calculus ii fall 2014 - university of colorado...
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Calculus II – Fall 2014
Eitan Angel
University of Colorado
Tuesday, November 18, 2014
E. Angel (CU) Calculus II 18 Nov 1 / 30
Introduction
We will gain intuition into functions of two variables graphically,numerically, and analytically.
A two variable function z = f(x, y) has two independent inputs, xand y, which may be varied separately.
We will discuss three dimensional coordinate systems in order tounderstand two variable functions.
Functions of two variables and their domains. Cross sections and levelcurves can be used to graph two variable functions.
E. Angel (CU) Calculus II 18 Nov 2 / 30
Introduction
We will gain intuition into functions of two variables graphically,numerically, and analytically.
A two variable function z = f(x, y) has two independent inputs, xand y, which may be varied separately.
We will discuss three dimensional coordinate systems in order tounderstand two variable functions.
Functions of two variables and their domains. Cross sections and levelcurves can be used to graph two variable functions.
E. Angel (CU) Calculus II 18 Nov 2 / 30
Introduction
We will gain intuition into functions of two variables graphically,numerically, and analytically.
A two variable function z = f(x, y) has two independent inputs, xand y, which may be varied separately.
We will discuss three dimensional coordinate systems in order tounderstand two variable functions.
Functions of two variables and their domains. Cross sections and levelcurves can be used to graph two variable functions.
E. Angel (CU) Calculus II 18 Nov 2 / 30
Introduction
We will gain intuition into functions of two variables graphically,numerically, and analytically.
A two variable function z = f(x, y) has two independent inputs, xand y, which may be varied separately.
We will discuss three dimensional coordinate systems in order tounderstand two variable functions.
Functions of two variables and their domains. Cross sections and levelcurves can be used to graph two variable functions.
E. Angel (CU) Calculus II 18 Nov 2 / 30
Graphical Example: A Weather Map
The weather map below displays temperature, T . What are theindependent variables? We may consider T (x, y).
E. Angel (CU) Calculus II 18 Nov 3 / 30
Graphical Example: A Weather Map
The weather map below displays temperature, T . What are theindependent variables? We may consider T (x, y).
E. Angel (CU) Calculus II 18 Nov 3 / 30
Numerical Example: Veggie Burger Consumption
Suppose you distribute veggie burger patties and you want to understandyour market. The quantity of burger patties bought by a household in amonth, C, depends on the price of patties, p, as well as the householdincome, I. So some values of C(I, p) (in lbs.) are
Household income I (in $1000)
Price of patties, p ($ / lb.)
3.00 3.50 4.00 4.50
20 2.65 2.59 2.51 2.43
40 4.14 4.05 3.94 3.88
60 5.11 5.00 4.97 4.84
80 5.35 5.29 5.19 5.07
100 5.79 5.77 5.60 5.53
E. Angel (CU) Calculus II 18 Nov 4 / 30
Numerical Example: Veggie Burger Consumption
We can visualize this data with a bar graph
215
Transparency Master for bar graph of Table 12.1 in the text
3.0
3.5
4.0
4.5
20
40
60
80
1001
2
3
4
5
6
price
income
pounds
A three-dimensional bar graphE. Angel (CU) Calculus II 18 Nov 5 / 30
Numerical Example: Veggie Burger Consumption
Given more refined data:
217
3.0
3.5
4.0
4.5
20
40
60
80
1001
2
3
4
5
6
price
income
pounds
More and more refined beef dataE. Angel (CU) Calculus II 18 Nov 6 / 30
Analytic Examples
If M is the amount of money in a bank account t years after aninvestment of P dollars, and if interest is accrued at a rate of 5% peryear compounded annually, then
M = f(P, t) = P (1.05)t.
If P is the population of rabbits whose growth rate is a continuous3% per year, and P0 is the initial population of rabbits, then
P = f(P0, t) = P0e0.03t
A cone has radius r and height h. If its volume is V we can give aformula for the volume by
V (r, h) =1
3πr2h
E. Angel (CU) Calculus II 18 Nov 7 / 30
Analytic Examples
If M is the amount of money in a bank account t years after aninvestment of P dollars, and if interest is accrued at a rate of 5% peryear compounded annually, then
M = f(P, t) = P (1.05)t.
