MATH253 Final Exam Practice Problems

Your in-class final exam will consist entirely of problems similar to the following, with numbers/functions changed but worded the same way.  It is assumed you will thoroughly master all the problems on this handout and errors in understanding on your in-class final will earn less part credit than on the previous exams.

MATH253 - Formulas

Linearization: L(x, y) = f x (x 0 , y 0) (x – x 0) + f y (x 0 , y 0) (y – y 0) + z 0

Total Differential: df = f x (x 0 , y 0) dx + f y (x 0 , y 0) dy

Chain Rule for Functions of Two Independent Variables, w = f(x,y):

Tangent Line to a Level Curve: f x (x 0 , y 0) (x – x 0) + f y (x 0 , y 0) (y – y 0) = 0

Tangent Plane (explicit): f x (x 0 , y 0) (x – x 0) + f y (x 0 , y 0) (y – y 0) – (z – z 0) = 0

Tangent Plane (implicit): f x (P 0) (x – x 0) + f y (P 0) (y – y 0) + f z (P 0) (z – z 0) = 0

Normal Line: x = f x (x 0 , y 0) t + x 0 , y = f y (x 0 , y 0) t + y 0 , z = - t + z 0

D u f = Ñf • u      

Error in Standard Linear Approximation: | E(x, y) | < ½ M( | x – x 0 | + | y – y 0 | ) 2 where M is any upper bound for | f xx | , | f yy | , | f xy | on R.

dy/dx = - F x / F y
2nd derivative test: f has a local max at (a, b) if f xx < 0 and f xx f yy - f xy 2 > 0 at (a, b)
f has a local min at (a, b) if f xx > 0 and f xx f yy - f xy 2 > 0 at (a, b)
f has a saddle at (a,b) if f xx f yy - f xy 2 < 0 at (a, b)
inconclusive if f xx f yy - f xy 2 = 0 at (a, b)

Length of a curve:

Center of Mass:

If A = áa 1, a 2, a 3ñ , B = áb 1, b 2, b 3ñ then A·B = |A||B|cos θ = a 1b 1 + a 2b 2 + a 3b 3

Unit Vector A= unit vector in the direction of A

Direction cosines: cos α = , cos β = , cosγ =



distance between a point (x 0, y 0) and a line ax + by + c = 0   

Work = | F |cos θ ||

A × B =

|A × B| = area of the parallelogram determined by A and B.

|A·(B × C)| = volume of a parallelpiped

Parametric equations of a line: x = at + x 0, y = bt + y 0 , z = ct + z 0

Equation of a plane: ax + by + cz = d

Green’s Theorem:

Stoke’s Theorem:

Divergence Theorem:
CONVERSION FORMULAS
(r, θ, z) → (x, y, z)
(x, y, z) → (r, θ, z)		 x = r cos θ, y = r sin θ, z = z
r = , tan θ = y/x, z = z
(ρ, θ, Φ) → (r, θ, z)
(r, θ, z) → (ρ, θ, Φ)		 r = ρ sin Φ, θ = θ, z = ρ cos Φ
ρ = , θ = θ, tan Φ = r/z
(ρ, θ, Φ) → (x, y, z)
(x, y, z) → (ρ, θ, φ)		 x = ρ sin Φ cos θ, y = ρ sin Φ sin θ, z = ρ cos φ
ρ = , tan θ = y/x, cos φ = z /

