James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

982 Chapter 14 Partial Derivatives
TRUE-FALSE QUIZ
Determine whether the statement is true or false. If it is true, explain
why. If it is false, explain why or give an example that disproves the
statement.
1. f ysa, bd − lim
y l b
f sa, yd 2 f sa, bd
y 2 b
2. There exists a function f with continuous second-order
partial derivatives such that f xsx, yd − x 1 y 2 and
f ysx, yd − x 2 y 2 .
3. f xy − −2 f
−x −y
4. D k f sx, y, zd − f zsx, y, zd
5. If f sx, yd l L as sx, yd l sa, bd along every straight line
through sa, bd, then lim sx, yd l sa, bd f sx, yd − L.
6. If f xsa, bd and f ysa, bd both exist, then f is differentiable
at sa, bd.
7. If f has a local minimum at sa, bd and f is differentiable at
sa, bd, then =f sa, bd − 0.
8. If f is a function, then
lim fsx, yd − fs2, 5d
sx, yd l s2, 5d
9. If f sx, yd − ln y, then =f sx, yd − 1yy.
10. If s2, 1d is a critical point of f and
f xxs2, 1d f yys2, 1d , f f x ys2, 1dg 2
then f has a saddle point at s2, 1d.
11. If f sx, yd − sin x 1 sin y, then 2s2 < D u f sx, yd < s2 .
12. If f sx, yd has two local maxima, then f must have a local
minimum.
EXERCISES
1–2 Find and sketch the domain of the function.
1. f sx, yd − lnsx 1 y 1 1d
2. f sx, yd − s4 2 x 2 2 y 2 1 s1 2 x 2
(b) Is f x s3, 2d positive or negative? Explain.
(c) Which is greater, f y s2, 1d or f y s2, 2d? Explain.
y
5
3–4 Sketch the graph of the function.
4
3. f sx, yd − 1 2 y 2
4. f sx, yd − x 2 1 sy 2 2d 2
5–6 Sketch several level curves of the function.
5. f sx, yd − s4x 2 1 y 2
3
2
1
80
70
60
50
40
30
20
6. f sx, yd − e x 1 y
0
1 2 3 4
x
7. Make a rough sketch of a contour map for the function whose
graph is shown.
z
9–10 Evaluate the limit or show that it does not exist.
9. lim
sx, yd l s1, 1d
2xy
10. lim
x 2 2
1 2y sx, yd l s0, 0d
2xy
x 2 1 2y 2
2
x
8. The contour map of a function f is shown.
(a) Estimate the value of f s3, 2d.
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2 y
11. A metal plate is situated in the xy-plane and occupies the
rectangle 0 < x < 10, 0 < y < 8, where x and y are measured
in meters. The temperature at the point sx, yd in the plate is
Tsx, yd, where T is measured in degrees Celsius. Temperatures
at equally spaced points were measured and recorded in the
table.
(a) Estimate the values of the partial derivatives T xs6, 4d
and T ys6, 4d. What are the units?
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