If P is the population of rabbits whose growth rate is a continuous3% per year, and P0 is the initial population of rabbits, then
P = f(P0, t) = P0e0.03t
A cone has radius r and height h. If its volume is V we can give aformula for the volume by
V (r, h) =1
3πr2h
E. Angel (CU) Calculus II 18 Nov 7 / 30
Analytic Examples
If M is the amount of money in a bank account t years after aninvestment of P dollars, and if interest is accrued at a rate of 5% peryear compounded annually, then
M = f(P, t) = P (1.05)t.
If P is the population of rabbits whose growth rate is a continuous3% per year, and P0 is the initial population of rabbits, then
P = f(P0, t) = P0e0.03t
A cone has radius r and height h. If its volume is V we can give aformula for the volume by
V (r, h) =1
3πr2h
E. Angel (CU) Calculus II 18 Nov 7 / 30
Coordinate Systems
Given a fixed point O, a pair of realnumbers describes a point on a plane.The x-axis and y-axis are called thecoordinate axes.
A triple of real numbers describes apoint in space. The x-axis, y-axis, andz-axis are called the coordinate axes.
E. Angel (CU) Calculus II 18 Nov 9 / 30
Coordinate Systems
Given a fixed point O, a pair of realnumbers describes a point on a plane.The x-axis and y-axis are called thecoordinate axes.
A triple of real numbers describes apoint in space. The x-axis, y-axis, andz-axis are called the coordinate axes.
E. Angel (CU) Calculus II 18 Nov 9 / 30
Handedness
Left-handed system Right-handed system
We will only use right-handed coordinate systems in this course.
E. Angel (CU) Calculus II 18 Nov 10 / 30
Coordinate Planes
The three illustratedcoordinate planes dividespace into eight parts, calledoctants. The first octant, inthe foreground, is determinedby the positive axes.
E. Angel (CU) Calculus II 18 Nov 11 / 30
Point
As mentioned before, a point P isspecified by a triple of real numbers,(a, b, c). To locate P ,
First travel a distance a along thex-axis.
Next travel a distance b parallel to they-axis.
Finally travel a distance c parallel tothe z-axis.
We call a the x-coordinate of P , b the y-coordinate of P , and c thez-coordinate of P .
E. Angel (CU) Calculus II 18 Nov 12 / 30
Point
As mentioned before, a point P isspecified by a triple of real numbers,(a, b, c). To locate P ,
First travel a distance a along thex-axis.
Next travel a distance b parallel to they-axis.
Finally travel a distance c parallel tothe z-axis.
We call a the x-coordinate of P , b the y-coordinate of P , and c thez-coordinate of P .
E. Angel (CU) Calculus II 18 Nov 12 / 30
Point
As mentioned before, a point P isspecified by a triple of real numbers,(a, b, c). To locate P ,
First travel a distance a along thex-axis.
Next travel a distance b parallel to they-axis.
Finally travel a distance c parallel tothe z-axis.
We call a the x-coordinate of P , b the y-coordinate of P , and c thez-coordinate of P .
E. Angel (CU) Calculus II 18 Nov 12 / 30
Point
As mentioned before, a point P isspecified by a triple of real numbers,(a, b, c). To locate P ,
First travel a distance a along thex-axis.
Next travel a distance b parallel to they-axis.
Finally travel a distance c parallel tothe z-axis.
We call a the x-coordinate of P , b the y-coordinate of P , and c thez-coordinate of P .
E. Angel (CU) Calculus II 18 Nov 12 / 30
Point
As mentioned before, a point P isspecified by a triple of real numbers,(a, b, c). To locate P ,
First travel a distance a along thex-axis.
Next travel a distance b parallel to they-axis.
Finally travel a distance c parallel tothe z-axis.
