HW 5.1 – Surfaces, Functions, Level curves - solutions
Match the function & graph & contour map
1) z=sin(xy)
2) z=sin(x-y)
3) z=(1-x2)(1-y2)
4) z=excos(y)
5) z=sin(x)-sin(y)
6)
z=
x− y
1+ x 2+ y 2
1Cii, 2Fi, 3Bvi 4Aiv, 5Eiii, 6Dv
7) One contour map is for a cone, the other is for a paraboloid. Identify them, give your reason
.
I-parabola; II-cone [level curves evenly spaced]
8) On each contour map, rank the points in order from flattest to steepest
map #1
map #2
map #3
B,A,C
C,B,A
A,B,C
9) Biologists often model the surface area of the human body with the function
S=f(w,h)=0.1091w0.425h0.725 where w=weight (in pounds) h=height (in inches) S=area (in sq feet)
a) calculate f(160,70) and state the meaning in a sentence.
A person at 160 lbs and 70 inches has a surface area of 20.5 sq ft
b) what is your surface area (according to the model)?
c) define S70 = f(w,70) … i) sketch the graph ii) what does it mean?
S70 = 2.37w0.425 … its the relationship between weight and surface area for someone 70” tall
10) f(x,y) = 1+√x-y2
a) calculate f(3,1)
=1+√2
b) find the domain of f and sketch it
D: x>y2
c) find the range of f
R: z ≥ 1
11) f(x,y) = √16-(x-1)2-(y-2)2
a) calculate f(3,1)
= √11
b) find the domain of f and sketch it
D: circle r=4 centered at 1,2 = {-3<x<5, 2-√16-(x-1)2 < y < 2+√16-(x-1)2 }
c) find the range of f
R: 0<z<4
12)
day 160: 11o … day 180: 19o
Draw a contour map with several level curves (at least four, include both positive and negative values)
13) f(x,y) = (y-2x)2
c=(y-2x)2 → y-2x=√c → y=2x+√c … level curves are lines
14) g(x,y) = ye-x
c=ye-x → y=cex … level curves are exponential graphs