Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Textbook Question

Chapter 1.5, Problem 24E

Explain, using Theorems 4, 5, 6, and 8, why the function is continuous at every number in its domain. State the domain.

24. F(x) = sin(cos(sin x))

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Chapter 1 Solutions

Essential Calculus: Early Transcendentals

Ch. 1.1 - 1. If f(x)=x+2x and g(u)=u+2u, is it true that f =...Ch. 1.1 - If f(x)=x2xx1andg(x)=x is it true that f = g?Ch. 1.1 - The graph of a function f is given. (a) State the...Ch. 1.1 - The graphs of f and g are given. (a) State the...Ch. 1.1 - Prob. 5E Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Prob. 9E Ch. 1.1 - The graph shows the height of the water in a...

Ch. 1.1 - Prob. 11E Ch. 1.1 - Sketch a rough graph of the number of hours of...Ch. 1.1 - Prob. 13E Ch. 1.1 - Sketch a rough graph of the market value of a new...Ch. 1.1 - Prob. 15E Ch. 1.1 - You place a frozen pie in an oven and bake it for...Ch. 1.1 - A homeowner mows the lawn every Wednesday...Ch. 1.1 - An airplane takes off from an airport and lands an...Ch. 1.1 - If f(x) = 3x2 x + 2, find f(2), f(2), f(a), f(a),...Ch. 1.1 - A spherical balloon with radius r inches has...Ch. 1.1 - Evaluate the difference quotient for the given...Ch. 1.1 - Evaluate the difference quotient for the given...Ch. 1.1 - Evaluate the difference quotient for the given...Ch. 1.1 - Prob. 24E Ch. 1.1 - Find the domain of the function. 31. f(x)=x+4x29 Ch. 1.1 - Prob. 26E Ch. 1.1 - Prob. 28E Ch. 1.1 - Prob. 29E Ch. 1.1 - Find the domain of the function. 37. F(p)=2p Ch. 1.1 - Find the domain and range and sketch the graph of...Ch. 1.1 - Prob. 31E Ch. 1.1 - Prob. 34E Ch. 1.1 - Find the domain and sketch the graph of the...Ch. 1.1 - Prob. 33E Ch. 1.1 - Prob. 35E Ch. 1.1 - Prob. 36E Ch. 1.1 - Find the domain and sketch the graph of the...Ch. 1.1 - Prob. 38E Ch. 1.1 - Prob. 39E Ch. 1.1 - Prob. 40E Ch. 1.1 - Prob. 41E Ch. 1.1 - Find the domain and sketch the graph of the...Ch. 1.1 - Find an expression for the function whose graph is...Ch. 1.1 - Prob. 44E Ch. 1.1 - Prob. 45E Ch. 1.1 - Prob. 46E Ch. 1.1 - Prob. 47E Ch. 1.1 - Find a formula for the described function and...Ch. 1.1 - Prob. 49E Ch. 1.1 - Find a formula for the described function and...Ch. 1.1 - Find a formula for the described function and...Ch. 1.1 - A cell phone plan has a basic charge of 35 a...Ch. 1.1 - In a certain country, income tax is assessed as...Ch. 1.1 - The functions in Example 6 and Exercises 52 and...Ch. 1.1 - Graphs of f and g are shown. Decide whether each...Ch. 1.1 - Graphs of f and g are shown. Decide whether each...Ch. 1.1 - (a) If the point (5, 3) is on the graph of an even...Ch. 1.1 - A function f has domain [5, 5] and a portion of...