We call a the x-coordinate of P , b the y-coordinate of P , and c thez-coordinate of P .
E. Angel (CU) Calculus II 18 Nov 12 / 30
Points and Space
The Cartesian product
R× R× R = {(x, y, z) : x, y, z ∈ R}
is the set of all ordered triples of real numbers and is denoted by R3.
We have described a one-to-one correspondence between points P inspace and ordered triples (a, b, c) in R3.
This is called a three dimensional rectangular coordinate system.
Notice that the first octant can be described as points with allpositive coordinates.
E. Angel (CU) Calculus II 18 Nov 13 / 30
Points and Space
The Cartesian product
R× R× R = {(x, y, z) : x, y, z ∈ R}
is the set of all ordered triples of real numbers and is denoted by R3.
We have described a one-to-one correspondence between points P inspace and ordered triples (a, b, c) in R3.
This is called a three dimensional rectangular coordinate system.
Notice that the first octant can be described as points with allpositive coordinates.
E. Angel (CU) Calculus II 18 Nov 13 / 30
Points and Space
The Cartesian product
R× R× R = {(x, y, z) : x, y, z ∈ R}
is the set of all ordered triples of real numbers and is denoted by R3.
We have described a one-to-one correspondence between points P inspace and ordered triples (a, b, c) in R3.
This is called a three dimensional rectangular coordinate system.
Notice that the first octant can be described as points with allpositive coordinates.
E. Angel (CU) Calculus II 18 Nov 13 / 30
Points and Space
The Cartesian product
R× R× R = {(x, y, z) : x, y, z ∈ R}
is the set of all ordered triples of real numbers and is denoted by R3.
We have described a one-to-one correspondence between points P inspace and ordered triples (a, b, c) in R3.
This is called a three dimensional rectangular coordinate system.
Notice that the first octant can be described as points with allpositive coordinates.
E. Angel (CU) Calculus II 18 Nov 13 / 30
Distance
Theorem (Distance Formula)
The distance between a point P1(x1, y1, z1) and P2(x2, y2, z2) is given by
d =√
(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2.
This formula can be proven by repeated application of the Pythagoreantheorem.
E. Angel (CU) Calculus II 18 Nov 14 / 30
Distance
Draw a box where P1 and P2
form the main diagonal.
Label A(x2, y1, z1),B(x2, y2, z1), a = |x2 − x1|,b = |y2 − y1|, c = |z2 − z1|.|P1B|2 = a2 + b2 (Pythagoras)
Using Pythagoras again:
d2 = |P1P2|2
= |P1B|2 + c2
= a2 + b2 + c2
= (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2.
E. Angel (CU) Calculus II 18 Nov 15 / 30
Distance
Draw a box where P1 and P2
form the main diagonal.
Label A(x2, y1, z1),B(x2, y2, z1), a = |x2 − x1|,b = |y2 − y1|, c = |z2 − z1|.|P1B|2 = a2 + b2 (Pythagoras)
Using Pythagoras again:
d2 = |P1P2|2
= |P1B|2 + c2
= a2 + b2 + c2
= (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2.
E. Angel (CU) Calculus II 18 Nov 15 / 30
Distance
Draw a box where P1 and P2
form the main diagonal.
Label A(x2, y1, z1),B(x2, y2, z1), a = |x2 − x1|,b = |y2 − y1|, c = |z2 − z1|.|P1B|2 = a2 + b2 (Pythagoras)
Using Pythagoras again:
d2 = |P1P2|2
= |P1B|2 + c2
= a2 + b2 + c2
= (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2.
E. Angel (CU) Calculus II 18 Nov 15 / 30
Distance
Draw a box where P1 and P2
form the main diagonal.
Label A(x2, y1, z1),B(x2, y2, z1), a = |x2 − x1|,b = |y2 − y1|, c = |z2 − z1|.|P1B|2 = a2 + b2 (Pythagoras)
Using Pythagoras again:
d2 = |P1P2|2
= |P1B|2 + c2
= a2 + b2 + c2
= (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2.
