Calculus: Early Transcendentals (2nd Edition)
Calculus: Early Transcendentals (2nd Edition)
2nd Edition
ISBN: 9780321947345
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
Publisher: PEARSON
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Concept explainers

Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
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Textbook Question
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Chapter 4.1, Problem 69E

Critical points and extreme values

  1. a. Find the critical points of the following functions on the given interval.
  2. b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.
  3. c. Find the absolute maximum and minimum values on the given interval when they exist.

69. f(θ) = 2 sin θ + cos θ on [−2π, 2π]

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Chapter 4 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Sketch the graph of a function that has an...Ch. 4.1 - What is a critical point of a function?Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Local and absolute extreme values Use the...
Ch. 4.1 - Local and absolute extreme values Use the...Ch. 4.1 - Local and absolute extreme values Use the...Ch. 4.1 - Local and absolute extreme values Use the...Ch. 4.1 - Locating critical points a. Find the critical...Ch. 4.1 - Prob. 24ECh. 4.1 - Locating critical points a. Find the critical...Ch. 4.1 - Locating critical points a. Find the critical...Ch. 4.1 - Prob. 27ECh. 4.1 - Prob. 28ECh. 4.1 - Prob. 29ECh. 4.1 - Prob. 30ECh. 4.1 - Prob. 31ECh. 4.1 - Locating critical points a. Find the critical...Ch. 4.1 - Prob. 33ECh. 4.1 - Locating critical points a. Find the critical...Ch. 4.1 - Prob. 35ECh. 4.1 - Prob. 36ECh. 4.1 - Absolute maxima and minima a. Find the critical...Ch. 4.1 - Prob. 38ECh. 4.1 - Absolute maxima and minima a. Find the critical...Ch. 4.1 - Prob. 40ECh. 4.1 - Prob. 41ECh. 4.1 - Prob. 42ECh. 4.1 - Absolute maxima and minima a. Find the critical...Ch. 4.1 - Prob. 44ECh. 4.1 - Prob. 45ECh. 4.1 - Prob. 46ECh. 4.1 - Prob. 47ECh. 4.1 - Prob. 48ECh. 4.1 - Prob. 49ECh. 4.1 - Prob. 50ECh. 4.1 - Trajectory high point A stone is launched...Ch. 4.1 - Maximizing revenue A sales analyst determines that...Ch. 4.1 - Maximizing profit Suppose a tour guide has a bus...Ch. 4.1 - Maximizing rectangle perimeters All rectangles...Ch. 4.1 - Explain why or why not Determine whether the...Ch. 4.1 - Prob. 56ECh. 4.1 - Absolute maxima and minima a. Find the critical...Ch. 4.1 - Prob. 58ECh. 4.1 - Prob. 59ECh. 4.1 - Absolute maxima and minima a. Find the critical...Ch. 4.1 - Absolute maxima and minima a. Find the critical...Ch. 4.1 - Absolute maxima and minima a. Find the critical...Ch. 4.1 - Absolute maxima and minima a. Find the critical...Ch. 4.1 - Critical points of functions with unknown...Ch. 4.1 - Prob. 65ECh. 4.1 - Prob. 66ECh. 4.1 - Prob. 67ECh. 4.1 - Critical points and extreme values a. Find the...Ch. 4.1 - Critical points and extreme values a. Find the...Ch. 4.1 - Critical points and extreme values a. Find the...Ch. 4.1 - Prob. 71ECh. 4.1 - Prob. 72ECh. 4.1 - Prob. 73ECh. 4.1 - Absolute value functions Graph the following...Ch. 4.1 - Prob. 75ECh. 4.1 - Minimum surface area box All boxes with a square...Ch. 4.1 - Every second counts You must get from a point P on...Ch. 4.1 - Prob. 78ECh. 4.1 - Values of related functions Suppose f is...Ch. 4.1 - Extreme values of parabolas Consider the function...Ch. 4.1 - Prob. 81ECh. 4.1 - Prob. 82ECh. 4.1 - Proof of the Local Extreme Value Theorem Prove...Ch. 4.2 - Explain how the first derivative of a function...Ch. 4.2 - Explain how to apply the First Derivative Test.Ch. 4.2 - Sketch the graph of a function that has neither a...Ch. 4.2 - Prob. 4ECh. 4.2 - Suppose f exists and is positive on an interval I....Ch. 4.2 - Sketch a function that changes from concave up to...Ch. 4.2 - Prob. 7ECh. 4.2 - Give a function that does not have an inflection...Ch. 4.2 - Is it possible for a function to satisfy f(x) 0,...Ch. 4.2 - Suppose f is continuous on an interval containing...Ch. 4.2 - Sketches from properties Sketch a graph of a...Ch. 4.2 - f(x) 0 on (, 2); f(x) 0 on (2, 5); f(x) 0 on...Ch. 4.2 - Sketches from properties Sketch a graph of a...Ch. 4.2 - Sketches from properties Sketch a graph of a...Ch. 4.2 - Functions from derivatives The following figures...Ch. 4.2 - Functions from derivatives The following figures...Ch. