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92
Chapter 9
Multivariable Calculus
Wewill look at the calculus of functions with several variables.
9.1 Functions of Several Variables
Equation z = f(x;y) is a function of two variables if there is a unique z from each
ordered pair (x;y) whose graph is an example of a surface. Pair (x;y) are independent
variables; z is a dependent variable; set of all (x;y) is domain; set of all z = f(x;y) is
range. These definitions extend naturally to more than two dimensions. Graph
ax+by+cz=d
is a plane if a;b;c are all not 0. Traces take “coordinate axes plane slices” through
surfaces; level curves are ”slices” of planes parallel to coordinate axes” through sur-
faces. There are three types of traces for the z = f(x;y) surface: xy-trace, yx-trace
and xz-trace. Four common equations are
2 2
• paraboloid: z = x +y
• ellipsoid: x2 + y2 + z2 = 1
2 2 2
a b c
2 2
• hyperbolic paraboloid: z = x − y
2 2 2
• hyperboloid of two sheets: −x −y +z = 1
Although an ellipsoid is not a function, since there is more than one z for different
(x;y), it is possible in this case to treat the ellipsoid as a level surface for a function
of three variables,
2 2 2
w(x;y;z) = x + y + z
2 2 2
a b c
where w = 1.
Exercise 9.1 (Functions of Several Variables)
93
94 Chapter 9. Multivariable Calculus (LECTURE NOTES 6)
1. Multivariate function evaluation
(a) f(x;y) = 3x+4y
For x = 3, y = 5, f(x;y) = f(3;5) = 3(3)+4(5) = (i) 28 (ii) 29 (iii) 30
Multivariate function calculations not available on calculator, so awkward to deal with:
Y1 =X, Y2 =X, 2nd QUIT, 3 VARS Y-VARS ENTER Y1(3) + 4 VARS Y-VARS ENTER Y2(5)
OR, easiest to just type 3 × 3 +4 × 5 = 29
Different Notation.
For x = 3, y = 5, z = 3x+4y = (i) 28 (ii) 29 (iii) 30
For x = −3, y = 17, z = 3x +4y = (i) 28 (ii) 29 (iii) 59
For x = −3:2, y = −7:5, z = 3x+4y = (i) −28:3 (ii) −39:6 (iii) −59
2
(b) f(x;y) = 3x +4y
f(3;5) = (i) 38 (ii) 44 (iii) 47
√ 2
(c) f(x;y) = 3x +4y
f(3;5) = (i) 3:86 (ii) 6:86 (iii) 7:32
2
(d) f(x;y;z) = 3x +4y +3z.
f(3;5;−8) = (i) 22 (ii) 23 (iii) 24
2
(e) f(x;y;z) = 3x +lny +3z.
2
f(3;e ;−8) = (i) 2 (ii) 4 (iii) 5
2
(f) f(x;y;z) = 3x (lny)z.
2
f(3;e ;−8) = (i) −245 (ii) −432 (iii) −1296
2
(g) f(a;b;c) = 3a (lnb)c.
2
f(3;e ;−8) = (i) −245 (ii) −432 (iii) −1296
(h) f(u;v;w) = 3.
2
f(3;e ;−8) = (i) 3 (ii) −456 (iii) −1296
x2 x3
(i) f(x ;x ;x ;x ) = 3x + 2.
1 2 3 4 1 3x
4
f(3;2;8;5) = (i) 26:23 (ii) 27:11 (iii) 28:03
x2
(j) f(x ;x ;x ;x ) = 3x +5.
1 2 3 4 1
f(3;2;8;5) = (i) 26 (ii) 29 (iii) 32
Section 1. Functions of Several Variables (LECTURE NOTES 6) 95
2 2
(k) Let f(x;y) = 3x +2y
2 2 2 2
f(x+h;y)−f(x;y) = (3(x+h) +2y )−(3x +2y )
h h
2 2 2 2 2
= (3(x +2xh+h )+2y )−3x −2y
h
2 2 2 2 2
= 3x +6xh+3h +2y −3x −2y =
h
(i) 6x + 2h (ii) 6x + 3h (iii) 6xh + 3h2
2. Social Science Application: Teaching
Ateacher’s rating, f, is given by
a √ 2
f(n;p;a;t) = 3n + tp
where n is number of students, p is teacher preparedness, a is student atten-
dance and t is teacher–student interaction.
So, f(30;5;0:85;5) = (i) 36:23 (ii) 40:05 (iii) 55:99
3. Biology Application: Virus
Avirus’s infection rate, f, is given by
p 2
f(L;p;R;r;v) = R−r
4Lv
where the L is length of incubation period, p is blood pressure, R is radius of
virus, r is time between infections, and v is viscosity.
So, f(10;120;0:001;3;12) = (i) 2:25 (ii) 3:05 (iii) 8:03
4. Linear equations geometrically: planes in three–dimensional space.
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