E. Angel (CU) Calculus II 18 Nov 15 / 30
Distance
Distance
What is the distance between (2, 7, 3) and (1, 9, 5)?
d =√(1− 2)2 + (9− 7)2 + (5− 3)2 = 3
Sphere
What is the equation of a sphere with radius r and center C(x0, y0, z0)?
A sphere is the set of all pointsP (x, y, z) whose distance from the Cis r, i.e., |PC| = r. Upon squaringboth sides,
r2 = |PC|2
= (x− x0)2 + (y − y0)2 + (z − z0)2
E. Angel (CU) Calculus II 18 Nov 16 / 30
Distance
Distance
What is the distance between (2, 7, 3) and (1, 9, 5)?
d =√(1− 2)2 + (9− 7)2 + (5− 3)2 = 3
Sphere
What is the equation of a sphere with radius r and center C(x0, y0, z0)?
A sphere is the set of all pointsP (x, y, z) whose distance from the Cis r, i.e., |PC| = r. Upon squaringboth sides,
r2 = |PC|2
= (x− x0)2 + (y − y0)2 + (z − z0)2
E. Angel (CU) Calculus II 18 Nov 16 / 30
Distance
Distance
What is the distance between (2, 7, 3) and (1, 9, 5)?
d =√(1− 2)2 + (9− 7)2 + (5− 3)2 = 3
Sphere
What is the equation of a sphere with radius r and center C(x0, y0, z0)?
A sphere is the set of all pointsP (x, y, z) whose distance from the Cis r, i.e., |PC| = r. Upon squaringboth sides,
r2 = |PC|2
= (x− x0)2 + (y − y0)2 + (z − z0)2E. Angel (CU) Calculus II 18 Nov 16 / 30
Distance
Distance
What is the distance between (2, 7, 3) and (1, 9, 5)?
d =√(1− 2)2 + (9− 7)2 + (5− 3)2 = 3
Sphere
What is the equation of a sphere with radius r and center C(x0, y0, z0)?
A sphere is the set of all pointsP (x, y, z) whose distance from the Cis r, i.e., |PC| = r. Upon squaringboth sides,
r2 = |PC|2
= (x− x0)2 + (y − y0)2 + (z − z0)2E. Angel (CU) Calculus II 18 Nov 16 / 30
Spheres
The previous example implies a standard way to describe a sphere.
The standard equation of a sphere with radius r and centerC(x0, y0, z0) is
r2 = (x− x0)2 + (y − y0)2 + (z − z0)2.
If the terms the standard equation of a sphere are multiplied out and liketerms are collected, we arrive at the general equation of the sphere
x2 + y2 + z2 +Ax+By + Cz = D.
E. Angel (CU) Calculus II 18 Nov 17 / 30
Spheres
Example
Show thatx2 + y2 + z2 + 4x− 2y + 6z − 2 = 0
is the equation of a sphere, and find the center and radius.
Complete the square (thrice!)
(x2 + 4x+ 4) + (y2 − 2y + 1) + (z2 + 6z + 9)− 2 = 4 + 1 + 9
(x+ 2)2 + (y − 1)2 + (z + 3)2 = 16
which implies the radius is 4 and the center is (−2, 1,−3).
Q: Could the equation fail to describe a sphere?A: Sure. What if the right hand side is not positive?
E. Angel (CU) Calculus II 18 Nov 18 / 30
Spheres
Example
Show thatx2 + y2 + z2 + 4x− 2y + 6z − 2 = 0
is the equation of a sphere, and find the center and radius.
Complete the square (thrice!)
(x2 + 4x+ 4) + (y2 − 2y + 1) + (z2 + 6z + 9)− 2 = 4 + 1 + 9
(x+ 2)2 + (y − 1)2 + (z + 3)2 = 16
which implies the radius is 4 and the center is (−2, 1,−3).
Q: Could the equation fail to describe a sphere?A: Sure. What if the right hand side is not positive?