PROBLEMS

1.   [Answer]Write a triple integral which represents the volume of the first octant solid bounded by the coordinate planes and the graphs of z = 3x2 , z = 4 – x2, and y - z = 3. An unlabeled diagram is given as an aid. Do not evaluate the integral.
2.   [Answer]An equation of the surface of a mountain is z = 1900 – 3xy, where distance is measured in meters, the positive x-axis points to the west, and the positive y-axis points to the south. A mountain climber is at the point corresponding to (50, 10, 400).
  1. What is the direction of steepest ascent?
  2. Is the climber ascending or descending when headed north?
  3. In what direction should the climber head to travel a level path?
3.   [Answer]Ike, the inchworm, travels along R(t) = e –t i + t j + e t k starting at the point for which t = 0 and stopping at the point where t = ln 4. (distance measured in cm, time in minutes.)
  1. How far did Ike travel?
  2. How fast was Ike traveling at t = ln 4?
4.   [Answer]Ike’s cousin, Izzy, is moving in space along the path given by , 0 ≤ t ≤ π. (distance in cm, time in minutes)
  1. What is Izzy’s maximum height during this time?
  2. For what value(s) of t will Izzy experience maximum acceleration and how great is that maximum acceleration
5.   [Answer]Let C be the curve with equations x = 2 – t 3 , y = 2t – 1, z = ln t.
  1. Does this curve intersect the xz-plane? If so, find the point of intersection. If not, explain why it doesn’t.
  2. Find the parametric equations of the tangent line at the point (1, 1, 0).
6.   [Answer]A particle starts at the origin with initial velocity i + 2j + k. Its acceleration is a(t) = ti + j + t 2 k. Find its position vector.
7.   [Answer]Suppose z = f(x, y), where x = g(s, t), y = h(s, t), g(1, 2) = 3, g s (1, 2) = -1, g t(1, 2) = 4, h(1, 2) = 6, h s(1, 2) = -5, h t(1, 2) = 10, f x(3, 6) = 7, and f y(3, 6) = 8. Find when s = 1 and t = 2.
8.   [Answer] Let be a differentiable function and let .
  1. Show that .
  2. Find the direction of maximal increase of at  in terms of .
9.   [Answer]A painting contractor charges $4 per square meter for painting the four walls and ceiling of a room. The dimensions of the ceiling are measured to be 4 m and 5 m, the height of the room is measured to be 3 m, and these measurements are correct to 2.0 cm. Use differentials to approximate the greatest error in estimating the cost of the job from these measurements.
10.  [Answer]Use polar coordinates to combine the sum below into one integral. Then evaluate this integral.
11.  [Answer]It is known that a certain curve f(x, y) = c, in two dimensions has slope: . It is also known that (1,0) satisfies f(x, y) = c. Determine the equation f(x, y) = c. Hint: cross multiply and move all terms to the left-hand side of the equation, then use the techniques we just studied.
12.  [Answer]Find the most general M(x,y) for which M(x,y) dx + (2ye x + 4x) dy will be an exact differential.
13.  [Answer]Evaluate , showing all steps.
14.  [Answer]Evaluate each line integral on the specified path.
  1. along the path from (0, 0) to (1, 2) defined by y = 2x 2.
  2. along the line segments:
  3. along the line segments:
15.  [Answer]This sum of two double integrals may be written as one double integral. What is this one double integral?
16.  [Answer]Find every point on the given surface at which the tangent line is horizontal.