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Prob. 18ECh. 4.2 - Prob. 19ECh. 4.2 - Prob. 20ECh. 4.2 - Prob. 21ECh. 4.2 - Prob. 22ECh. 4.2 - Prob. 23ECh. 4.2 - Prob. 24ECh. 4.2 - Prob. 25ECh. 4.2 - Prob. 26ECh. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Prob. 31ECh. 4.2 - Prob. 32ECh. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - Increasing and decreasing functions Find the...Ch. 4.2 - First Derivative Test a. Locale the critical...Ch. 4.2 - First Derivative Test a. Locale the critical...Ch. 4.2 - First Derivative Test a. Locale the critical...Ch. 4.2 - First Derivative Test a. Locale the critical...Ch. 4.2 - First Derivative Test a. Locale the critical...Ch. 4.2 - First Derivative Test a. Locale the critical...Ch. 4.2 - First Derivative Test a. Locale the critical...Ch. 4.2 - First Derivative Test a. Locale the critical...Ch. 4.2 - First Derivative Test a. Locale the critical...Ch. 4.2 - Prob. 48ECh. 4.2 - Absolute extreme values Verify that the following...Ch. 4.2 - Absolute extreme values Verify that the following...Ch. 4.2 - Absolute extreme values Verify that the following...Ch. 4.2 - Absolute extreme values Verify that the following...Ch. 4.2 - Prob. 53ECh. 4.2 - Prob. 54ECh. 4.2 - Prob. 55ECh. 4.2 - Prob. 56ECh. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Prob. 66ECh. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Prob. 68ECh. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Concavity Determine the intervals on which the...Ch. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Prob. 74ECh. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Prob. 76ECh. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Prob. 78ECh. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Explain why or why not Determine whether the...Ch. 4.2 - Is it possible? Determine whether the following...Ch. 4.2 - Prob. 87ECh. 4.2 - Prob. 88ECh. 4.2 - Prob. 89ECh. 4.2 - Designer functions Sketch the graph of a function...Ch. 4.2 - Prob. 91ECh. 4.2 - Designer functions Sketch the graph of a function...Ch. 4.2 - Designer functions Sketch the graph of a function...Ch. 4.2 - Graph carefully Graph the function f(x) = 60x5 ...Ch. 4.2 - Interpreting the derivative The graph of f on the...Ch. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Second Derivative Test Locate the critical points...Ch. 4.2 - Prob. 99ECh. 4.2 - Concavity of parabolas Consider the general...Ch. 4.2 - Prob. 101ECh. 4.2 - Prob. 102ECh. 4.2 - Population models The population of a species is...Ch. 4.2 - Tangent lines and concavity Give an argument to...Ch. 4.2 - Symmetry of cubics Consider the general cubic...Ch. 4.2 - Properties of cubics Consider the general cubic...Ch. 4.2 - Prob. 107ECh. 4.2 - Even quartics Consider the quartic (fourth-degree)...Ch. 4.2 - General quartic Show that the general quartic...Ch. 4.3 - Why is it important to determine the domain of f...Ch. 4.3 - Prob. 2ECh. 4.3 - Prob. 3ECh. 4.3 - Where are the vertical asymptotes of a rational...Ch. 4.3 - How do you find the absolute maximum and minimum...Ch. 4.3 - Describe the possible end behavior of a...Ch. 4.3 - Shape of the curve Sketch a curve with the...Ch. 4.3 - Shape of the curve Sketch a curve with the...Ch. 4.3 - Graphing polynomials Sketch a graph of the...Ch. 4.3 - Prob. 10ECh. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Prob. 13ECh. 4.3 - Prob. 14ECh. 4.3 - Graphing rational functions Use the guidelines of...Ch. 4.3 - Graphing rational functions Use the guidelines of...Ch. 4.3 - Graphing rational functions Use the guidelines of...Ch. 4.3 - Graphing rational functions Use the guidelines of...Ch. 4.3 - Graphing rational functions Use the guidelines of...Ch. 4.3 - Prob. 20ECh. 4.3 - More graphing Make a complete graph of the...Ch. 4.3 - Prob. 22ECh. 4.3 - Prob. 23ECh. 4.3 - More graphing Make a complete graph of the...Ch. 4.3 - Prob. 25ECh. 4.3 - Prob. 26ECh. 4.3 - More graphing Make a complete graph of the...Ch. 4.3 - Prob. 28ECh. 4.3 - Prob. 29ECh. 4.3 - Prob. 30ECh. 4.3 - More graphing Make a complete graph of the...Ch. 4.3 - Prob. 32ECh. 4.3 - Prob. 33ECh. 4.3 - Prob. 34ECh. 4.3 - Prob. 35ECh. 4.3 - More graphing Make a complete graph of the...Ch. 4.3 - Graphing with technology Make a complete graph of...Ch. 4.3 - Graphing with technology Make a complete graph of...Ch. 4.3 - Graphing with technology Make a complete graph of...Ch. 4.3 - Graphing with technology Make a complete graph of...Ch. 4.3 - Graphing with technology Make a complete graph of...Ch. 4.3 - Graphing with technology Make a complete graph of...Ch. 4.3 - Explain why or why not Determine whether the...Ch. 4.3 - Functions from derivatives Use the derivative f to...Ch. 4.3 - Functions from derivatives Use the derivative f to...Ch. 4.3 - Functions from derivatives Use the derivative f to...Ch. 4.3 - Functions from derivatives Use the derivative f to...Ch. 4.3 - Functions from graphs Use the graphs of f and f to...Ch. 4.3 - Functions from graphs Use the graphs of f and f to...Ch. 4.3 - Nice cubics and quartics The following third- and...Ch. 4.3 - Prob. 51ECh. 4.3 - Nice cubics and quartics The following third- and...Ch. 4.3 - Prob. 53ECh. 4.3 - Oscillations Consider the function f(x) = cos (ln...Ch. 4.3 - Local max/min of x1/x Use analytical methods to...Ch. 4.3 - Local max/min of xx Use analytical methods to find...Ch. 4.3 - Designer functions Sketch a continuous function f...Ch. 4.3 - Designer functions Sketch a continuous function f...Ch. 4.3 - Designer functions Sketch a continuous function f...Ch. 4.3 - Designer functions Sketch a continuous function f...Ch. 4.3 - Prob. 61ECh. 4.3 - Prob. 62ECh. 4.3 - Prob. 63ECh. 4.3 - Prob. 64ECh. 4.3 - Prob. 65ECh. 4.3 - Prob. 66ECh. 4.3 - Prob. 67ECh. 4.3 - Prob. 68ECh. 4.3 - Prob. 69ECh. 4.3 - Prob. 70ECh. 4.3 - Prob. 72ECh. 4.3 - Derivative information Suppose a continuous...Ch. 4.3 - e e Prove that e e by first finding the maximum...Ch. 4.3 - Special curves The following classical curves have...Ch. 4.3 - Special curves The following classical curves have...Ch. 4.3 - Special curves The following classical curves have...Ch. 4.3 - Prob. 78ECh. 4.3 - Prob. 79ECh. 4.3 - Special curves The following classical curves have...Ch. 4.3 - Special curves The following classical curves have...Ch. 4.3 - Prob. 82ECh. 4.3 - Prob. 83ECh. 4.3 - Prob. 84ECh. 4.3 - Prob. 85ECh. 4.3 - Prob. 86ECh. 4.4 - Fill in the blanks: The goal of an optimization...Ch. 4.4 - Prob. 2ECh. 4.4 - Suppose the objective function is Q = x2y and you...Ch. 4.4 - Suppose you wish to minimize a continuous...Ch. 4.4 - Maximum area rectangles Of all rectangles with a...Ch. 4.4 - Maximum area rectangles Of all rectangles with a...Ch. 4.4 - Minimum perimeter rectangles Of all rectangles of...Ch. 4.4 - Minimum perimeter rectangles Of all rectangles...Ch. 4.4 - Maximum product What two nonnegative real numbers...Ch. 4.4 - Sum of squares What two nonnegative real numbers a...Ch. 4.4 - Minimum sum What two positive real numbers whose...Ch. 4.4 - Maximum product Find numbers x and y satisfying...Ch. 4.4 - Minimum sum Find positive numbers x and y...Ch. 4.4 - Pen problems a. A rectangular pen is built with...Ch. 4.4 - Prob. 15ECh. 4.4 - Maximum-volume box Suppose an airline policy...Ch. 4.4 - Shipping crates A square-based, box-shaped...Ch. 4.4 - Minimum distance Find the point P on the line y =...Ch. 4.4 - Prob. 20ECh. 4.4 - Walking and rowing A boat on the ocean is 4 mi...Ch. 4.4 - Shortest ladder A 10-ft-tall fence runs parallel...Ch. 4.4 - Shortest laddermore realistic An 8-ft-tall fence...Ch. 4.4 - Prob. 24ECh. 4.4 - Rectangles beneath a semicircle A rectangle is...Ch. 4.4 - Circle and square A piece of wire of length 60 is...Ch. 4.4 - Maximum-volume cone A cone is constructed by...Ch. 4.4 - Covering a marble Imagine a flat-bottomed...Ch. 4.4 - Optimal garden A rectangular flower garden with an...Ch. 4.4 - Rectangles beneath a line a. A rectangle is...Ch. 4.4 - Keplers wine barrel Several mathematical stories...Ch. 4.4 - Folded boxes a. Squares with sides of length x are...Ch. 4.4 - Making silos A grain silo consists of a...Ch. 4.4 - Suspension system A load must be suspended 6 m...Ch. 4.4 - Light sources The intensity of a light source at a...Ch. 4.4 - Crease-length problem A rectangular sheet of paper...Ch. 4.4 - Laying cable An island is 3.5 mi from the nearest...Ch. 4.4 - Laying cable again Solve the problem in Exercise...Ch. 4.4 - Sum of isosceles distances a. An isosceles...Ch. 4.4 - Prob. 40ECh. 4.4 - Prob. 41ECh. 4.4 - Prob. 42ECh. 4.4 - Crankshaft A crank of radius r rotates with an...Ch. 4.4 - Metal rain gutters A rain gutter is made from...Ch. 4.4 - Optimal soda can a. Classical problem Find the...Ch. 4.4 - Cylinder and cones (Putnam Exam 1938) Right...Ch. 4.4 - Viewing angles An auditorium with a flat floor has...Ch. 4.4 - Searchlight problemnarrow beam A searchlight is...Ch. 4.4 - Watching a Ferris wheel An observer stands 20 m...Ch. 4.4 - Maximum angle Find the value of x that maximizes ...Ch. 4.4 - Maximum-volume cylinder in a sphere Find the...Ch. 4.4 - Rectangles in triangles Find the dimensions and...Ch. 4.4 - Prob. 53ECh. 4.4 - Maximizing profit Suppose you own a tour bus and...Ch. 4.4 - Cone in a cone A right circular cone is inscribed...Ch. 4.4 - Another pen problem A rancher is building a horse...Ch. 4.4 - Minimum-length roads A house is located at each...Ch. 4.4 - Light transmission A window consists of a...Ch. 4.4 - Slowest shortcut Suppose you are standing in a...Ch. 4.4 - The arbelos An arbelos is the region enclosed by...Ch. 4.4 - Proximity questions a. What point on the line y =...Ch. 4.4 - Turning a corner with a pole a. What is the length...Ch. 4.4 - Travel costs A simple model for travel costs...Ch. 4.4 - Do dogs know calculus? A mathematician stands on a...Ch. 4.4 - Fermats Principle a. Two poles of heights m and n...Ch. 4.4 - Prob. 66ECh. 4.4 - Tree notch (Putnam Exam 1938, rephrased) A notch...Ch. 4.4 - Gliding mammals Many species of small mammals...Ch. 4.4 - A challenging pen problem Two triangular pens are...Ch. 4.4 - Prob. 70ECh. 4.5 - Sketch the graph of a smooth function f and label...Ch. 4.5 - Suppose you find the linear approximation to a...Ch. 4.5 - How is linear approximation used to approximate...Ch. 4.5 - How can linear approximation be used to...Ch. 4.5 - Given a function f that is differentiable on its...Ch. 4.5 - Does the differential dy represent the change in f...Ch. 4.5 - Estimating speed Use the linear approximation...Ch. 4.5 - Prob. 8ECh. 4.5 - Prob. 9ECh. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Prob. 12ECh. 4.5 - Linear approximation a. Write the equation of the...Ch. 4.5 - Linear approximation a. Write the equation of the...Ch. 4.5 - Linear approximation a. Write the equation of the...Ch. 4.5 - Prob. 16ECh. 4.5 - Linear approximation a. Write the equation of the...Ch. 4.5 - Prob. 18ECh. 4.5 - Linear approximation a. Write the equation of the...Ch. 4.5 - Linear approximation a. Write the equation of the...Ch. 4.5 - Estimations with linear approximation Use linear...Ch. 4.5 - Prob. 22ECh. 4.5 - Estimations with linear approximation Use linear...Ch. 4.5 - Estimations with linear approximation Use linear...Ch. 4.5 - Estimations with linear approximation Use linear...Ch. 4.5 - Estimations with linear approximation Use linear...Ch. 4.5 - Estimations with linear approximation Use linear...Ch. 4.5 - Estimations with linear approximation Use linear...Ch. 4.5 - Estimations with linear approximation Use linear...Ch. 4.5 - Prob. 30ECh. 4.5 - Linear approximation and concavity Carry out the...Ch. 4.5 - Linear approximation and concavity Carry out the...Ch. 4.5 - Prob. 33ECh. 4.5 - Linear approximation and concavity Carry out the...Ch. 4.5 - Approximating changes 35. Approximate the change...Ch. 4.5 - Approximating changes 36. Approximate the change...Ch. 4.5 - Approximating changes 37. Approximate the change...Ch. 4.5 - Approximating changes 38. Approximate the change...Ch. 4.5 - Approximating changes 39. Approximate the change...Ch. 4.5 - Approximating changes 40. Approximate the change...