E. Angel (CU) Calculus II 18 Nov 18 / 30
Spheres
Example
Show thatx2 + y2 + z2 + 4x− 2y + 6z − 2 = 0
is the equation of a sphere, and find the center and radius.
Complete the square (thrice!)
(x2 + 4x+ 4) + (y2 − 2y + 1) + (z2 + 6z + 9)− 2 = 4 + 1 + 9
(x+ 2)2 + (y − 1)2 + (z + 3)2 = 16
which implies the radius is 4 and the center is (−2, 1,−3).
Q: Could the equation fail to describe a sphere?A: Sure. What if the right hand side is not positive?
E. Angel (CU) Calculus II 18 Nov 18 / 30
Spheres
Example
Show thatx2 + y2 + z2 + 4x− 2y + 6z − 2 = 0
is the equation of a sphere, and find the center and radius.
Complete the square (thrice!)
(x2 + 4x+ 4) + (y2 − 2y + 1) + (z2 + 6z + 9)− 2 = 4 + 1 + 9
(x+ 2)2 + (y − 1)2 + (z + 3)2 = 16
which implies the radius is 4 and the center is (−2, 1,−3).
Q: Could the equation fail to describe a sphere?A: Sure. What if the right hand side is not positive?
E. Angel (CU) Calculus II 18 Nov 18 / 30
Spheres
Thus the relevant theorem:
Theorem
An equation of the form
x2 + y2 + z2 +Ax+By + Cz = D
represents a sphere, a point, or has no graph.
So just because an equation has the general form of a sphere, it may bedegenerate or have no solution.
E. Angel (CU) Calculus II 18 Nov 19 / 30
Functions of Two Variables
The temperature T at a point on the Earth’s surface depends on thelatitude x and the longitude y of that point. So we can think oftemperature as a function of two variables, T (x, y).
The volume Vcyl of a cylinder depends on its radius r and its heighth. We say Vcyl is a function of r and h, and is given by the equationVcyl(r, h) = πr2h.
Definition
Let D ⊂ R2. A function f of two variables is a rule f : D → R thatassigns to each ordered pair (x, y) in D a unique real number f(x, y)denoted (x, y) 7→ f(x, y). The set D is called the domain of f and itsrange is the set of values that f takes on, i.e., {f(x, y)|(x, y) ∈ D}.
We often write z = f(x, y) to make explicit the value taken on by f at thepoint (x, y). The variables x and y are independent variables and z isthe dependent variable.
E. Angel (CU) Calculus II 18 Nov 20 / 30
Functions of Two Variables
The temperature T at a point on the Earth’s surface depends on thelatitude x and the longitude y of that point. So we can think oftemperature as a function of two variables, T (x, y).
The volume Vcyl of a cylinder depends on its radius r and its heighth. We say Vcyl is a function of r and h, and is given by the equationVcyl(r, h) = πr2h.
Definition
Let D ⊂ R2. A function f of two variables is a rule f : D → R thatassigns to each ordered pair (x, y) in D a unique real number f(x, y)denoted (x, y) 7→ f(x, y). The set D is called the domain of f and itsrange is the set of values that f takes on, i.e., {f(x, y)|(x, y) ∈ D}.
We often write z = f(x, y) to make explicit the value taken on by f at thepoint (x, y). The variables x and y are independent variables and z isthe dependent variable.
E. Angel (CU) Calculus II 18 Nov 20 / 30
Functions of Two Variables
The temperature T at a point on the Earth’s surface depends on thelatitude x and the longitude y of that point. So we can think oftemperature as a function of two variables, T (x, y).
The volume Vcyl of a cylinder depends on its radius r and its heighth. We say Vcyl is a function of r and h, and is given by the equationVcyl(r, h) = πr2h.
Definition
Let D ⊂ R2. A function f of two variables is a rule f : D → R thatassigns to each ordered pair (x, y) in D a unique real number f(x, y)denoted (x, y) 7→ f(x, y). The set D is called the domain of f and itsrange is the set of values that f takes on, i.e., {f(x, y)|(x, y) ∈ D}.