z = 3x 2 + 12x + 4y 3 - 6y 2 + 5
17.  [Answer]A surveyor wants to find the area in acres of a certain field (1 acre is 43560 ft 2). She measures two different sides, finding them to be a = 500 feet and b = 700 feet, with a possible error of as much as one foot in each measurement. She finds the angle between the two sides to be 30° with a possible error of as much as .25° . The field is triangular, so its area is given by A = ½ ab sin θ. Use differentials to find the maximum resulting error, in acres, in computing the area of this field using this formula.
18.  [Answer]Evaluate the given integral by first converting it to cylindrical coordinates.
19.  [Answer]Let r(t) = á sin 2t, 3t, cos 2t ñ, -π ≤ t ≤ π. Is there any time t for which r(t) and the velocity vector are perpendicular? If so, find all such values.
20.  [Answer]Suppose that f(x,y) = e x – y and f (ln 2, ln 2) = 1. Use the technique of linear approximation to estimate f (ln 2 + 0.1, ln 2 + 0.04).
21.  [Answer]Suppose u = á 1, 0 ñ, v = , Duf(a,b) = 3 and Dvf(a,b) = .
  1. Find Ñf(a, b) = áf 1, f 2ñ.
  2. What is the maximum possible value of Dwf(a,b) for any w? Why?
  3. Find a unit vector w = á w 1, w 2 ñ such that Dwf(a,b) = 0.
22.  [Answer]Consider the two surfaces ρ = 3 csc φ and r = 3. Are they the same surface, or are they different surfaces? Explain your answer.
23.  [Answer]Evaluate , where R consists of the set of points such that 1 ≤ x 2 + y 2 ≤ 4.
24.  [Answer]Consider the triple integral representing a solid S. Let R be the projection of S onto the plane z = 0.
  1. Draw the region R.
  2. Rewrite this integral as .
25.  [Answer]Compute the work done by the vector field F(x,y) = (sin x + xy 2)i + (e y + ½ x 2) j , where C is the path that goes around the unit square twice, counterclockwise.
26.  [Answer]Consider the vector field F(x,y,z) = 2xi + 2yj + 2zk. If C is any path from (0, 0, 0) to (a 1, a 2, a 3) and a = a 1 i + a 2 j + a 3 k, show that = a•a.
27.  [Answer]Find the volume of the region bounded above by the sphere x 2 + y 2 + z 2 = 1 and below by the sphere x 2 + y 2 + (z - 1) 2 = 1.
28.  [Answer]Express the vector a = 2i + 4j + 5k as the sum of a vector p which is parallel to the vector b = 2i – j – 2k and a vector n which is normal to b.
29.  [Answer]Evaluate along the straight line joining these points. Use any method that works.
30.  [Answer]Find the x coordinate of the center of mass of the triangular lamina with vertices (0, 0), (1,0) and (0, 2) and density function ρ(x, y) = 1 + 3x + 3y .
31.  [Answer]Suppose it is known that along the line segment joining (2, 1) to (K, 3). What is the value of K? Work must be shown to receive full credit.
32.  [Answer]Evaluate where C is one counterclockwise trip around the circle x 2 + y 2 = 4.
33.  [Answer]The gravitational constant G is often measured at different points on the surface of the earth with a simple pendulum and the formula (L in feet, T in seconds). Suppose it is known that the length L of a pendulum = 2 feet and that one period T = 2 seconds. A technician is able to measure L with error of ± one inch and T with an error of ± one tenth of a second. Approximate the maximum percentage error in the measurement of G which might be caused by the technician’s errors. Use differentials.
34.  [Answer]The point (2, 3, 6) lies on the surface .
Compute the numerical value of at (2, 3, 6).
35.  [Answer]Find all the local maxima, local minima, and the saddle points of the function
.
36.  [Answer]Find unit tangent and unit normal vectors to the curve r(t) = t3i + 2t2j at t = 1. Sketch a portion of the curve showing these vectors.
37.  [Answer]A particle moves through 3-space in such a way that its velocity is v(t) = 2i - 4t3j + 6√t k. Find the coordinates of the particle at t = 1, if the particle was initially at (1,5,3) at t = 0.
38.  [Answer]Let F (x, y, z) = (z + y 2)i + 2xyj + (x + y)k
  1. Find the value of the curl of F at the point (1, 1, 1).
  2. Find the value of the divergence of F at the point (1, 2, 3).
39.  [Answer]For what value of the constant b is the vector field F = bxy 2i + x 2yj irrotational?
40.  [Answer]Find the area of the portion of the surface x2 – 2z = 0 that lies above the triangle bounded by the lines x = , y = 0, and y = x in the xy-plane.
41.  [Answer]Let S be the cylinder x2 + y2 = a2 , 0 ≤ z ≤ h, together with its top, x2 + y2 ≤ a2, z = h. Let F = -y i + x j + x2 k. Use Stoke’s theorem to calculate the flux of Ñ × F outward through S.
42.  [Answer]Use the Divergence theorem to find the outward flux of the field F = 3xz2 i + y j - z3 k across the surface of the solid in the first octant that is bounded by the cylinder x2 + 4y2 = 16, and the planes y = 2z, x = 0, and z = 0.

ANSWERS

1.   back
2.  backa) á-30, -150ñ     b) ascending     c)
3.  backa) 15/4 cm     b) 17/4 cm/min
4.  backa) 2π cm     b) max acc = 2 cm/min2 when t = 0 or π min
5.  backa) yes (15/8, 0, - ln 2)     b) x = -3t + 1, y = 2t + 1, z = t
6.  backr =
7.  back-47
8.  back (a) 

(b) 

9.  back$3.12
10.  back
11.  backsin xy – xe2y – y2 = -1
12.  backM(x, y) = y2ex + 4y + G(x)
13.  back245.925
14.  backa) -1.9     b) - 4     c) 0
15.  back
16.  back(-2, 0, -7) and (-2, 1, -9)
17.  back.022 acres
18.  back = (π/2)×(ln 17)
19.  backr ^ v when t = 0
20.  back1.06
21.  backa) á 3, -1ñ     b) , max value = |Ñf(a, b)|     c)
22.  backsame surface, cylinder of radius 3
23.  back(sin 4 – sin 1)π ≈ -5.02
24.  backa)     b)
25.  back0
26.  backHint: let r = áa1t, a2t, a3tñ, 0 < t < 1 (the straight line path)
27.  back5π/12
28.  back
29.  back5
30.  back1/3
31.  back-6
32.  back16π
33.  back14.17%
34.  back-9
35.  backsaddle (0, 0); local max (2, 2)
36.  back
37.  back(3, 4, 7)
38.  backa) i     b) 2
39.  back1
40.  back7/3
41.  back2πa2
42.  back8/3