Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
25,23
Transcribed Image Text:7),
6,
e differen-
2) = 7,
-2. Find
values to
1
he given
=(u. 1)
p+q
p+r
ƏN ƏN ƏN
when u = 2, v = 3, w = 4
ди dv dw
26. u = xe", x=a²ß, y = B²y, t = y²a;
ди ди ди
Әв ду
when a = -1, ß = 2, y = 1
да
p=u+vw, q=v+uw, r = w + uv;
25. N =
27-30 Use Equation 6 to find dy/dx.
27. y cos x = x² + y²
29. tan ¹(x²y) = x + xy²
28. cos(xy) = 1 + sin y
30. e' sin x = x + xy
31-34 Use Equations 7 to find az/ax and az/ay.
31. x² + 2y² + 3z² = 1
33. e² = xyz
32. x² - y² + 2² - 2z = 4
34. yz + x ln y = z²
given by x =
35. The temperature at a point (x, y) is 7(x, y), measured in degrees
Celsius. A bug crawls so that its position after t seconds is
√1+1, y = 2 + t, where x and y are mea-
sured in centimeters. The temperature function satisfies
T.(2, 3) = 4 and T, (2, 3) = 3. How fast is the temperature
rising on the bug's path after 3 seconds?
36. Wheat production W in a given year depends on the average
temperature T and the annual rainfall R. Scientists estimate
that the average temperature is rising at a rate of 0.15°C/year
and rainfall is decreasing at a rate of 0.1 cm/year. They also
estimate that at current
Transcribed Image Text:(0, 0)
(1, 2)
r =
3
6
6
3
16. Suppose f is a differentiable function of x and y, and
g(r, s) = f(2r-s, s² - 4r). Use the table of values in
Exercise 15 to calculate g,(1, 2) and g, (1, 2).
V
2u + v
ƏT ƏT ƏT
др
да ər
22. T =
4
2
17-20 Use a tree diagram to write out the Chain Rule for the given
case. Assume all functions are differentiable.
17. u=f(x, y),
18. w=f(x, y, z),
19. T = F(p, q, r),
= r(x, y, z)
20. R = F(t, u) where t = t (w, x, y, z), u = u(w, x, y, z)
8
5
where x = = x(r, s, t), y = y(r, s, t)
where x = x(u, v), y = y(u, v), z =
where p = p(x, y, z), q = q (x, y, z),
21-26 Use the Chain Rule to find the indicated partial derivatives.
21. z = x² + x²y, x = s + 2t - u, y = stu²;
дz дz дz
as' dt' du
when s = 4, t = 2, u = 1
u = pq√r, v= p√√ar;
when p = 2, q = 1, r = 4
= z(u, v)
23. w = xy + yz + zx, x = r cos 0, y = r sin 0, z = r0;
δω δω
when r = 2, 0 = π/2
dr 20
24. P = √² + v² + w², u = xe', v=ye, w = exy;
JP JP
when x = 0, y = 2
дх' ду
35. The
Celsius. A b
given by x =
sured in cen
T,(2, 3) ==
rising on th
36. Wheat proc
temperature
that the ave
and rainfal
estimate th
and aW/OF
(a) What
deriva
(b) Estim-
dW/d
37. The speed
35 parts p
C = 1
where Ci
temperatu
ocean su
into the c
water tem
graphs. E
20
DA
the speec
diver 20
15-
10
5
10
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