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Differentials Consider the following functions and...Ch. 4.5 - Explain why or why not Determine whether the...Ch. 4.5 - Linear approximation Estimate f(5.1) given that...Ch. 4.5 - Linear approximation Estimate f(3.85) given that...Ch. 4.5 - Linear approximation a. Write an equation of the...Ch. 4.5 - Linear approximation a. Write an equation of the...Ch. 4.5 - Linear approximation a. Write an equation of the...Ch. 4.5 - Prob. 57ECh. 4.5 - Ideal Gas Law The pressure P, temperature T, and...Ch. 4.5 - Prob. 59ECh. 4.5 - Prob. 60ECh. 4.5 - Prob. 61ECh. 4.5 - Errors in approximations Suppose f(x) = 1/(1 + x)...Ch. 4.5 - Prob. 63ECh. 4.6 - Explain Rolles Theorem with a sketch.Ch. 4.6 - Draw the graph of a function for which the...Ch. 4.6 - Explain why Rolles Theorem cannot be applied to...Ch. 4.6 - Explain the Mean Value Theorem with a sketch.Ch. 4.6 - Draw the graph of a function for which the...Ch. 4.6 - At what points c does the conclusion of the Mean...Ch. 4.6 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.6 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.6 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.6 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.6 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.6 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.6 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.6 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.6 - Lapse rates in the atmosphere Concurrent...Ch. 4.6 - Drag racer acceleration The fastest drag racers...Ch. 4.6 - Prob. 17ECh. 4.6 - Prob. 18ECh. 4.6 - Prob. 19ECh. 4.6 - Prob. 20ECh. 4.6 - Prob. 21ECh. 4.6 - Mean Value Theorem a. Determine whether the Mean...Ch. 4.6 - Mean Value Theorem a. Determine whether the Mean...Ch. 4.6 - Prob. 24ECh. 4.6 - Explain why or why not Determine whether the...Ch. 4.6 - Questions about derivatives 26. Without evaluating...Ch. 4.6 - Questions about derivatives 27. Without evaluating...Ch. 4.6 - Questions about derivatives 28. Find all functions...Ch. 4.6 - Mean Value Theorem and graphs By visual...Ch. 4.6 - Mean Value Theorem and graphs Find all points on...Ch. 4.6 - Mean Value Theorem and graphs Find all points on...Ch. 4.6 - Avalanche forecasting Avalanche forecasters...Ch. 4.6 - Mean Value Theorem and the police A state patrol...Ch. 4.6 - Prob. 34ECh. 4.6 - Running pace Explain why if a runner completes a...Ch. 4.6 - Mean Value Theorem for linear functions Interpret...Ch. 4.6 - Mean Value Theorem for quadratic functions...Ch. 4.6 - Means a. Show that the point c guaranteed to exist...Ch. 4.6 - Equal derivatives Verify that the functions f(x) =...Ch. 4.6 - Prob. 40ECh. 4.6 - 100-m speed The Jamaican sprinter Usain Bolt set a...Ch. 4.6 - Prob. 42ECh. 4.6 - Generalized Mean Value Theorem Suppose the...Ch. 4.7 - Explain with examples what is meant by the...Ch. 4.7 - Why are special methods, such as lHpitals Rule,...Ch. 4.7 - Explain the steps used to apply lHpitals Rule to a...Ch. 4.7 - Prob. 4ECh. 4.7 - Explain how to convert a limit of the form 0 to...Ch. 4.7 - Give an example of a limit of the form / as x 0.Ch. 4.7 - Explain why the form 1 is indeterminate and cannot...Ch. 4.7 - Give the two-step method for attacking an...Ch. 4.7 - In terms of limits, what does it mean for f to...Ch. 4.7 - In terms of limits, what does it mean for the...Ch. 4.7 - Rank the functions x3, ln x, xx, and 2x in order...Ch. 4.7 - Rank the functions x100, ln x10, xx, and 10x in...Ch. 4.7 - 0/0 form Evaluate the following limits using...Ch. 4.7 - 0/0 form Evaluate the following limits using...Ch. 4.7 - 0/0 form Evaluate the following limits using...Ch. 4.7 - 0/0 form Evaluate the following limits using...Ch. 4.7 - 0/0 form Evaluate the following limits using...Ch. 4.7 - 0/0 form Evaluate the following limits using...Ch. 4.7 - Prob. 19ECh. 4.7 - 0/0 form Evaluate the following limits using...Ch. 4.7 - 0/0 form Evaluate the following limits using...Ch. 4.7 - 0/0 form Evaluate the following limits using...Ch. 4.7 - 0/0 form Evaluate the following limits. 23....Ch. 4.7 - 0/0 form Evaluate the following limits. 24....Ch. 4.7 - 0/0 form Evaluate the following limits. 25....Ch. 4.7 - 0/0 form Evaluate the following limits. 26....Ch. 4.7 - 0/0 form Evaluate the following limits. 27....Ch. 4.7 - 0/0 form Evaluate the following limits. 28....Ch. 4.7 - Prob. 29ECh. 4.7 - 0/0 form Evaluate the following limits. 