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - Determine whether f is even, odd, or neither. If...Ch. 1.1 - If f and g are both even functions, is f + g even?...Ch. 1.1 - If f and g are both even functions, is the product...Ch. 1.2 - (a) Find an equation for the family of linear...Ch. 1.2 - What do all members of the family of linear...Ch. 1.2 - What do all members of the family of linear...Ch. 1.2 - Find expressions for the quadratic functions whose...Ch. 1.2 - Prob. 5E Ch. 1.2 - Prob. 6E Ch. 1.2 - Prob. 7E Ch. 1.2 - Prob. 8E Ch. 1.2 - Prob. 9E Ch. 1.2 - Prob. 10E Ch. 1.2 - Prob. 11E Ch. 1.2 - Prob. 12E Ch. 1.2 - Prob. 13E Ch. 1.2 - The monthly cost of driving a car depends on the...Ch. 1.2 - Prob. 15E Ch. 1.2 - Prob. 16E Ch. 1.2 - Prob. 17E Ch. 1.2 - Explain how each graph is obtained from the graph...Ch. 1.2 - Prob. 19E Ch. 1.2 - Prob. 20E Ch. 1.2 - Prob. 21E Ch. 1.2 - Prob. 22E Ch. 1.2 - Prob. 23E Ch. 1.2 - Prob. 24E Ch. 1.2 - Prob. 25E Ch. 1.2 - Prob. 30E Ch. 1.2 - Prob. 27E Ch. 1.2 - Prob. 32E Ch. 1.2 - Prob. 26E Ch. 1.2 - Prob. 28E Ch. 1.2 - Prob. 31E Ch. 1.2 - Prob. 29E Ch. 1.2 - Prob. 34E Ch. 1.2 - Prob. 33E Ch. 1.2 - Prob. 36E Ch. 1.2 - Prob. 35E Ch. 1.2 - Prob. 37E Ch. 1.2 - Prob. 38E Ch. 1.2 - Prob. 40E Ch. 1.2 - Prob. 39E Ch. 1.2 - Prob. 41E Ch. 1.2 - Prob. 42E Ch. 1.2 - Prob. 43E Ch. 1.2 - Prob. 44E Ch. 1.2 - Prob. 45E Ch. 1.2 - Prob. 46E Ch. 1.2 - Prob. 47E Ch. 1.2 - Prob. 48E Ch. 1.2 - Prob. 49E Ch. 1.2 - Express the function in the form f g. 48....Ch. 1.2 - Prob. 51E Ch. 1.2 - Prob. 52E Ch. 1.2 - Prob. 53E Ch. 1.2 - Prob. 54E Ch. 1.2 - Prob. 55E Ch. 1.2 - Prob. 57E Ch. 1.2 - Prob. 56E Ch. 1.2 - Prob. 58E Ch. 1.2 - Prob. 59E Ch. 1.2 - Prob. 60E Ch. 1.2 - Prob. 61E Ch. 1.2 - Prob. 62E Ch. 1.2 - Prob. 63E Ch. 1.2 - Prob. 64E Ch. 1.2 - Prob. 65E Ch. 1.2 - Prob. 66E Ch. 1.3 - If a ball is thrown into the air with a velocity...Ch. 1.3 - If a rock is thrown upward on the planet Mars with...Ch. 1.3 - Use the given graph of f to state the value of...Ch. 1.3 - For the function f whose graph is given, state the...Ch. 1.3 - Prob. 5E Ch. 1.3 - Prob. 6E Ch. 1.3 - Prob. 7E Ch. 1.3 - Prob. 8E Ch. 1.3 - Prob. 9E Ch. 1.3 - Sketch the graph of an example of a function f...Ch. 1.3 - Prob. 11E Ch. 1.3 - Guess the value of the limit (if it exists) by...Ch. 1.3 - Prob. 13E Ch. 1.3 - Guess the value of the limit (if it exists) by...Ch. 1.3 - Prob. 15E Ch. 1.3 - Prob. 16E Ch. 1.3 - Prob. 17E Ch. 1.3 - Prob. 18E Ch. 1.3 - Prob. 19E Ch. 1.3 - Prob. 20E Ch. 1.3 - Prob. 21E Ch. 1.3 - Prob. 22E Ch. 1.3 - Prob. 23E Ch. 1.3 - Use the given graph of f(x) =x2 to find a number ...Ch. 1.3 - Prob. 25E Ch. 1.3 - Use a graph to find a number such that if...Ch. 1.3 - Prob. 27E Ch. 1.3 - Prob. 28E Ch. 1.3 - Prob. 29E Ch. 1.3 - Prove the statement using the , definition of a...Ch. 1.3 - Prob. 31E Ch. 1.3 - Prove the statement using the , definition of a...Ch. 1.3 - Prob. 33E Ch. 1.3 - Prob. 34E Ch. 1.3 - Prob. 35E Ch. 1.3 - Prob. 36E Ch. 1.3 - Prob. 37E Ch. 1.3 - Prob. 38E Ch. 1.3 - Prob. 39E Ch. 1.3 - Prove the statement using the , definition of a...Ch. 1.3 - Prob. 41E Ch. 1.3 - Prob. 42E Ch. 1.3 - Prob. 43E Ch. 1.3 - Prob. 44E Ch. 1.3 - Prob. 46E Ch. 1.4 - Given that limx2f(x)=4limx2g(x)=2limx2h(x)=0 find...Ch. 1.4 - The graphs of f and g are given. Use them to...Ch. 1.4 - Evaluate the limit and justify each step by...Ch. 1.4 - Prob. 4E Ch. 1.4 - Prob. 5E Ch. 1.4 - Prob. 6E Ch. 1.4 - Evaluate the limit and justify each step by...Ch. 1.4 - Prob. 8E Ch. 1.4 - Prob. 9E Ch. 1.4 - (a) What is wrong with the following equation?...Ch. 1.4 - Prob. 11E Ch. 1.4 - Evaluate the limit, if it exists. limx4x24xx23x4 Ch. 1.4 - Evaluate the limit, if it exists. limx5x25x+6x5 Ch. 1.4 - Evaluate the limit, if it exists. limx1x24xx23x4 Ch. 1.4 - Prob. 15E Ch. 1.4 - Prob. 16E Ch. 1.4 - Prob. 17E Ch. 1.4 - Evaluate the limit, if it exists. limh0(2+h)38h Ch. 1.4 - Prob. 19E Ch. 1.4 - Prob. 20E Ch. 1.4 - Evaluate the limit, if it exists. limh09+h3h Ch. 1.4 - Evaluate the limit, if it exists. limu24u+13u2 Ch. 1.4 - Prob. 25E Ch. 1.4 - Evaluate the limit, if it exists. limt0(1t1t2+t)Ch. 1.4 - Prob. 23E Ch. 1.4 - Evaluate the limit, if it exists. limx4x2+95x+4 Ch. 1.4 - Prob. 27E Ch. 1.4 - Evaluate the limit, if it exists. limh01(xh)21x2h Ch. 1.4 - Prob. 29E Ch. 1.4 - Prob. 30E Ch. 1.4 - Prob. 31E Ch. 1.4 - Use the Squeeze Theorem to show that...Ch. 1.4 - Prob. 33E Ch. 1.4 - If 2x g(x) x4 x2 + 2 for all x, evaluate...Ch. 1.4 - Prove that limx0x4cos2x=0.Ch. 1.4 - Prove that limx0+x[1+sin2(2/x)]=0.Ch. 1.4 - Prob. 37E Ch. 1.4 - Find the limit, if it exists. If the limit does...Ch. 1.4 - Prob. 39E Ch. 1.4 - Find the limit, if it exists. If the limit does...Ch. 1.4 - Find the limit, if it exists. If the limit does...Ch. 1.4 - Prob. 42E Ch. 1.4 - Let g(x)=x2+x6x2 (a) Find (i) limx2+g(x) (ii)...Ch. 1.4 - Prob. 44E Ch. 1.4 - Prob. 45E Ch. 1.4 - Prob. 46E Ch. 1.4 - Prob. 47E Ch. 1.4 - Prob. 48E Ch. 1.4 - Prob. 49E Ch. 1.4 - Find the