Essential Calculus: Early Transcendentals
2nd Edition
ISBN: 9781133112280
Author: James Stewart
Publisher: Cengage Learning
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Chapter 1.6, Problem 51E
How close to –3 do we have to take x so that
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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. 5ECh. 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. 9ECh. 1.1 - The graph shows the height of the water in a...
Ch. 1.1 - Prob. 11ECh. 1.1 - Sketch a rough graph of the number of hours of...Ch. 1.1 - Prob. 13ECh. 1.1 - Sketch a rough graph of the market value of a new...Ch. 1.1 - Prob. 15ECh. 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. 24ECh. 1.1 - Find the domain of the function. 31. f(x)=x+4x29Ch. 1.1 - Prob. 26ECh. 1.1 - Prob. 28ECh. 1.1 - Prob. 29ECh. 1.1 - Find the domain of the function. 37. F(p)=2pCh. 1.1 - Find the domain and range and sketch the graph of...Ch. 1.1 - Prob. 31ECh. 1.1 - Prob. 34ECh. 1.1 - Find the domain and sketch the graph of the...Ch. 1.1 - Prob. 33ECh. 1.1 - Prob. 35ECh. 1.1 - Prob. 36ECh. 1.1 - Find the domain and sketch the graph of the...Ch. 1.1 - Prob. 38ECh. 1.1 - Prob. 39ECh. 1.1 - Prob. 40ECh. 1.1 - Prob. 41ECh. 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. 44ECh. 1.1 - Prob. 45ECh. 1.1 - Prob. 46ECh. 1.1 - Prob. 47ECh. 1.1 - Find a formula for the described function and...Ch. 1.1 - Prob. 49ECh. 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. 5ECh. 1.2 - Prob. 6ECh. 1.2 - Prob. 7ECh. 1.2 - Prob. 8ECh. 1.2 - Prob. 9ECh. 1.2 - Prob. 10ECh. 1.2 - Prob. 11ECh. 1.2 - Prob. 12ECh. 1.2 - Prob. 13ECh. 1.2 - The monthly cost of driving a car depends on the...Ch. 1.2 - Prob. 15ECh. 1.2 - Prob. 16ECh. 1.2 - Prob. 17ECh. 1.2 - Explain how each graph is obtained from the graph...Ch. 1.2 - Prob. 19ECh. 1.2 - Prob. 20ECh. 1.2 - Prob. 21ECh. 1.2 - Prob. 22ECh. 1.2 - Prob. 23ECh. 1.2 - Prob. 24ECh. 1.2 - Prob. 25ECh. 1.2 - Prob. 30ECh. 1.2 - Prob. 27ECh. 1.2 - Prob. 32ECh. 1.2 - Prob. 26ECh. 1.2 - Prob. 28ECh. 1.2 - Prob. 31ECh. 1.2 - Prob. 29ECh. 1.2 - Prob. 34ECh. 1.2 - Prob. 33ECh. 1.2 - Prob. 36ECh. 1.2 - Prob. 35ECh. 1.2 - Prob. 37ECh. 1.2 - Prob. 38ECh. 1.2 - Prob. 40ECh. 1.2 - Prob. 39ECh. 1.2 - Prob. 41ECh. 1.2 - Prob. 42ECh. 1.2 - Prob. 43ECh. 1.2 - Prob. 44ECh. 1.2 - Prob. 45ECh. 1.2 - Prob. 46ECh. 1.2 - Prob. 47ECh. 1.2 - Prob. 48ECh. 1.2 - Prob. 49ECh. 1.2 - Express the function in the form f g. 48....Ch. 1.2 - Prob. 51ECh. 1.2 - Prob. 52ECh. 1.2 - Prob. 53ECh. 1.2 - Prob. 54ECh. 1.2 - Prob. 55ECh. 1.2 - Prob. 57ECh. 1.2 - Prob. 56ECh. 1.2 - Prob. 58ECh. 1.2 - Prob. 59ECh. 1.2 - Prob. 60ECh. 1.2 - Prob. 61ECh. 1.2 - Prob. 62ECh. 1.2 - Prob. 63ECh. 1.2 - Prob. 64ECh. 1.2 - Prob. 65ECh. 1.2 - Prob. 66ECh. 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. 5ECh. 1.3 - Prob. 6ECh. 1.3 - Prob. 7ECh. 1.3 - Prob. 8ECh. 1.3 - Prob. 9ECh. 1.3 - Sketch the graph of an example of a function f...Ch. 1.3 - Prob. 11ECh. 1.3 - Guess the value of the limit (if it exists) by...Ch. 1.3 - Prob. 13ECh. 1.3 - Guess the value of the limit (if it exists) by...Ch. 1.3 - Prob. 15ECh. 1.3 - Prob. 16ECh. 1.3 - Prob. 17ECh. 1.3 - Prob. 18ECh. 1.3 - Prob. 19ECh. 1.3 - Prob. 20ECh. 1.3 - Prob. 21ECh. 1.3 - Prob. 22ECh. 1.3 - Prob. 23ECh. 1.3 - Use the given graph of f(x) =x2 to find a number ...Ch. 1.3 - Prob. 25ECh. 1.3 - Use a graph to find a number such that if...Ch. 1.3 - Prob. 27ECh. 1.3 - Prob. 28ECh. 1.3 - Prob. 29ECh. 1.3 - Prove the statement using the , definition of a...Ch. 1.3 - Prob. 31ECh. 1.3 - Prove the statement using the , definition of a...Ch. 1.3 - Prob. 33ECh. 1.3 - Prob. 34ECh. 1.3 - Prob. 35ECh. 1.3 - Prob. 36ECh. 1.3 - Prob. 37ECh. 1.3 - Prob. 38ECh. 1.3 - Prob. 39ECh. 1.3 - Prove the statement using the , definition of a...Ch. 1.3 - Prob. 41ECh. 1.3 - Prob. 42ECh. 1.3 - Prob. 43ECh. 1.3 - Prob. 44ECh. 1.3 - Prob. 46ECh. 