U 39. y = 1/2+ +==x-x² U
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Ex 3.2
Q39,45&Q53 needed
These are easy questions please solve all mentioned above
Needed to be solved correctly in 30 minutes and get the thumbs up please show me neat and clean work for it by hand solution needed
Transcribed Image Text:In Exercises 21-26, a function h(x) is defined in terms of a differen-
tiable f(x). Find an expression for h'(x).
21. h(x)=f(x²)
23. h(x) = -f(-x)
S(x²)
25. h(x)=1
26. h(x)=√(x²)
27. Sketch the graph of y= 4x/(x + 1)², x>-1.
28. Sketch the graph of y = 2/(1+x²).
d
mpute f(g(x)), where f(x) and g(x) are the following:
f(x)= x³, g(x) = 6x - 1
30. f(x)=√x, g(x)=x² + 1
f(x) = g(x)=1-x²
1 + √² 8(x) = !
x
33. f(x)=x²-x², g(x)=x²-4
34. f(x)= =+ x², g(x) = 1 - x4
35. f(x) = (x³ + 1)², g(x) = x² + 5
36. f(x)= x(x-2), g(x) = x³
31.
32. f(x) =
dy
Compute
dx
in terms of x only.
37. y=u²³/2, u = 4x + 1
38. y Vu+ I, u = 2x²
+=₁u=x-x²
39. y=
u 2
2
40. y=-
u
u²+2u
u+1
using the chain rule in formula (1). State your answer
Compute
43.
22. h(x)=2f(2x + 1)
24. h(x)=f(f(x))
di
-to
u= x(x + 1)
41. y=x²-3x, x= 1² +3, to=0
42. y = (x² - 2x + 4)², x=-
1
1+1'
to=1
v = x + 1₁ x = — -₁ 10 =
x-l'
44. y = √x + 1, x = V1 + 1, to = 0
45. Find the equation of the line tangent to the graph of
y = 2x(x-4)6 at the point (5, 10).
46. Find the equation of the line tangent to the graph of
at the point (1, 1).
y =
V2-x²
47. Find the x-coordinates of all points on the curve
y=(-x² + 4x - 3)³ with a horizontal tangent line.
48. The function f(x)=√x² - 6x + 10 has one relative mini-
mum point for x ≥ 0. Find it.
The length, x, of the edge of a cube is increasing.
(a) Write the chain rule for
dV
dt
volume of the cube.
the time rate of change of the
dV
(b) For what value of x is equal to 12 times the rate of
dt
increase of x?
50. Allometric Equation Many relations in biology are expressed
by power functions, known as allometric equations, of the form
y = kx", where k and a are constants. For example, the weight
of a male hognose snake is approximately 446x³ grams, where
x is its length in meters. If a snake has length .4 meters and is
growing at the rate of .2 meters per year, at what rate is the
snake gaining weight? (Source: Museum of Natural History.)
51. Suppose that P, y, and t are variables, where P is a function
of y and y is a function of 1.
(a) Write the derivative symbols for the following quantities:
(i) the rate of change of y with respect to t; (ii) the rate of
change of P with respect to y; (iii) the rate of change of P
with respect to 1. Select your answers from the following:
dP dy dy dP dt
dy
dP'
dp di dt
dP
dt
(b) Write the chain rule for
3.2 The Chain Rule 207
dy dx do dx do
dx' dy' dx' do' dy'
(b) Write the chain rule for
52. Suppose that Q, x, and y are variables, where Q is a function
of x and x is a function of y. (Read this carefully.)
(a) Write the derivative symbols for the following quantities:
(i) the rate of change of x with respect to y; (ii) the rate of
change of Q with respect to y; (iii) the rate of change of Q
with respect to x. Select your answers from the following:
do
dy
and
(c) How fast
1 = 82
dt
dy
and
53. Marginal Profit and Time Rate of Change When a company
produces and sells x thousand units per week, its total weekly
profit is P thousand dollars, where
P =
do
200x
100+x²
The production level at t weeks from the present is x = 4 + 2t.
dP
dx
dC
(a) Find the marginal cost, dx
(a) Find the marginal profit,
dP
(b) Find the time rate of change of profit, t
respect to time) are profits changing when
54. Marginal Cost and Time Rate of Change The cost of manufac-
turing x cases of cereal is C dollars, where C = 3x + 4√x + 2.
Weekly production at t weeks from the present is estimated to
be x = 6200 + 100r cases.
dC
(b) Find the time rate of change of cost, dt
(c) How fast (with respect to time) are costs rising when t = 2?
55. A Model for Carbon Monoxide Levels Ecologists estimate
that, when the population of a certain city is x thousand
persons, the average level L of carbon monoxide in the
air above the city will be L ppm (parts per million), where
L = 10 + 4x + .0001x². The population of the city is
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