30....Ch. 4.7 - 0/0 form Evaluate the following limits. 31....Ch. 4.7 - 0/0 form Evaluate the following limits. 32....Ch. 4.7 - 0/0 form Evaluate the following limits. 33....Ch. 4.7 - 0/0 form Evaluate the following limits. 34....Ch. 4.7 - 0/0 form Evaluate the following limits. 35....Ch. 4.7 - 0/0 form Evaluate the following limits. 36....Ch. 4.7 - Prob. 37ECh. 4.7 - / form Evaluate the following limits. 38....Ch. 4.7 - / form Evaluate the following limits. 39....Ch. 4.7 - Prob. 40ECh. 4.7 - / form Evaluate the following limits. 41....Ch. 4.7 - / form Evaluate the following limits. 42....Ch. 4.7 - Prob. 43ECh. 4.7 - Prob. 44ECh. 4.7 - 0 form Evaluate the following limits. 45....Ch. 4.7 - 0 form Evaluate the following limits. 46....Ch. 4.7 - 0 form Evaluate the following limits. 47....Ch. 4.7 - 0 form Evaluate the following limits. 48....Ch. 4.7 - 0 form Evaluate the following limits. 49....Ch. 4.7 - 0 form Evaluate the following limits. 50....Ch. 4.7 - form Evaluate the following limits. 51....Ch. 4.7 - Prob. 52ECh. 4.7 - form Evaluate the following limits. 53....Ch. 4.7 - Prob. 54ECh. 4.7 - Prob. 55ECh. 4.7 - Prob. 56ECh. 4.7 - Prob. 57ECh. 4.7 - 1, 00, 0 forms Evaluate the following limits or...Ch. 4.7 - 1, 00, 0 forms Evaluate the following limits or...Ch. 4.7 - Prob. 60ECh. 4.7 - Prob. 61ECh. 4.7 - Prob. 62ECh. 4.7 - 1, 00, 0 forms Evaluate the following limits or...Ch. 4.7 - 1, 00, 0 forms Evaluate the following limits or...Ch. 4.7 - Prob. 65ECh. 4.7 - 1, 00, 0 forms Evaluate the following limits or...Ch. 4.7 - Prob. 67ECh. 4.7 - Prob. 68ECh. 4.7 - Comparing growth rates Use limit methods to...Ch. 4.7 - Comparing growth rates Use limit methods to...Ch. 4.7 - Comparing growth rates Use limit methods to...Ch. 4.7 - Comparing growth rates Use limit methods to...Ch. 4.7 - Comparing growth rates Use limit methods to...Ch. 4.7 - Comparing growth rates Use limit methods to...Ch. 4.7 - Comparing growth rates Use limit methods to...Ch. 4.7 - Prob. 76ECh. 4.7 - Prob. 77ECh. 4.7 - Prob. 78ECh. 4.7 - Comparing growth rates Use limit methods to...Ch. 4.7 - Prob. 80ECh. 4.7 - Explain why or why not Determine whether the...Ch. 4.7 - Two methods Evaluate the following limits in two...Ch. 4.7 - Two methods Evaluate the following limits in two...Ch. 4.7 - Prob. 84ECh. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 4.7 - Limits with parameters Evaluate the following...Ch. 4.7 - Limits with parameters Evaluate the following...Ch. 4.7 - Limits with parameters Evaluate the following...Ch. 4.7 - Limits with parameters Evaluate the following...Ch. 4.7 - An optics limit The theory of interference of...Ch. 4.7 - Compound interest Suppose you make a deposit of P...Ch. 4.7 - Algorithm complexity The complexity of a computer...Ch. 4.7 - LHpital loops Consider the limit limx0ax+bcx+d,...Ch. 4.7 - General result Let a and b be positive real...Ch. 4.7 - Exponential functions and powers Show that any...Ch. 4.7 - Exponentials with different bases Show that f(x) =...Ch. 4.7 - Logs with different bases Show that f(x) = loga x...Ch. 4.7 - Factorial growth rate The factorial function is...Ch. 4.7 - A geometric limit Let f() be the area of the...Ch. 4.7 - Exponentials vs. super exponentials Show that xx...Ch. 4.7 - Exponential growth rates a. For what values of b ...Ch. 4.8 - Give a geometric explanation of Newtons method.Ch. 4.8 - Prob. 2ECh. 4.8 - How do you decide when to terminate Newtons...Ch. 4.8 - Give the formula for Newtons method for the...Ch. 4.8 - Formulating Newtons method Write the formula for...Ch. 4.8 - Formulating Newtons method Write the formula for...Ch. 4.8 - Formulating Newtons method Write the formula for...Ch. 4.8 - Formulating Newtons method Write the formula for...Ch. 4.8 - Finding roots with Newtons method Use a calculator...Ch. 4.8 - Finding roots with Newtons method Use a calculator...Ch. 4.8 - Finding roots with Newtons method Use a calculator...Ch. 4.8 - Finding roots with Newtons method Use a calculator...Ch. 4.8 - Finding roots with Newtons method Use a calculator...Ch. 4.8 - Finding roots with Newtons method Use a calculator...Ch. 4.8 - Finding intersection points Use Newtons method to...Ch. 4.8 - Prob. 16ECh. 4.8 - Finding