We often write z = f(x, y) to make explicit the value taken on by f at thepoint (x, y). The variables x and y are independent variables and z isthe dependent variable.
E. Angel (CU) Calculus II 18 Nov 20 / 30
Functions of Two Variables
The temperature T at a point on the Earth’s surface depends on thelatitude x and the longitude y of that point. So we can think oftemperature as a function of two variables, T (x, y).
The volume Vcyl of a cylinder depends on its radius r and its heighth. We say Vcyl is a function of r and h, and is given by the equationVcyl(r, h) = πr2h.
Definition
Let D ⊂ R2. A function f of two variables is a rule f : D → R thatassigns to each ordered pair (x, y) in D a unique real number f(x, y)denoted (x, y) 7→ f(x, y). The set D is called the domain of f and itsrange is the set of values that f takes on, i.e., {f(x, y)|(x, y) ∈ D}.
We often write z = f(x, y) to make explicit the value taken on by f at thepoint (x, y). The variables x and y are independent variables and z isthe dependent variable.
E. Angel (CU) Calculus II 18 Nov 20 / 30
Domain of Functions of Two Variables
Example
Find the domain of the following and evaluate f(3, 2).
f(x, y) =
√x+ y + 1
x− 1
E. Angel (CU) Calculus II 18 Nov 21 / 30
Domain of Functions of Two Variables
Example
Find the domain of the following and evaluate f(3, 2).
f(x, y) =
√x+ y + 1
x− 1
f(3, 2) =
√3 + 2 + 1
3− 1=
√6
2
The expression for f makes sense ifthe denominator is not 0 and thequantity under the square root sign isnonnegative. So the domain of f is
D = {(x, y) |x+ y + 1 ≥ 0, x 6= 1}
E. Angel (CU) Calculus II 18 Nov 21 / 30
Domain of Functions of Two Variables
Example
Find the domain of the following.
f(x, y) = x ln(y2 − x)
E. Angel (CU) Calculus II 18 Nov 22 / 30
Domain of Functions of Two Variables
Example
Find the domain of the following.
f(x, y) = x ln(y2 − x)
Since ln(y2 − x) is defined only wheny2 − x > 0, that is, x < y2, thedomain of f is
D = {(x, y) |x < y2}
In other words, the set of points tothe left of the parabola x = y2.
E. Angel (CU) Calculus II 18 Nov 22 / 30
Graphs of Functions of Two Variables
Definition
If f is a function of two variables with domain D, the graph of f is the set
S = {(x, y, z) ∈ R3 | z = f(x, y), (x, y) ∈ D}
The graph of a function f of onevariable is a curve C with equationy = f(x). The graph of a function fof two variables is a surface S withequation z = f(x, y). We canvisualize the graph S of f as lyingdirectly above or below its domain Din the xy-plane.
E. Angel (CU) Calculus II 18 Nov 23 / 30
Graphs of Functions of Two Variables
Definition
If f is a function of two variables with domain D, the graph of f is the set
S = {(x, y, z) ∈ R3 | z = f(x, y), (x, y) ∈ D}
The graph of a function f of onevariable is a curve C with equationy = f(x). The graph of a function fof two variables is a surface S withequation z = f(x, y). We canvisualize the graph S of f as lyingdirectly above or below its domain Din the xy-plane.
E. Angel (CU) Calculus II 18 Nov 23 / 30
Graphs of Functions of Two Variables
There are a couple of basic techniques to graph a two variable functionf(x, y).
We can numerically plot points. This consists of choosing a list ofvalues for x and y, computing f(x, y) for these chosen values, thenplotting in 3-space.
We can take cross-sections. This consists of holding one dependentvariable constant while allowing the other to vary.
We will give an example of both techniques.
E. Angel (CU) Calculus II 18 Nov 24 / 30
Graphs of Functions of Two Variables
There are a couple of basic techniques to graph a two variable functionf(x, y).
We can numerically plot points. This consists of choosing a list ofvalues for x and y, computing f(x, y) for these chosen values, thenplotting in 3-space.