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. 4ECh. 1.4 - Prob. 5ECh. 1.4 - Prob. 6ECh. 1.4 - Evaluate the limit and justify each step by...Ch. 1.4 - Prob. 8ECh. 1.4 - Prob. 9ECh. 1.4 - (a) What is wrong with the following equation?...Ch. 1.4 - Prob. 11ECh. 1.4 - Evaluate the limit, if it exists. limx4x24xx23x4Ch. 1.4 - Evaluate the limit, if it exists. limx5x25x+6x5Ch. 1.4 - Evaluate the limit, if it exists. limx1x24xx23x4Ch. 1.4 - Prob. 15ECh. 1.4 - Prob. 16ECh. 1.4 - Prob. 17ECh. 1.4 - Evaluate the limit, if it exists. limh0(2+h)38hCh. 1.4 - Prob. 19ECh. 1.4 - Prob. 20ECh. 1.4 - Evaluate the limit, if it exists. limh09+h3hCh. 1.4 - Evaluate the limit, if it exists. limu24u+13u2Ch. 1.4 - Prob. 25ECh. 1.4 - Evaluate the limit, if it exists. limt0(1t1t2+t)Ch. 1.4 - Prob. 23ECh. 1.4 - Evaluate the limit, if it exists. limx4x2+95x+4Ch. 1.4 - Prob. 27ECh. 1.4 - Evaluate the limit, if it exists. limh01(xh)21x2hCh. 1.4 - Prob. 29ECh. 1.4 - Prob. 30ECh. 1.4 - Prob. 31ECh. 1.4 - Use the Squeeze Theorem to show that...Ch. 1.4 - Prob. 33ECh. 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. 37ECh. 1.4 - Find the limit, if it exists. If the limit does...Ch. 1.4 - Prob. 39ECh. 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. 42ECh. 1.4 - Let g(x)=x2+x6x2 (a) Find (i) limx2+g(x) (ii)...Ch. 1.4 - Prob. 44ECh. 1.4 - Prob. 45ECh. 1.4 - Prob. 46ECh. 1.4 - Prob. 47ECh. 1.4 - Prob. 48ECh. 1.4 - Prob. 49ECh. 1.4 - Find the limit. limx0sin4xsin6xCh. 1.4 - Find the limit. limt0tan6tsin2tCh. 1.4 - Prob. 52ECh. 1.4 - Find the limit. limx0sin3x5x34xCh. 1.4 - Prob. 54ECh. 1.4 - Prob. 55ECh. 1.4 - Find the limit. limx0sin(x2)xCh. 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. 64ECh. 1.4 - Prove that if limxag(x)=0 and limxaf(x) exists and...Ch. 1.4 - Prob. 65ECh. 1.4 - Prob. 66ECh. 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. 27ECh. 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. 43ECh. 1.5 - Prob. 44ECh. 1.5 - Prob. 45ECh. 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. 48ECh. 1.5 - Prob. 49ECh. 1.5 - Prob. 50ECh. 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. 52ECh. 1.6 - Prob. 53ECh. 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)2Ch. 1.6 - Find the limit. limx2x22xx24x+4Ch. 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. 24ECh. 1.6 - Prob. 13ECh. 1.6 - Find the limit. limx3x+2x+3Ch. 1.6 - Prob. 25ECh. 1.6 - Prob. 26ECh. 1.6 - Find the limit or show that it does not exist....Ch. 1.6 - Prob. 31ECh. 1.6 - Prob. 32ECh. 1.6 - Prob. 30ECh. 1.6 - Prob. 17ECh. 1.6 - Prob. 33ECh. 1.6 - Prob. 28ECh. 1.6 - Prob. 16ECh. 1.6 - Prob. 29ECh. 1.6 - Prob. 37ECh. 1.6 - Prob. 38ECh. 1.6 - Prob. 36ECh. 1.6 - Find the horizontal and vertical asymptotes of...Ch. 1.6 - Prob. 39ECh. 1.6 - Prob. 34ECh. 1.6 - Let P and Q be polynomials. Find limxP(x)Q(x) if...Ch. 1.6 - Prob. 46ECh. 1.6 - Prob. 41ECh. 1.6 - Prob. 40ECh. 1.6 - Evaluate the limits. (a) limxxsin1x (b) limxxsin1xCh. 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. 