Math 230 Final Exam Solutions correct answers

Math 230  Final Exam Solutions correct answers

Instructions: • Read each problem carefully. • Write legibly. • Show all your work on these sheets. • Make sure that your final answer is clearly indicated. If two answers are presented, the average of the points for each answer will be given. • This exam has 16 pages and 9 problems. Before starting the exam, please check that your copy contains all of them and obtain a new copy of the exam immediately if it does not. • You may not use books, notes, or calculators.

Mark your section and write your NetID in the upper right corner of this page. Do NOT write your name on this exam

Question 1 (36 points, 3 points each).

PART A True or False? Circle the correct answer.

(a) The line x = 3 + 2t, y = 4 − t, z = 1 + 3t intersects the y-axis. 

(b) The function f(x, y) = p x 2 + y 2 is continuous on the entire xy-plane.

(c) Let g(x, y, z) be a continuous function of three variables. The level surface g = 1 must not intersect the level surface g = 2. 

(d) If z = h(x + y) for some differentiable function h(u), then ∂z ∂x and ∂z ∂y must be equal for all values of x and y.

(e) Every ellipse has constant curvature.

(f ) The point (x, y, z) = (√ 3, 3, 6) lies on the surface described in spherical coordinates by φ = π/3. 

(g) Let i, j, and k be the unit vectors along the three-dimensional rectangular coordinate axes. The vectors i and k satisfy (i×i)×k = i×(i×k).

(h) A curve represented by a vector-valued function r(t) lies entirely on a surface. If r(0) = h1, 2, 3i, then the tangent vector r 0 (0) can be taken as a normal vector for the tangent plane to the surface at (1, 2, 3).

PART B Suppose that f : R 2 → R has continuous second partial derivatives. A table of values at four points is given. 

What does f have at each of these four points? Mark your answers.

Question 2 (15 points). Consider the two planes x − y + z = 3 and x + 2y + z = 3

Are they parallel? If so, find the distance between them. If not, find parametric equations for the line where they intersect

Question 3 (15 points). A leprechaun is walking along the curve given by

r(t) = hcost, sin t, ti 0 6 t < ∞

where the components are measured in meters. If his pot of gold is located 10 meters along the path from his starting location, at what time will he reach it?

Question 4 (12 points, 6 points each). Find the following limits or show that they do not exist. Regardless of whether the limit does or does not exist show your work.

(a) lim (x,y)→(0,0) xy x 4 + x 2 + y 2 

(b) lim (x,y)→(0,0) x 2 y 2 x 4 + x 2 + y 2 

Question 5 (27 points). Consider the function f(x, y) = e 2x+y .

(a) (8 points) Compute the gradient vector of f at the point (0, 0). 

(b) (4 points) Find the directions in which the directional derivative of f at (0, 0) has the value 1

(c) (12 points) Find a quadratic function g(x, y) that best approximates f(x, y) = e 2x+y near (0, 0).

(d) (3 points) Use g to estimate f(0.2, −0.1).

Question 6 (15 points). Let f(x, y) = x 2 + 2y 2 − 2x − 4y. Find the absolute maximum and minimum values of f over the closed triangular region with vertices (0, 0), (0, 2), and (4, 2).

Question 7 (18 points). Consider the surface z(x 2 + y 2 ) = y 3 .

(a) (15 points) Find an equation for the plane tangent to the surface at (−2, 2, 1). 

(b) (3 points) The trace of this surface in the plane z = 1 is a curve described by the equation x 2 + y 2 = y 3 . This equation implicitly defines y as a function of x. Compute y 0 (x) when x = −2 and y = 2

Question 8 (35 points). A surface is defined by x 2 − y 2 + z 2 4 = −1.

(a) (16 points, 4 points each) Sketch its traces in the following planes. Label the points of intersection with the coordinate axes.

(b) (4 points) Set up the coordinate axes correctly and sketch this surface in R 3 

(c) (15 points) Find the points on the surface x 2 − y 2 + z 2 4 = −1 that are closest to the point (0, 0, 1)

Question 9 (25 points). A snowball with mass 0.4 kg is thrown northward into the air with a speed of 20m/s at an angle of 45◦ from the ground. A wind applies a steady force of 4N to the ball in a westerly direction. The magnitude of the acceleration due to gravity is given by 10 m/s2

(a) (8 points) Find the ball’s acceleration and initial velocity vectors. Hint: You may find it convenient to set up a three-dimensional coordinate system with the x-axis pointing east and the y-axis pointing north

(b) (12 points) Find parametric equations for the line tangent to the path of the ball when it reaches the highest point.

(c) (4 points) Find the tangential component of the ball’s acceleration at the highest point, that is, find the component of the acceleration vector along the tangent vector you computed in part (b).

(d) (1 point) How fast is its speed changing at the instant when the ball reaches the highest point?