Calculus, Single Variable: Early Transcendentals (3rd Edition)
3rd Edition
ISBN: 9780134766850
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Chapter 7.3, Problem 78E
(a)
To determine
To evaluate: The value of
(b)
To determine
To evaluate: The value of
(c)
To determine
To evaluate: The value of
(d)
To determine
To evaluate: The value of
(e)
To determine
To evaluate: The value of
(f)
To determine
To evaluate: The value of
(g)
To determine
To evaluate: The value of
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Evaluate the integral using integration by parts.
Stan (13y)dy
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Consider the sequences of functions f₁: [-π, π] → R,
sin(n²x)
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f pointwise as
(i) Find a function ƒ : [-T,π] → R such that fn
n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞.
[20 Marks]
(ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]?
Justify your answer.
[10 Marks]
1. (i) Give the definition of a metric on a set X.
[5 Marks]
(ii) Let X = {a, b, c} and let a function d : XxX → [0, ∞) be defined
as d(a, a) = d(b,b) = d(c, c) 0, d(a, c) = d(c, a) 1, d(a, b) = d(b, a) = 4,
d(b, c) = d(c,b) = 2. Decide whether d is a metric on X. Justify your answer.
=
(iii) Consider a metric space (R, d.), where
=
[10 Marks]
0
if x = y,
d* (x, y)
5
if xy.
In the metric space (R, d*), describe:
(a) open ball B2(0) of radius 2 centred at 0;
(b) closed ball B5(0) of radius 5 centred at 0;
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Chapter 7 Solutions
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Ch. 7.1 - What is the domain of ln |x|?Ch. 7.1 - Quick Check 2 Simplify e ln 2x, ln (e2x), e2 ln x,...Ch. 7.1 - Prob. 3QCCh. 7.1 - Prob. 4QCCh. 7.1 - Prob. 1ECh. 7.1 - Prob. 2ECh. 7.1 - Evaluate 4xdx.Ch. 7.1 - Prob. 4ECh. 7.1 - Express 3x, x, and xsin x using the base e.Ch. 7.1 - Prob. 6E
Ch. 7.1 - Derivatives Evaluate the following derivatives...Ch. 7.1 - Prob. 8ECh. 7.1 - Derivatives with ln x Evaluate the following...Ch. 7.1 - Derivatives with ln x Evaluate the following...Ch. 7.1 - Derivatives with ln x Evaluate the following...Ch. 7.1 - Prob. 12ECh. 7.1 - Derivatives Evaluate the derivatives of the...Ch. 7.1 - Prob. 14ECh. 7.1 - Prob. 15ECh. 7.1 - Derivatives Evaluate the derivatives of the...Ch. 7.1 - Derivatives Evaluate the derivatives of the...Ch. 7.1 - Derivatives Evaluate the derivatives of the...Ch. 7.1 - Derivatives Evaluate the derivatives of the...Ch. 7.1 - Derivatives Evaluate the derivatives of the...Ch. 7.1 - Miscellaneous derivatives Compute the following...Ch. 7.1 - Miscellaneous derivatives Compute the following...Ch. 7.1 - Miscellaneous derivatives Compute the following...Ch. 7.1 - Miscellaneous derivatives Compute the following...Ch. 7.1 - Miscellaneous derivatives Compute the following...Ch. 7.1 - Prob. 26ECh. 7.1 - Miscellaneous derivatives Compute the following...Ch. 7.1 - Prob. 28ECh. 7.1 - Integrals with ln x Evaluate the following...Ch. 7.1 - Integrals Evaluate the following integrals....Ch. 7.1 - Integrals with ln x Evaluate the following...Ch. 7.1 - Prob. 32ECh. 7.1 - Integrals with ln x Evaluate the following...Ch. 7.1 - Prob. 34ECh. 7.1 - Integrals with ln x Evaluate the following...Ch. 7.1 - Prob. 36ECh. 7.1 - Integrals with ex Evaluate the following...Ch. 7.1 - Prob. 38ECh. 7.1 - Prob. 39ECh. 7.1 - Integrals with ex Evaluate the following...Ch. 7.1 - Integrals with ex Evaluate the following...Ch. 7.1 - Prob. 42ECh. 7.1 - Integrals with general bases Evaluate the...Ch. 7.1 - Integrals with general bases Evaluate the...Ch. 7.1 - Integrals with general bases Evaluate the...Ch. 7.1 - Integrals with general bases Evaluate the...Ch. 7.1 - Integrals with general bases Evaluate the...Ch. 7.1 - Prob. 48ECh. 7.1 - Integrals Evaluate the following integrals....Ch. 7.1 - Miscellaneous integrals Evaluate the following...Ch. 7.1 - Prob. 51ECh. 7.1 - Prob. 52ECh. 7.1 - Miscellaneous integrals Evaluate the following...Ch. 7.1 - Prob. 54ECh. 7.1 - Miscellaneous integrals Evaluate the following...Ch. 7.1 - Prob. 56ECh. 7.1 - Miscellaneous integrals Evaluate the following...Ch. 7.1 - Miscellaneous integrals Evaluate the following...Ch. 7.1 - Prob. 59ECh. 7.1 - Integrals Evaluate the following integrals....Ch. 7.1 - Integrals Evaluate the following integrals....Ch. 7.1 - Miscellaneous integrals Evaluate the following...Ch. 7.1 - Prob. 63ECh. 7.1 - Calculator limits Use a calculator to make a table...Ch. 7.1 - Prob. 65ECh. 7.1 - Calculator limits Use a calculator to make a table...Ch. 7.1 - Prob. 67ECh. 7.1 - Logarithm properties Use the integral definition...Ch. 7.1 - Prob. 69ECh. 7.1 - Prob. 70ECh. 7.1 - Prob. 71ECh. 7.1 - Derivative of ln |x| Differentiate ln x for x 0...Ch. 7.1 - Prob. 73ECh. 7.1 - ln x is unbounded Use the following argument to...Ch. 7.1 - Prob. 75ECh. 7.1 - Alternative proof of product property Assume that...Ch. 7.1 - Harmonic sum In Chapter 10, we will encounter the...Ch. 7.1 - Probability as an integral Two points P and Q are...Ch. 7.2 - Population A increases at a constant rate of...Ch. 7.2 - Prob. 2QCCh. 7.2 - Assume y() 100e0.005, 3y (exactly) what...Ch. 7.2 - If a quantity decreases by a factor of 8 every 30...Ch. 7.2 - In terms of relative growth rate, what is the...Ch. 7.2 - Prob. 2ECh. 7.2 - Explain the meaning of doubling time.Ch. 7.2 - Explain the meaning of half-life.Ch. 7.2 - Prob. 5ECh. 7.2 - Prob. 6ECh. 7.2 - Suppose a quantity described by the function y(t)...Ch. 7.2 - Suppose a quantity is described by the function...Ch. 7.2 - Give two examples of processes that are modeled by...Ch. 7.2 - Give two examples of processes that are modeled by...Ch. 7.2 - Prob. 11ECh. 7.2 - Prob. 12ECh. 7.2 - Absolute and