We can take cross-sections. This consists of holding one dependentvariable constant while allowing the other to vary.
We will give an example of both techniques.
E. Angel (CU) Calculus II 18 Nov 24 / 30
Graphs of Functions of Two Variables
Example
Sketch the graph of the function h(x, y) = 4x2 + y2.
E. Angel (CU) Calculus II 18 Nov 25 / 30
Graphs of Functions of Two Variables
Example
Sketch the graph of the function h(x, y) = 4x2 + y2.Let’s compute f(x, y) for some points.
x
y
-3 -2 -1 0 1 2 3
-3 45 40 37 36 37 40 45
-2 25 20 17 16 17 20 25
-1 13 8 5 4 5 8 13
0 9 4 1 0 1 4 9
1 13 8 5 4 5 8 13
2 25 20 17 16 17 20 25
3 45 40 37 36 37 40 45
E. Angel (CU) Calculus II 18 Nov 25 / 30
Graphs of Functions of Two Variables
Example
Sketch the graph of the function h(x, y) = 4x2 + y2.Now let’s plot these points:
E. Angel (CU) Calculus II 18 Nov 25 / 30
Graphs of Functions of Two Variables
Example
Sketch the graph of the function h(x, y) = 4x2 + y2.We can roughly guess how they connect with a wireframe:
E. Angel (CU) Calculus II 18 Nov 25 / 30
Graphs of Functions of Two Variables
Example
Sketch the graph of the function h(x, y) = 4x2 + y2.We can roughly guess how they connect with a wireframe:
E. Angel (CU) Calculus II 18 Nov 25 / 30
Graphs of Functions of Two Variables
Example
Sketch the graph of the function h(x, y) = 4x2 + y2.
E. Angel (CU) Calculus II 18 Nov 26 / 30
Graphs of Functions of Two Variables
Example
Sketch the graph of the function h(x, y) = 4x2 + y2.The other strategy is to take cross sections, that is fix one variable whileletting the other vary.
x = −2 h(−2, y) = 16 + y2
x = −1 h(−1, y) = 4 + y2
x = 0 h(0, y) = y2
x = 1 h(1, y) = 4 + y2
x = 2 h(2, y) = 16 + y2
y = −2 h(x,−2) = 4x2 + 4
y = −1 h(x,−1) = 4x2 + 1
y = 0 h(x, 0) = 4x2
y = 1 h(x, 1) = 4x2 + 1
y = 2 h(x, 2) = 4x2 + 4
E. Angel (CU) Calculus II 18 Nov 26 / 30
Graphs of Functions of Two Variables
Example
Sketch the graph of the function h(x, y) = 4x2 + y2.We obtain families of parabolas which we can graph
E. Angel (CU) Calculus II 18 Nov 26 / 30
Graphs of Functions of Two Variables
Example
Sketch the graph of the function h(x, y) = 4x2 + y2.Filling in we obtain the graph of an elliptic paraboloid:
E. Angel (CU) Calculus II 18 Nov 26 / 30
Graphs of Functions of Two Variables
Example
Find the domain and range and sketch the graph of
g(x, y) =√9− x2 − y2
E. Angel (CU) Calculus II 18 Nov 27 / 30
Graphs of Functions of Two Variables
Example
Find the domain and range and sketch the graph of
g(x, y) =√9− x2 − y2
The domain of g is
D = {(x, y) | 9− x2 − y2 ≥ 0} = {(x, y) |x2 + y2 ≤ 9}
which is the disk with radius 3 and center (0, 0). The range of g is
{z | z =√
9− x2 − y2, (x, y) ∈ D}
Since z is a positive square root, z ≥ 0. Also,√
9− x2 − y2 ≤ 3. So therange is
{z | 0 ≤ z ≤ 3} = [0, 3]
E. Angel (CU) Calculus II 18 Nov 27 / 30
Graphs of Functions of Two Variables
Cross-sections off(x, y) = sinx+ sin y
E. Angel (CU) Calculus II 18 Nov 28 / 30