44ECh. 1.6 - Prob. 47ECh. 1.6 - Prob. 49ECh. 1.6 - Prob. 55ECh. 1.6 - Prob. 54ECh. 1.6 - Prob. 56ECh. 1.6 - Prob. 57ECh. 1.6 - Prob. 58ECh. 1.6 - Prove that limxf(x)=limt0+f(1/t) and...Ch. 1 - Prob. 1RCCCh. 1 - Prob. 2RCCCh. 1 - Prob. 3RCCCh. 1 - Prob. 4RCCCh. 1 - Prob. 5RCCCh. 1 - Prob. 6RCCCh. 1 - Prob. 7RCCCh. 1 - Prob. 8RCCCh. 1 - Prob. 9RCCCh. 1 - Prob. 10RCCCh. 1 - Prob. 11RCCCh. 1 - Prob. 1RQCh. 1 - Prob. 2RQCh. 1 - Prob. 3RQCh. 1 - Prob. 4RQCh. 1 - Prob. 5RQCh. 1 - Prob. 6RQCh. 1 - Prob. 19RQCh. 1 - Prob. 1RECh. 1 - Prob. 2RECh. 1 - Prob. 3RECh. 1 - Prob. 4RECh. 1 - Prob. 5RECh. 1 - Prob. 6RECh. 1 - Prob. 7RECh. 1 - Prob. 8RECh. 1 - Prob. 9RECh. 1 - Prob. 10RECh. 1 - Prob. 11RECh. 1 - Use transformations to sketch the graph of the...Ch. 1 - Prob. 13RECh. 1 - Prob. 14RECh. 1 - Prob. 15RECh. 1 - Prob. 16RECh. 1 - Prob. 17RECh. 1 - Prob. 18RECh. 1 - Prob. 12RCCCh. 1 - Prob. 13RCCCh. 1 - Prob. 14RCCCh. 1 - Prob. 15RCCCh. 1 - Prob. 18RCCCh. 1 - Prob. 16RCCCh. 1 - Prob. 17RCCCh. 1 - Prob. 7RQCh. 1 - Prob. 8RQCh. 1 - Prob. 9RQCh. 1 - Prob. 10RQCh. 1 - Prob. 11RQCh. 1 - Determine whether the statement is true or false....Ch. 1 - Prob. 13RQCh. 1 - Prob. 14RQCh. 1 - Prob. 15RQCh. 1 - Prob. 16RQCh. 1 - Prob. 17RQCh. 1 - If f and g are polynomials and g(2) = 0, then the...Ch. 1 - Prob. 20RQCh. 1 - Prob. 21RQCh. 1 - Prob. 22RQCh. 1 - Prob. 23RQCh. 1 - Determine whether the statement is true or false....Ch. 1 - Prob. 25RQCh. 1 - Prob. 26RQCh. 1 - Prob. 19RECh. 1 - Prob. 20RECh. 1 - Prob. 21RECh. 1 - Prob. 22RECh. 1 - Prob. 23RECh. 1 - Prob. 24RECh. 1 - Find the limit. limh0(h1)3+1hCh. 1 - Prob. 26RECh. 1 - Prob. 27RECh. 1 - Prob. 28RECh. 1 - Prob. 29RECh. 1 - Prob. 30RECh. 1 - Prob. 31RECh. 1 - Prob. 32RECh. 1 - Prob. 33RECh. 1 - Prob. 35RECh. 1 - Prob. 36RECh. 1 - Prob. 34RECh. 1 - Prob. 37RECh. 1 - Prob. 38RECh. 1 - Prob. 39RECh. 1 - Prob. 40RECh. 1 - Prob. 41RECh. 1 - Prob. 42RECh. 1 - Prob. 43RECh. 1 - Prob. 44RECh. 1 - Prob. 45RECh. 1 - Prob. 46RECh. 1 - Prob. 47RECh. 1 - Prob. 48RE
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- Suppose an oil spill covers a circular area and the radius, r, increases according to the graph shown below where t represents the number of minutes since the spill was first observed. Radius (feet) 80 70 60 50 40 30 20 10 0 r 0 10 20 30 40 50 60 70 80 90 Time (minutes) (a) How large is the circular area of the spill 30 minutes after it was first observed? Give your answer in terms of π. square feet (b) If the cost to clean the oil spill is proportional to the square of the diameter of the spill, express the cost, C, as a function of the radius of the spill, r. Use a lower case k as the proportionality constant. C(r) = (c) Which of the following expressions could be used to represent the amount of time it took for the radius of the spill to increase from 20 feet to 60 feet? r(60) - r(20) Or¹(80-30) r(80) - r(30) r-1(80) - r−1(30) r-1(60) - r¹(20)arrow_forward6. Graph the function f(x)=log3x. Label three points on the graph (one should be the intercept) with corresponding ordered pairs and label the asymptote with its equation. Write the domain and range of the function in interval notation. Make your graph big enough to see all important features.arrow_forwardFind the average value gave of the function g on the given interval. gave = g(x) = 8√√x, [8,64] Need Help? Read It Watch Itarrow_forward
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