relative growth rates Two functions f...Ch. 7.2 - Absolute and relative growth rates Two functions f...Ch. 7.2 - Designing exponential growth functions Complete...Ch. 7.2 - Designing exponential growth functions Complete...Ch. 7.2 - Designing exponential growth functions Complete...Ch. 7.2 - Prob. 18ECh. 7.2 - Designing exponential growth functions Complete...Ch. 7.2 - Designing exponential growth functions Complete...Ch. 7.2 - Determining APY Suppose 1000 is deposited in a...Ch. 7.2 - Tortoise growth In a study conducted at University...Ch. 7.2 - Projection sensitivity According to the 2014...Ch. 7.2 - Prob. 24ECh. 7.2 - Population of Texas Texas was the third fastest...Ch. 7.2 - Prob. 26ECh. 7.2 - Designing exponential decay functions Devise an...Ch. 7.2 - Designing exponential decay functions Devise an...Ch. 7.2 - Designing exponential decay functions Devise an...Ch. 7.2 - Designing exponential decay functions Devise an...Ch. 7.2 - Population of West Virginia The population of West...Ch. 7.2 - Prob. 32ECh. 7.2 - Atmospheric pressure The pressure of Earths...Ch. 7.2 - Prob. 34ECh. 7.2 - Uranium dating Uranium-238 (U-238) has a half-life...Ch. 7.2 - Prob. 36ECh. 7.2 - Caffeine After an individual drinks a beverage...Ch. 7.2 - Caffeine After an individual drinks a beverage...Ch. 7.2 - LED lighting LED (light-emitting diode) bulbs are...Ch. 7.2 - Prob. 40ECh. 7.2 - Tumor growth Suppose the cells of a tumor are...Ch. 7.2 - Prob. 42ECh. 7.2 - Explain why or why not Determine whether the...Ch. 7.2 - Prob. 44ECh. 7.2 - Prob. 45ECh. 7.2 - Overtaking City A has a current population of...Ch. 7.2 - Prob. 47ECh. 7.2 - Prob. 48ECh. 7.2 - Prob. 49ECh. 7.2 - Prob. 50ECh. 7.2 - Prob. 51ECh. 7.2 - Prob. 52ECh. 7.2 - Prob. 53ECh. 7.2 - Prob. 54ECh. 7.2 - Constant doubling time Prove that the doubling...Ch. 7.3 - Use the definition of the hyperbolic sine to show...Ch. 7.3 - Prob. 2QCCh. 7.3 - Prob. 3QCCh. 7.3 - Prob. 4QCCh. 7.3 - Prob. 5QCCh. 7.3 - Prob. 6QCCh. 7.3 - Explain why longer waves travel faster than...Ch. 7.3 - Prob. 1ECh. 7.3 - Prob. 2ECh. 7.3 - Prob. 3ECh. 7.3 - Prob. 4ECh. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - Prob. 7ECh. 7.3 - Prob. 8ECh. 7.3 - Prob. 9ECh. 7.3 - Prob. 10ECh. 7.3 - Prob. 11ECh. 7.3 - Prob. 12ECh. 7.3 - Verifying identities Verify each identity using...Ch. 7.3 - Verifying identities Verify each identity using...Ch. 7.3 - Prob. 15ECh. 7.3 - Prob. 16ECh. 7.3 - Verifying identities Use the given identity to...Ch. 7.3 - Prob. 18ECh. 7.3 - Prob. 19ECh. 7.3 - Prob. 20ECh. 7.3 - Prob. 21ECh. 7.3 - Prob. 22ECh. 7.3 - Prob. 23ECh. 7.3 - Prob. 24ECh. 7.3 - Prob. 25ECh. 7.3 - Prob. 26ECh. 7.3 - Prob. 27ECh. 7.3 - Prob. 28ECh. 7.3 - Prob. 29ECh. 7.3 - Prob. 30ECh. 7.3 - Prob. 31ECh. 7.3 - Prob. 32ECh. 7.3 - Prob. 33ECh. 7.3 - Prob. 34ECh. 7.3 - Prob. 35ECh. 7.3 - Prob. 36ECh. 7.3 - Prob. 37ECh. 7.3 - Prob. 