limit. limx0sin4xsin6x Ch. 1.4 - Find the limit. limt0tan6tsin2t Ch. 1.4 - Prob. 52E Ch. 1.4 - Find the limit. limx0sin3x5x34x Ch. 1.4 - Prob. 54E Ch. 1.4 - Prob. 55E Ch. 1.4 - Find the limit. limx0sin(x2)x Ch. 1.4 - If p is a polynomial, Show that limxa p(x) = p(a)Ch. 1.4 - If r is a rational function. use Exercise 57 to...Ch. 1.4 - If limx1f(x)8x1=10, find limx1f(x).Ch. 1.4 - To prove that sine has the Direct Substitution...Ch. 1.4 - Prove that cosine has the Direct Substitution...Ch. 1.4 - Show by means of an example that limxa[f(x)+g(x)]...Ch. 1.4 - Prob. 64E Ch. 1.4 - Prove that if limxag(x)=0 and limxaf(x) exists and...Ch. 1.4 - Prob. 65E Ch. 1.4 - Prob. 66E Ch. 1.5 - Write an equation that expresses the fact that a...Ch. 1.5 - If f is continuous on ( , ).what can you say about...Ch. 1.5 - (a) From the graph of f , state the numbers at...Ch. 1.5 - From the graph of g, state the intervals on which...Ch. 1.5 - Sketch the graph of a function f that is...Ch. 1.5 - Sketch the graph of a function f that is...Ch. 1.5 - Sketch the graph of a function f that is...Ch. 1.5 - Sketch the graph of a function f that is...Ch. 1.5 - The toll T charged for driving on a certain...Ch. 1.5 - Explain why each function is continuous or...Ch. 1.5 - Use the definition of continuity and the...Ch. 1.5 - Use the definition of continuity and the...Ch. 1.5 - Use the definition of continuity and the...Ch. 1.5 - Explain why the function is discontinuous at the...Ch. 1.5 - Explain why the function is discontinuous at the...Ch. 1.5 - Explain why the function is discontinuous at the...Ch. 1.5 - Explain why the function is discontinuous at the...Ch. 1.5 - Explain, using Theorems 4, 5, 6, and 8, why the...Ch. 1.5 - Explain, using Theorems 4, 5, 6, and 8, why the...Ch. 1.5 - Explain, using Theorems 4, 5, 6, and 8, why the...Ch. 1.5 - Explain, using Theorems 4, 5, 6, and 8, why the...Ch. 1.5 - Explain, using Theorems 4, 5, 6, and 8, why the...Ch. 1.5 - Explain, using Theorems 4, 5, 6, and 8, why the...Ch. 1.5 - Locate the discontinuities of the function and...Ch. 1.5 - Locate the discontinuities of the function and...Ch. 1.5 - Prob. 27E Ch. 1.5 - Use continuity to evaluate the limit....Ch. 1.5 - Show that f is continuous on (, )....Ch. 1.5 - Show that f is continuous on ( , )....Ch. 1.5 - Find the numbers at which the function...Ch. 1.5 - The gravitational force exerted by the planet...Ch. 1.5 - For what value of the constant c is the function f...Ch. 1.5 - Find the values of a and h that make f continuous...Ch. 1.5 - Suppose f and g are continuous functions such that...Ch. 1.5 - Which of the following functions .f has a...Ch. 1.5 - Suppose that a function f is continuous on [0, 1]...Ch. 1.5 - If f(x) = x2 + 10 sin x, show that there is a...Ch. 1.5 - Suppose f is continuous on [1, 5] and the only...Ch. 1.5 - Use the Intermediate Value Theorem to show that...Ch. 1.5 - Use the Intermediate Value Theorem to show that...Ch. 1.5 - Use the Intermediate Value Theorem to show that...Ch. 1.5 - Use the Intermediate Value Theorem to show that...Ch. 1.5 - Prob. 43E Ch. 1.5 - Prob. 44E Ch. 1.5 - Prob. 45E Ch. 1.5 - (a) Prove that the equation has at least one real...Ch. 1.5 - Is there a number that is exactly 1 more than its...Ch. 1.5 - Prob. 48E Ch. 1.5 - Prob. 49E Ch. 1.5 - Prob. 50E Ch. 1.5 - A Tibetan monk leaves the monastery at 7:00 AM and...Ch. 1.6 - How close to 3 do we have to take x so that...Ch. 1.6 - Prob. 52E Ch. 1.6 - Prob. 53E Ch. 1.6 - For the function f whose graph is given, state the...Ch. 1.6 - For the function g whose graph is given, state the...Ch. 1.6 - Sketch the graph of an example of a function f...Ch. 1.6 - Sketch the graph of an example of a function f...Ch. 1.6 - Sketch the graph of an example of a function f...Ch. 1.6 - Sketch the graph of an example of a function f...Ch. 1.6 - Sketch the graph of an example of a function f...Ch. 1.6 - Sketch the graph of an example of a function f...Ch. 1.6 - Guess the value of the limit limxx22x by...Ch. 1.6 - Determine limx11x31 and limx1+1x31 (a) by...Ch. 1.6 - Use a graph to estimate all the vertical and...Ch. 1.6 - (a) Use a graph of f(x)=(12x)x to estimate the...Ch. 1.6 - Find the limit or show that it does not exist....Ch. 1.6 - Find the limit or show that it does not exist....Ch. 1.6 - Find the limit. limx12x(x1)2 Ch. 1.6 - Find the limit. limx2x22xx24x+4 Ch. 1.6 - Find the limit or show that it does not exist....Ch. 1.6 - Find the limit or show that it does not exist....Ch. 1.6 - Find the limit or show that it does not exist....Ch. 1.6 - Prob. 24E Ch. 1.6 - Prob. 13E Ch. 1.6 - Find the limit. limx3x+2x+3 Ch. 1.6 - Prob. 25E Ch. 1.6 - Prob. 26E Ch. 1.6 - Find the limit or show that it does not exist....Ch. 1.6 - Prob. 31E Ch. 1.6 - Prob. 32E Ch. 1.6 - Prob. 30E Ch. 1.6 - Prob. 17E Ch. 1.6 - Prob. 33E Ch. 1.6 - Prob. 28E Ch. 1.6 - Prob. 16E Ch. 1.6 - Prob. 29E Ch. 1.6 - Prob. 37E Ch. 1.6 - Prob. 38E Ch. 1.6 - Prob. 36E Ch. 1.6 - Find the horizontal and vertical asymptotes of...Ch. 1.6 - Prob. 39E Ch. 1.6 - Prob. 34E Ch. 1.6 - Let P and Q be polynomials. Find limxP(x)Q(x) if...Ch. 1.6 - Prob. 46E Ch. 1.6 - Prob. 41E Ch. 1.6 - Prob. 40E Ch. 1.6 - Evaluate