Math Vector Calcu

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Math 254 Written Homework 6 – Spring 2018
Due May 10, 2018
Page 1 of 2
Instructions: Put your solutions on a separate piece(s) of paper. Bring your completed
paper copy to Recitation. You may submit it to Canvas up until 11:59 pm on May 10.
After 11:59 pm on Thursday it will be late. Late homework is not accepted.
To submit it to Canvas, scan your document as a pdf file. Canvas will only accept pdf files.
(1) Find the domain of the following functions. Sketch the domain.
1
(a) f (x, y) = p
,
2
x y 2 â?? 25
(b) f (x, y) = ln(x2 â?? y).
(2) At what points of R2 is the following function continuous?

xy

if (x, y) 6= (0, 0),
2
x y2
f (x, y) =
 0
if (x, y) = (0, 0).
(3) Let z = f (x, y) = x2 y â?? xy 3 and suppose
(
x = x(t) = t2
y = y(t) = tâ??2 .
dz
. Express your answer in terms of
Use the Chain Rule to compute the derivative
dt
the independent variable t.
(4) When two electrical resistors with resistance R1 > 0 and R2 > 0 are wired in parallel
in a circuit, the combined resistance R, measured in ohms (â?¦), is given by
1
1
1
=

R
R1 R2
â??R
â??R
(a) Find
and
after solving for R (e.g., R = . . . ).
â??R1
â??R2
(b) Describe how an increase in R1 with R2 held constant affects R. (Will R increase
or decrease?)
(c) Describe how a decrease in R2 with R1 held constant affects R. (Will R increase
or decrease?)
(5) A grocery store sells two kinds of citrus fruit: oranges and grapefruit. These are
competing â?? when the price of one increases, it causes more of the other to be sold.
Let Q(g, r) represent the quantity (in kilograms) of grapefruit sold with g being the
price of grapefruit ($/kg) and r being the price of oranges ($/kg).
Interpret the equation
â??Q
= Qr (3, 4) = 6.
â??r (3,4)
Use the contextual words or phrases â??quantity of grapefruit soldâ?, â??dollarsâ?, â??kilogramsâ?, etc… in your interpretation.
Math 254 Written Homework 6 – Spring 2018
Due May 10, 2018
Page 2 of 2
(6) For each of the following, compute the second partial derivatives fxx , fyy , fxy , and fyx .
(a) f (x, y) = cos(xy)
(b) f (x, y) = ye3x
1
(c) f (x, y) = â??
xy
(7) Consider the surface z = x2 2 . Suppose you are walking on this surface directly
above a curve C in the xy-plane, where the parameterized curve C is given by
C : x = cos t y = sin t.
Find the values of t for which you are walking uphill (that is, z is increasing).
(Assume you are walking above the curve C in the direction of positive orientation.
The direction of positive orientation for the plane curve C is indicated by its tangent
vectors.)

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