Since it approaches dierent values from dierent
directions, the limit doesnt exist.
ex ey
10.
lim
.
(x,y)(0,0) x + y + 2
All the functions are continuous and the denominator approaches 2, so the limit can be found by
1
evaluating the function. f (0, 0) =
Then take their derivatives
2f
= y 2 cos xy
x2
Section 2.4 selected answers
2f
2f
=
= sin xy xy cos xy
yx
xy
Math 131 Multivariate Calculus
D Joyce, Spring 2014
2f
= x2 cos xy
y 2
Exercises 1, 2, 911, 20, 28, 29a.
2. Verify the sum rule for derivative
14. F (x, y, z) = eax cos by + eaz sin bx.
F
x
F
y
F
z
Section 2.3 selected answers
Math 131 Multivariate Calculus
D Joyce, Spring 2014
Exercises 17, 1214, 1921, 29, 30, 3436.
For the rst few exercises, compute the partial
derivatives.
= aeax cos by + bea
therefore
Df Dx
1 1
2s 2t
Section 2.5 selected answers
=
y cos xy x cos xy
Math 131 Multivariate Calculus
D Joyce, Spring 2014
=
y cos xy + 2sx cos xy y cos xy + 2tx cos xy
Thus,
Exercises 1, 2, 5, 8, 11, 15, 19, 22, 23.
f
= y cos xy + 2sx cos xy
s
2. If
That is the opposite direction, the negation of u.
c. desires her temperature not to change?
That is a direction orthogonal to u, and that
2
3
would be ,
.
13 13
Section 2.6 selected answers
Math 131 Multivariate Calculus
D Joyce, Spring 2014
15. Igor, t
8. Calculate the velocity, speed and acceleration
of the path
x(t) = 5 cos t i + 3 sin t j.
Section 3.1 answers
Math 131 Multivariate Calculus
D Joyce, Spring 2014
Since the position at time t is
x(t) = (x, y) = (5 cos t, 3 sin t),
Exercises from section
20. Calculate the ow line x(t) of the vector eld
F(x, y) = (x, y) if x(0) = (2, 1).
The ow line, also called a ow path, x(t) has to
have its derivatives in the vector eld, that is
Section 3.3 selected answers
Math 131 Multivariate Calculus
D Joyce, Spring
You can use integral tables or techniques of integration to nish this exercise. One way to nd
this integral is by the substitution 2t = tan .
Then 1 + 4t2 = 1 + tan2 = sec , and 2dt =
sec2 d. The integral becomes
Section 3.2 selected answers
Math 131 Mult
Denition 2. We dene the curl of a vector eld
in space, F : R3 R3 , as
=
F
, ,
x y z
i
j k
=
x
curl F =
Gradient, divergence, and curl
Math 131 Multivariate Calculus
D Joyce, Spring 2014
The del operator . First, well start by
stracting the gradient
to a