the limits. (a) limxxsin1x (b) limxxsin1x Ch. 1.6 - In the theory of relativity, the mass of a...Ch. 1.6 - (a) Show that limx4x25x2x2+1=2. (b) By graphing...Ch. 1.6 - A function f is a ratio of quadratic functions and...Ch. 1.6 - Prob. 44E Ch. 1.6 - Prob. 47E Ch. 1.6 - Prob. 49E Ch. 1.6 - Prob. 55E Ch. 1.6 - Prob. 54E Ch. 1.6 - Prob. 56E Ch. 1.6 - Prob. 57E Ch. 1.6 - Prob. 58E Ch. 1.6 - Prove that limxf(x)=limt0+f(1/t) and...Ch. 1 - Prob. 1RCC Ch. 1 - Prob. 2RCC Ch. 1 - Prob. 3RCC Ch. 1 - Prob. 4RCC Ch. 1 - Prob. 5RCC Ch. 1 - Prob. 6RCC Ch. 1 - Prob. 7RCC Ch. 1 - Prob. 8RCC Ch. 1 - Prob. 9RCC Ch. 1 - Prob. 10RCC Ch. 1 - Prob. 11RCC Ch. 1 - Prob. 1RQ Ch. 1 - Prob. 2RQ Ch. 1 - Prob. 3RQ Ch. 1 - Prob. 4RQ Ch. 1 - Prob. 5RQ Ch. 1 - Prob. 6RQ Ch. 1 - Prob. 19RQ Ch. 1 - Prob. 1RE Ch. 1 - Prob. 2RE Ch. 1 - Prob. 3RE Ch. 1 - Prob. 4RE Ch. 1 - Prob. 5RE Ch. 1 - Prob. 6RE Ch. 1 - Prob. 7RE Ch. 1 - Prob. 8RE Ch. 1 - Prob. 9RE Ch. 1 - Prob. 10RE Ch. 1 - Prob. 11RE Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Prob. 13RE Ch. 1 - Prob. 14RE Ch. 1 - Prob. 15RE Ch. 1 - Prob. 16RE Ch. 1 - Prob. 17RE Ch. 1 - Prob. 18RE Ch. 1 - Prob. 12RCC Ch. 1 - Prob. 13RCC Ch. 1 - Prob. 14RCC Ch. 1 - Prob. 15RCC Ch. 1 - Prob. 18RCC Ch. 1 - Prob. 16RCC Ch. 1 - Prob. 17RCC Ch. 1 - Prob. 7RQ Ch. 1 - Prob. 8RQ Ch. 1 - Prob. 9RQ Ch. 1 - Prob. 10RQ Ch. 1 - Prob. 11RQ Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Prob. 13RQ Ch. 1 - Prob. 14RQ Ch. 1 - Prob. 15RQ Ch. 1 - Prob. 16RQ Ch. 1 - Prob. 17RQ Ch. 1 - If f and g are polynomials and g(2) = 0, then the...Ch. 1 - Prob. 20RQ Ch. 1 - Prob. 21RQ Ch. 1 - Prob. 22RQ Ch. 1 - Prob. 23RQ Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Prob. 25RQ Ch. 1 - Prob. 26RQ Ch. 1 - Prob. 19RE Ch. 1 - Prob. 20RE Ch. 1 - Prob. 21RE Ch. 1 - Prob. 22RE Ch. 1 - Prob. 23RE Ch. 1 - Prob. 24RE Ch. 1 - Find the limit. limh0(h1)3+1h Ch. 1 - Prob. 26RE Ch. 1 - Prob. 27RE Ch. 1 - Prob. 28RE Ch. 1 - Prob. 29RE Ch. 1 - Prob. 30RE Ch. 1 - Prob. 31RE Ch. 1 - Prob. 32RE Ch. 1 - Prob. 33RE Ch. 1 - Prob. 35RE Ch. 1 - Prob. 36RE Ch. 1 - Prob. 34RE Ch. 1 - Prob. 37RE Ch. 1 - Prob. 38RE Ch. 1 - Prob. 39RE Ch. 1 - Prob. 40RE Ch. 1 - Prob. 41RE Ch. 1 - Prob. 42RE Ch. 1 - Prob. 43RE Ch. 1 - Prob. 44RE Ch. 1 - Prob. 45RE Ch. 1 - Prob. 46RE Ch. 1 - Prob. 47RE Ch. 1 - Prob. 48RE

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Explain, using Theorems 4, 5, 6, and 8, why the function is continuous at every number in its domain. State the domain. 24. F ( x ) = sin(cos(sin x ))

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Chapter 1 Solutions