Multivariable Calculus Homework 3 Due Friday, June 7th

Homework 3

Math 230, Amine Benkiran

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Instructions: • Answers to these problems must be handed in at the beginning of the class and on a separate sheet of paper. Answers must be organized, legible, and all papers must be stapled. • Points will be deducted for incomplete reasoning and disorganized work (even if your answers are correct). • No late submissions will be accepted. • If you have any difficulties/questions, you are encouraged to discuss these problems with me.

I. Book Problems: Section 14.1 14.2 14.3 14.4 14.5

Problems 2, 5, 10, 16, 44, 46. 8, 12, 14, 19, 32, 38, 40, 41. 1, 22, 25, 32, 41, 44, 48, 64, 66, 68, 2, 5, 11, 16. 2, 6, 8, 9, 11, 15, 22, 24, 35, 41.

II. Let r(t) =< t2 , 4t − 3 >. Find T(t) and N(t), and tangential and normal components of acceleration. III. Sketch a contour map of f (x, y). 1. f (x, y) = 3x2 − 5y 2 . 1 2. f (x, y) = 2 . x + y2 + 1 IV. Find the limit or determine that it does not exist. 1.

x4 cos(πy) (x,y)→(−2,1) ex+y

2.

|x| (x,y)→(0,0) |x| + |y|

3.

lim

lim

lim

2

(x − 16) cos

x,y)→(4,0)

1 (x − 4)2 + y 2

y4 4. lim (x,y)→(0,0) x4 + y 4


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Homework 3


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V. The Laplace Operator ∆ is defined by ∆f = fxx + fyy . A function u(x, y) satisfying the Laplace equation ∆u = 0 is called a harmonic. Show that the following functions are harmonic: y 1. u(x, y) = tan−1 x 2. u(x, y) = ln(x2 + y 2 ) ∂z ∂ 2 z ∂z ∂ 2 z VI. If z = sin(x + sin t), show that = . ∂x ∂x∂t ∂t ∂x2 VII. The body mass index (BMI) of a person is defined by the equation B(m, h) =

m , h2

where m is the the person’s mass (in kilograms) and h is the height (in meters). So, B is a function of two variables m and h. Calculate the partial derivatives of B for a person with m = 64 kg and h = 1.68 m and interpret them. VIII. Find an equation of the tangent plane to the surface at the given point. 1. f (r, s) = r2 s−1/2 + s−3 at (2,1). 2. g(x, y) = ex/y at (−3, 1). IX. Find the linear approximation of the function f (x, y, z) = x3 (2, 3, 4) and use it to estimate the number p (1.98)3 (3.01)2 + (3.97)2 .

p y 2 + z 2 at the point

X. A model for the surface area of a human body is given by the function S = f (w, h) = 0.1091w0.425 h0.725 , where w is the weight (in pounds), h is the height (in inches), and S is measured in square feet. 1. Find the linearization, L(w, h), of the function when w = 160lb and h = 70in. 2. Compare the values of f (w, h) and L(w, h) when (w, h) = (162, 73) and when (w, h) = (170, 80). Comment on the relative accuracy of the approximations. XI. Find

∂z ∂z and at the given point. ∂s ∂t

1. z = x2 y 3 , x = s cos t, y = s sin t, when (s, t) = (1, π). s t 2. z = ex+2y , x = , y = , when (s, t) = (1, 1). t s


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