d. undefined

Question 59

Find the center of mass of the lamina of the unit square with mass density m(x,y)=(x+y) kg/m².

Choose one answer.

	a.
	b.
	c.
	d.

Question 60

Find the gradient vector field of U(x,y)= -x²-y².

Choose one answer.

	a. U= <-2x,-2y>
	b. U= <2x,2y>
	c. U= <-x,-y>
	d. U= <x+y,x-y>

Question 61

Find the volume of a solid between z=x and z=x-y over R: y = 0 and y = 1 and x = y³ and x = y.

Choose one answer.

	a.
	b.
	c.
	d.

Question 62

Find the volume of a solid S that is bounded by x²+y²+ z=16, the planes x=2 and y=2, and the three coordinate planes.

Choose one answer.

	a. 160
	b.
	c.
	d. 64

Question 63

Fubini’s theorem does NOT apply to

because of which reason?

Choose one answer.

	a. The function is not bounded on [0,1]x[0,1].
	b. The function is bounded on [0,1]x[0,1].
	c. The function is continuous on all points.
	d. The function is a constant.

Question 64

Fubini’s theorem states that if a function f(x, y) is continuous on the rectangle R = [a, b] x [c, d], then which of the following is true?

Choose one answer.

	a.
	b.
	c.
	d. All of the above

Question 65

is a constant function

, and

, then

= _________.

Choose one answer.

	a. k
	b. k((b - a) +(d - c))
	c. k(b - a)(d - c)
	d. 0

Question 66

Use Green’s theorem to evaluate

, where R is a unit square.

Choose one answer.

	a. -1
	b. 0
	c. 1
	d. 2

Question 67

Use Green’s theorem to find the area of a disk of radius 2.

Choose one answer.

	a. 2π
	b. 8π
	c. 4π
	d. π

Question 68

Use the properties of the double integrals

and

to evaluate

Choose one answer.

	a. 2
	b. 9
	c. 10
	d. 15

Question 69

Using Fubini’s theorem, calculate

, where R = [0, 1] x [0, 1].

Choose one answer.

	a.
	b. 2
	c.
	d.

Question 70

What is the line integral of a vector field V along a curve C?

Choose one answer.

	a.
	b.
	c.
	d.

Question 71

What is the mass of the lamina of the unit square with a height of 1 and a mass density of m(x, y) = (x + y) kg/m².

Choose one answer.

	a. 1 kg
	b. 2 kg
	c. 3 kg
	d. 4 kg

Question 72

What type of an integral is this:

Choose one answer.

	a. Undefined
	b. Type II
	c. Type I
	d. None of the above

Question 73

Which of the following describes the image of [0, 1] x [0, 2π] under the transformation T(u,v)= <2u cos⁡(v),2u sin⁡(v)>?

Choose one answer.

	a. Cube
	b. Sphere
	c. Ellipse
	d. None of the above

Question 74

Which of the following is a Type I integral?

Choose one answer.

	a. =dy dx$$
	b. =dydy$$
	c. =dxdx$$
	d. None of the above

Question 75

Which of the following is a Type II integral?

Choose one answer.

	a. = dx dy$$
	b. = dx dy$$
	c. = dy dx$$
	d. All of the above

Question 76

Which of the following statements is false?

Choose one answer.

	a. If f(x,y,z) has a continuous second or partial derivative, then curl(∇f)= 0.
	b. If is a conservative vector field, then curl.
	c. If is a conservative vector field, then .
	d.

Question 77

Compute the following integral:

Choose one answer.

	a. 60
	b. 64
	c. 32
	d. 16

Question 78

Apply Stokes’ theorem to evaluate the appropriate integral. Let S be the hemisphere x²+y²+z²=16, and let z ≥0 and F = < x,x,y>.

Choose one answer.

	a. 4π
	b. 8π
	c. 16π
	d. 64π

Question 79

Apply Stokes’ theorem to evaluate the appropriate integral. Let C be the circle of radius R centered at the origin, and let F = -y³ i+x³j be the vector field.

Choose one answer.

	a. π/2 R⁴
	b. π/3 R³
	c. 3π/4 R⁴
	d. 3π/2 R⁴

Question 80

Calculate the flux of F over S, where F is the vector field and S is the boundary of the region enclosed by the paraboloid z=1-x²-y²and the plane z=0.

Choose one answer.

	a.
	b.
	c.
	d.