intersection points Use Newtons method to...Ch. 4.8 - Prob. 18ECh. 4.8 - Finding intersection points Use Newtons method to...Ch. 4.8 - Prob. 20ECh. 4.8 - Prob. 21ECh. 4.8 - Newtons method and curve sketching Use Newtons...Ch. 4.8 - Newtons method and curve sketching Use Newtons...Ch. 4.8 - Prob. 24ECh. 4.8 - Prob. 25ECh. 4.8 - Slow convergence 26. Consider the function f(x) =...Ch. 4.8 - Prob. 27ECh. 4.8 - Fixed points An important question about many...Ch. 4.8 - Fixed points An important question about many...Ch. 4.8 - Fixed points An important question about many...Ch. 4.8 - Fixed points An important question about many...Ch. 4.8 - More root finding Find all the roots of the...Ch. 4.8 - More root finding Find all the roots of the...Ch. 4.8 - More root finding Find all the roots of the...Ch. 4.8 - More root finding Find all the roots of the...Ch. 4.8 - More root finding Find all the roots of the...Ch. 4.8 - More root finding Find all the roots of the...Ch. 4.8 - More root finding Find all the roots of the...Ch. 4.8 - Residuals and errors Approximate the root of f(x)...Ch. 4.8 - Approximating square roots Let a 0 be given and...Ch. 4.8 - Prob. 43ECh. 4.8 - Prob. 44ECh. 4.8 - Applications 45. A damped oscillator The...Ch. 4.8 - The sinc function The sinc function, sinc(x)=sinxx...Ch. 4.8 - Prob. 47ECh. 4.8 - Prob. 48ECh. 4.8 - Prob. 49ECh. 4.9 - Fill in the blanks with either of the words the...Ch. 4.9 - Describe the set of antiderivatives of f(x) = 0.Ch. 4.9 - Describe the set of antiderivatives of f(x) = 1.Ch. 4.9 - Why do two different antiderivatives of a function...Ch. 4.9 - Give the antiderivatives of xp. For what values of...Ch. 4.9 - Prob. 6ECh. 4.9 - Give the antiderivatives of 1/x.Ch. 4.9 - Prob. 8ECh. 4.9 - If F(x) = x2 3x + C and F(1) = 4, what is the...Ch. 4.9 - For a given function f, explain the steps used to...Ch. 4.9 - Finding antiderivatives Find all the...Ch. 4.9 - Finding antiderivatives Find all the...Ch. 4.9 - Prob. 13ECh. 4.9 - Prob. 14ECh. 4.9 - Prob. 15ECh. 4.9 - Prob. 16ECh. 4.9 - Finding antiderivatives Find all the...Ch. 4.9 - Finding antiderivatives Find all the...Ch. 4.9 - Finding antiderivatives Find all the...Ch. 4.9 - Finding antiderivatives Find all the...Ch. 4.9 - Prob. 21ECh. 4.9 - Prob. 22ECh. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Prob. 36ECh. 4.9 - Prob. 37ECh. 4.9 - Prob. 38ECh. 4.9 - Prob. 39ECh. 4.9 - Prob. 40ECh. 4.9 - Indefinite integrals involving trigonometric...Ch. 4.9 - Prob. 42ECh. 4.9 - Indefinite integrals involving trigonometric...Ch. 4.9 - Prob. 44ECh. 4.9 - Prob. 45ECh. 4.9 - Indefinite integrals involving trigonometric...Ch. 4.9 - Other indefinite integrate Determine the following...Ch. 4.9 - Prob. 48ECh. 4.9 - Other indefinite integrate Determine the following...Ch. 4.9 - Prob. 50ECh. 4.9 - Prob. 51ECh. 4.9 - Prob. 52ECh. 4.9 - Prob. 53ECh. 4.9 - Prob. 54ECh. 4.9 - Other indefinite integrate Determine the following...Ch. 4.9 - Prob. 56ECh. 4.9 - Other indefinite integrate Determine the following...Ch. 4.9 - Other indefinite integrate Determine the following...Ch. 4.9 - Prob. 59ECh. 4.9 - Prob. 60ECh. 4.9 - Prob. 61ECh. 4.9 - Prob. 62ECh. 4.9 - Prob. 63ECh. 4.9 - Particular antiderivatives For the following...Ch. 4.9 - Prob. 65ECh. 4.9 - Particular antiderivatives For the following...Ch. 4.9 - Solving initial value problems Find the solution...Ch. 4.9 - Solving initial value problems Find the solution...Ch. 4.9 - Solving initial value problems Find the solution...Ch. 4.9 - Prob. 70ECh. 4.9 - Prob. 71ECh. 4.9 - Prob. 72ECh. 4.9 - Prob. 73ECh. 4.9 - Prob. 74ECh. 4.9 - Prob. 75ECh. 4.9 - Prob. 76ECh. 4.9 - Graphing general solutions Graph several functions...Ch. 4.9 - Prob. 78ECh. 4.9 - Prob. 79ECh. 4.9 - Prob. 80ECh. 4.9 - Prob. 81ECh. 4.9 - Prob. 82ECh. 4.9 - Velocity to position Given the following velocity...Ch. 4.9 - Prob. 84ECh. 4.9 - Prob. 85ECh. 4.9 - Velocity to position Given the following velocity...Ch. 4.9 - Velocity to position Given the following velocity...Ch. 4.9 - Prob. 88ECh. 4.9 - Acceleration to position Given the following...Ch. 4.9 - Acceleration to position Given the following...Ch. 4.9 - Acceleration to position Given the following...Ch. 