Explanation Definition used: “Let f be a function defined on some open interval that contains the number a , except possibly at a itself. Then, the limit of f ( x ) as x approaches a is L , lim x → a f ( x ) = L if for every number ε > 0 there is a number δ > 0 such that if 0 < | x − a | < δ then | f ( x ) − L | < ε ”. Given: To find a number δ such that if | x − π 4 | < δ then | tan x − 1 | < 0.2 . Estimation: Obtain the number δ such that if | x − π 4 | < δ then | tan x − 1 | < 0.2 by using the graph. By the definition of limit, the values of a = π 4 , L = 1 , f ( x ) = tan x and ε = 0.2 Rewrite the inequality | tan x − 1 | < 0.2 as follows. − 0.2 < tan x − 1 < 0.2 − 0.2 + 1 < tan x − 1 + 1 < 0.2 + 1 0.8 < tan x < 1.2 The values of x for the function y = tan x lies between the horizontal lines y = 0.8 and y = 1.2 . Use the online graphing calculator to draw the function y = tan x , y = 0.8 and y = 1.2 near the point ( π 4 , 1 ) as shown below in Figure 1.

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Chapter 1.3, Problem 25E

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Chapter 1.3, Problem 24E

Chapter 1.3, Problem 26E

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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