Question 81

Calculate the flux of F(x,y,z) = <x,y,-2z> over the unit sphere.

Choose one answer.

	a. 0
	b.
	c.
	d. None of the above

Question 82

Calculate the flux of a vector field F = over S, where S is the boundary of the unit sphere.

Choose one answer.

	a.
	b.
	c.
	d.

Question 83

Calculate the surface area of sphere of radius R.

Choose one answer.

	a. πR²
	b. 4πR²
	c. 4R²
	d. 4π

Question 84

Calculate the surface area of the part xy + z that is in the plane x + y + z = 2 in the first octant.

Choose one answer.

	a. 2√3
	b. √3
	c. 4√3
	d. 2

Question 85

Compute the flux of the vector field

through the unit sphere

Choose one answer.

	a.
	b.
	c. 0
	d. 1

Question 86

Evaluate

over B = [1, 3] x [0, 1] x [0, 2].

Choose one answer.

	a. 16
	b. 32
	c. 48
	d. 64

Question 87

Fill in the blank. If div(V)=0, then there exists a vector field W such that _______________.

Choose one answer.

	a.
	b.
	c.
	d.

Question 88

Given the mass density, m(x,y,z)=3-z² over B = [0, 1] x [0, 1] x [0, 1], find the mass of the solid.

Choose one answer.

	a.
	b.
	c.
	d. 1

Question 89

If F(x,y,z) and G(x,y,z) are differentiable in each component, then which of the following is true?

Choose one answer.

	a. F'(x,y,z) = G'(x,y,z).
	b. div(F(x,y,z)+ G(x,y,z)) = F'(x,y,z) + G'(x,y,z).
	c. div(F(x,y,z)+ G(x,y,z))= div(F(x,y,z))+ div(G(x,y,z)).
	d. All of the above

Question 90

Let C be a simple closed curve and D be the region enclosed by C. Then, which of the following is true?

Choose one answer.

	a.
	b.
	c.
	d.

Question 91

If a solid has a mass density of m(x,y,z), then what is the total gravitational potential energy of the solid?

Choose one answer.

	a.
	b.
	c.
	d.

Question 92

Suppose a surface S is parameterized by (u,v)= <x(u,v),y(u,v),z(u,v)>, where (u, v) lies in a region R in the u-v plane. Then, we can rewrite

as which of the following?

Choose one answer.

	a.
	b.
	c.
	d.

Question 93

Suppose a vector field F represents the velocity of a fluid through a membrane represented by S. What does the flux represent?

Choose one answer.

	a. The rate of change in direction
	b. Surface area of S
	c. The volume of fluid passing through S
	d. None of the above

Question 94

The divergence theorem relates which of the following?

Choose one answer.

	a. The flux of a vector field to the flow of the vector field inside the surface
	b. The behavior of the vector field on the surface to its curvature
	c. The flux of a vector field through a surface to the behavior of the vector field inside the surface
	d. None of the above

Question 95

The surface integral in Stokes’ theorem does not change if which of the following occurs?

Choose one answer.

	a. The surface S is changed to any surface as long as the boundary of S is still the curve C.
	b. The surface S is changed to any surface as long as the boundary of S still intersects the curve C.
	c. The surface S is changed to a sphere.
	d. None of the above

Question 96

Use Gauss’s Law to calculate the electric field of an infinite sheet with a uniform charge density σ.

Choose one answer.

	a.
	b.
	c.
	d.

Question 97

Use the Fundamental Theorem for line integrals to evaluate

∇f ∙

, where C is the top half of the circle x²+y²=1, traversed counter clockwise, and f(x,y) =xy+x.

Choose one answer.

	a. -2
	b. 2
	c. 0
	d. None of the above

Question 98

Using Fubini’s theorem for double integrals, what can we say about Fubini’s theorem for triple integrals?

Choose one answer.

	a.
	b.
	c.
	d.

Question 99

Using the divergence theorem, compute the flux of the vector field F(x,y,z)= <x,y,z> through the unit sphere.

Choose one answer.

	a. 3π
	b. 4π
	c. 8π
	d. 16π

Question 100

What is the definition of the surface area differential for a surface S?

Choose one answer.

	a.
	b.
	c.
	d.

Question 101

What is the volume of a solid between x = y and x =0, y = 0, y = 2, z = 0, and z = 1?

Choose one answer.

	a. 2
	b. 4
	c. 1
	d. 6