38ECh. 7.3 - Indefinite integrals Determine each indefinite...Ch. 7.3 - Indefinite integrals Determine each indefinite...Ch. 7.3 - Prob. 41ECh. 7.3 - Prob. 42ECh. 7.3 - Definite integrals Evaluate each definite...Ch. 7.3 - Prob. 44ECh. 7.3 - Prob. 45ECh. 7.3 - Prob. 46ECh. 7.3 - Prob. 47ECh. 7.3 - Prob. 48ECh. 7.3 - Prob. 49ECh. 7.3 - Prob. 50ECh. 7.3 - Prob. 51ECh. 7.3 - Prob. 52ECh. 7.3 - Prob. 53ECh. 7.3 - Prob. 54ECh. 7.3 - Prob. 55ECh. 7.3 - Prob. 56ECh. 7.3 - Prob. 57ECh. 7.3 - Prob. 58ECh. 7.3 - Visual approximation a. Use a graphing utility to...Ch. 7.3 - Prob. 60ECh. 7.3 - Prob. 61ECh. 7.3 - Prob. 62ECh. 7.3 - Prob. 63ECh. 7.3 - Prob. 64ECh. 7.3 - Prob. 65ECh. 7.3 - Prob. 66ECh. 7.3 - Prob. 67ECh. 7.3 - Prob. 68ECh. 7.3 - Prob. 69ECh. 7.3 - Prob. 70ECh. 7.3 - Prob. 71ECh. 7.3 - Prob. 72ECh. 7.3 - Prob. 73ECh. 7.3 - Wave velocity Use Exercise 73 to do the following...Ch. 7.3 - Prob. 75ECh. 7.3 - Prob. 76ECh. 7.3 - Prob. 77ECh. 7.3 - Prob. 78ECh. 7.3 - Prob. 79ECh. 7.3 - Prob. 80ECh. 7.3 - Prob. 81ECh. 7.3 - Prob. 82ECh. 7.3 - Prob. 83ECh. 7.3 - Prob. 84ECh. 7.3 - Prob. 85ECh. 7.3 - Prob. 86ECh. 7.3 - LHpital loophole Explain why lHpitals Rule fails...Ch. 7.3 - Prob. 88ECh. 7.3 - Prob. 89ECh. 7.3 - Prob. 90ECh. 7.3 - Prob. 91ECh. 7.3 - Prob. 92ECh. 7.3 - Prob. 93ECh. 7.3 - Newtons method Use Newtons method to find all...Ch. 7.3 - Prob. 95ECh. 7.3 - Prob. 96ECh. 7.3 - Prob. 97ECh. 7.3 - Prob. 98ECh. 7.3 - Prob. 99ECh. 7.3 - Prob. 100ECh. 7.3 - Prob. 101ECh. 7.3 - Prob. 102ECh. 7.3 - Prob. 103ECh. 7.3 - Prob. 104ECh. 7.3 - Prob. 105ECh. 7.3 - Prob. 106ECh. 7.3 - Prob. 107ECh. 7.3 - Prob. 108ECh. 7.3 - Prob. 109ECh. 7.3 - Prob. 110ECh. 7.3 - Prob. 111ECh. 7.3 - Prob. 112ECh. 7 - Explain why or why not Determine whether the...Ch. 7 - Integrals Evaluate the following integrals. 56....Ch. 7 - Prob. 3RECh. 7 - Integrals Evaluate the following integrals. 58....Ch. 7 - Prob. 5RECh. 7 - Prob. 6RECh. 7 - Prob. 7RECh. 7 - Integrals Evaluate the following integrals. 62....Ch. 7 - Prob. 9RECh. 7 - Prob. 10RECh. 7 - Prob. 11RECh. 7 - Derivatives Find the derivatives of the following...Ch. 7 - Prob. 13RECh. 7 - Prob. 14RECh. 7 - Prob. 15RECh. 7 - Derivatives Find the derivatives of the following...Ch. 7 - Derivatives Find the derivatives of the following...Ch. 7 - Prob. 18RECh. 7 - Prob. 19RECh. 7 - Population growth The population of a large city...Ch. 7 - Prob. 21RECh. 7 - Prob. 22RECh. 7 - Prob. 23RECh. 7 - Radioactive decay The mass of radioactive material...Ch. 7 - Prob. 25RECh. 7 - Prob. 26RECh. 7 - Prob. 27RECh. 7 - Curve sketching Use the graphing techniques of...Ch. 7 - Prob. 29RECh. 7 - Prob. 30RECh. 7 - Prob. 31RECh. 7 - Limit Evaluate limx(tanhx)x.Ch. 7 - Derivatives of hyperbolic functions Compute the...Ch. 7 - Arc length Find the arc length of the curve y = ln...
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