4.9 - Acceleration to position Given the following...Ch. 4.9 - Prob. 93ECh. 4.9 - Prob. 94ECh. 4.9 - Races The velocity function and initial position...Ch. 4.9 - Prob. 96ECh. 4.9 - Prob. 97ECh. 4.9 - Motion with gravity Consider the following...Ch. 4.9 - Prob. 99ECh. 4.9 - Motion with gravity Consider the following...Ch. 4.9 - Explain why or why not Determine whether the...Ch. 4.9 - Miscellaneous indefinite integrals Determine the...Ch. 4.9 - Prob. 103ECh. 4.9 - Prob. 104ECh. 4.9 - Prob. 105ECh. 4.9 - Prob. 106ECh. 4.9 - Miscellaneous indefinite integrals Determine the...Ch. 4.9 - Miscellaneous indefinite integrals Determine the...Ch. 4.9 - Prob. 109ECh. 4.9 - Prob. 110ECh. 4.9 - Functions from higher derivatives Find the...Ch. 4.9 - Functions from higher derivatives Find the...Ch. 4.9 - Prob. 113ECh. 4.9 - Prob. 114ECh. 4.9 - How rate A large tank is filled with water when an...Ch. 4.9 - Prob. 116ECh. 4.9 - Prob. 117ECh. 4.9 - Verifying indefinite integrals Verify the...Ch. 4.9 - Prob. 119ECh. 4.9 - Prob. 120ECh. 4.9 - Prob. 121ECh. 4 - Explain why or why not Determine whether the...Ch. 4 - Locating extrema Consider the graph of a function...Ch. 4 - Designer functions Sketch the graph of a function...Ch. 4 - Designer functions Sketch the graph of a function...Ch. 4 - Prob. 5RECh. 4 - Prob. 6RECh. 4 - Prob. 7RECh. 4 - Prob. 8RECh. 4 - Prob. 9RECh. 4 - Prob. 10RECh. 4 - Absolute values Consider the function f(x) = |x ...Ch. 4 - Inflection points Does f(x) = 2x5 10x4 + 20x3 + x...Ch. 4 - Prob. 13RECh. 4 - Prob. 14RECh. 4 - Prob. 15RECh. 4 - Prob. 16RECh. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Prob. 18RECh. 4 - Prob. 19RECh. 4 - Prob. 20RECh. 4 - Optimization A right triangle has legs of length h...Ch. 4 - T 22. Rectangles beneath a curve A rectangle is...Ch. 4 - Maximum printable area A rectangular page in a...Ch. 4 - Nearest point What point on the graph of...Ch. 4 - Maximum area A line segment of length 10 joins the...Ch. 4 - Minimum painting surface A metal cistern in the...Ch. 4 - Linear approximation a. Find the linear...Ch. 4 - Linear approximation a. Find the linear...Ch. 4 - Estimations with linear approximation Use linear...Ch. 4 - Estimations with linear approximation Use linear...Ch. 4 - Change in elevation The elevation h (in feet above...Ch. 4 - Change in energy The energy E (in joules) released...Ch. 4 - Mean Value Theorem The population of a culture of...Ch. 4 - Growth rate of bamboo Bamboo belongs to the grass...Ch. 4 - Newtons method Use Newtons method to approximate...Ch. 4 - Prob. 36RECh. 4 - Newtons method Use Newtons method to approximate...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Prob. 46RECh. 4 - Prob. 47RECh. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Prob. 51RECh. 4 - Prob. 52RECh. 4 - Prob. 53RECh. 4 - Prob. 54RECh. 4 - Prob. 55RECh. 4 - Prob. 56RECh. 4 - Prob. 57RECh. 4 - Prob. 58RECh. 4 - Prob. 59RECh. 4 - Comparing growth rates Determine which of the two...Ch. 4 - Prob. 61RECh. 4 - Prob. 62RECh. 4 - Comparing growth rates Determine which of the two...Ch. 4 - Comparing growth rates Determine which of the two...Ch. 4 - Prob. 65RECh. 4 - Comparing growth rates Determine which of the two...Ch. 4 - Comparing growth rates Determine which of the two...Ch. 4 - Indefinite integrals Determine the following...Ch. 4 - Indefinite integrals Determine the following...Ch. 4 - Prob. 70RECh. 4 - Indefinite integrals Determine the following...Ch. 4 - Indefinite integrals Determine the following...Ch. 4 - Prob. 73RECh. 4 - Indefinite integrals Determine the following...Ch. 4 - Prob. 75RECh. 4 - Prob. 76RECh. 4 - Prob. 77RECh. 4 - Indefinite integrals Determine the following...Ch. 4 - Prob. 79RECh. 4 - Prob. 80RECh. 4 - Prob. 81RECh. 4 - Prob. 82RECh. 4 - Prob. 83RECh. 4 - Prob. 84RECh. 4 - Prob. 85RECh. 4 - Prob. 86RECh. 4 - Prob. 87RECh. 4 - Logs of logs Compare the growth rates of ln x, ln...Ch. 4 - Prob. 89RECh. 4 - Prob. 90RECh. 4 - Prob. 91RECh. 4 - Prob. 92RECh. 4 - Prob. 93RECh. 4 - Prob. 94RECh. 4 - Limits for e Consider the function g(x) = (1 +...Ch. 4 - A family of super-exponential functions Let f(x) =...

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