M ATHEMATICS 54 Elementary Analysis 2 Fourth Exam
First Semester AY 2010 - 2011
7 October 2010
Write all necessary solutions on your bluebooks and box all final answers. Present your solutions in the proper sequence. You have one hour and thirty minutes to finish this exam. Good luck! I. Vector Functions. Consider the space curve defined by ~r(t) = h3 sin 2t, 3 cos 2t, 8ti 1. Reparametrize ~r(t) with respect to arc length measured from the point (0, 3, 0) in the direction of increasing t. Where on the curve are we after traveling for a distance of 20π? (7 points) 2. Determine the moving trihedral and find equations of the osculating plane, rectifying and normal planes of ~r(t) at the point (0, 3, 8π). (15 points) 3. Find the curvature and radius of curvature of ~r(t) at the point (0, 3, 8π).
(6 points)
4. If a particle is moving along the space curve ~r(t), determine the tangential and normal components of acceleration at the point (0, 3, 8π). (6 points)
II. Limits. Determine whether the limit exists. If so, find its value. x3 y 1. lim (x,y)→(0,0) 3x6 + y 2
(7 points)
2.
lim
cos
−1
(x,y,z)→(0,1,2)
ex−2y+z (x − y + z) (6 points) x2 − y 2 − z 2 + 2yz
III. Partial Derivatives. Do as indicated. 1. Let f (x, y, z) = (3x + 4y − 5z)6 . Find fxyz (0, 1, 1). 2. Find
∂z if z 2 + ln(sinh(xy)) = exz tan(x + y 3 ). ∂y
3. Use a chain rule to find
∂w if w = zex/y ; x = r cos θ, y = r sin θ, z = r2 + θ2 . ∂r
(9 points) (11 points)
(11 points)
IV. Related rates, linear approximation, differentials. Solve the following completely. 1. At what rate is the area of a rectangle changing if its length is 8 ft and decreasing at 3 ft/s while its width is 5 ft and increasing at 2 ft/s? (6 points) p p 2. Find the linearization of f (x, y) = x2 + y 2 at the point (3, 4) and use it to estimate (3.01)2 + (3.95)2 . (10 points)
3. One leg of a right triangle increases from 3 cm to 3.2 cm, while the other leg decreases from 4 cm to 3.96 cm. Use a total differential to approximate the change in the length of the hypotenuse. (6 points)
“The more we know, the more we know we don’t know” - N ATIONAL G EOGRAPHIC
F E ND OF E XAM F T OTAL : 100 POINTS