MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)
3rd Edition
ISBN: 9780134856926
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Textbook Question
Chapter 4.7, Problem 109E
Graphing functions Make a complete graph of the following functions using the graphing guidelines outlined in Section 4.4.
109.
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Chapter 4 Solutions
MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)
Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Consider the function f(x) = x3. Where is the...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 Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Locating critical points Find the critical points...Ch. 4.1 - Prob. 42ECh. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Prob. 58ECh. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Prob. 64ECh. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Absolute maxima and minima Determine the location...Ch. 4.1 - Prob. 68ECh. 4.1 - Efficiency of wind turbines A wind Turbine...Ch. 4.1 - Derivation of wind turbine formula A derivation of...Ch. 4.1 - Suppose the position of an object moving...Ch. 4.1 - Minimum surface area box All boxes with a square...Ch. 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 - Absolute maxima and minima a. Find the critical...Ch. 4.1 - Absolute maxima and minima a. Find the critical...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. 82ECh. 4.1 - Prob. 83ECh. 4.1 - Absolute value functions Graph the following...Ch. 4.1 - Prob. 85ECh. 4.1 - Prob. 86ECh. 4.1 - Every second counts You must get from a point P on...Ch. 4.1 - Extreme values of parabolas Consider the function...Ch. 4.1 - Values of related functions Suppose f is...Ch. 4.1 - Prob. 90ECh. 4.1 - Proof of the Local Extreme Value Theorem Prove...Ch. 4.1 - Prob. 92ECh. 4.2 - Where on the interval [0, 4] does f(x) = 4x x2...Ch. 4.2 - Sketch the graph of a function that illustrates...Ch. 4.2 - Give two distinct linear functions f and g that...Ch. 4.2 - Explain Rolles Theorem with a sketch.Ch. 4.2 - Draw the graph of a function for which the...Ch. 4.2 - Explain why Rolles Theorem cannot be applied to...Ch. 4.2 - Explain the Mean Value Theorem with a sketch.Ch. 4.2 - For each function f and interval [a, b], a graph...Ch. 4.2 - For each function f and interval [a, b], a graph...Ch. 4.2 - For each function f and interval [a, b], a graph...Ch. 4.2 - At what points c does the conclusion of the Mean...Ch. 4.2 - Draw the graph of a function for which the...Ch. 4.2 - Letf(x)=x2/3. Show that there is no value of c in...Ch. 4.2 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.2 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.2 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.2 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.2 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.2 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.2 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.2 - Rolles Theorem Determine whether Rolles Theorem...Ch. 4.2 - Lapse rates in the atmosphere Concurrent...Ch. 4.2 - Drag racer acceleration The fastest drag racers...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Mean Value Theorem Consider the following...Ch. 4.2 - Explain why or why not Determine whether the...Ch. 4.2 - Prob. 34ECh. 4.2 - Another inverse tangent identity a.Use derivatives...Ch. 4.2 - Questions about derivatives 26. Without evaluating...Ch. 4.2 - Questions about derivatives 27. Without evaluating...Ch. 4.2 - Questions about derivatives 28. Find all functions...Ch. 4.2 - Mean Value Theorem and graphs By visual...Ch. 4.2 - Mean Value Theorem and graphs Find all points on...Ch. 4.2 - Mean Value Theorem and graphs Find all points on...Ch. 4.2 - Avalanche forecasting Avalanche forecasters...Ch. 4.2 - Mean Value Theorem and the police A state patrol...Ch. 4.2 - Mean Value Theorem and the police again Compare...Ch. 4.2 - Running pace Explain why if a runner completes a...Ch. 4.2 - Mean Value Theorem for linear functions Interpret...Ch. 4.2 - Mean Value Theorem for quadratic functions...Ch. 4.2 - Means a. Show that the point c guaranteed to exist...Ch. 4.2 - Equal derivatives Verify that the functions f(x) =...Ch. 4.2 - 100-m speed The Jamaican sprinter Usain Bolt set a...Ch. 4.2 - Verify the identity sec1x=cos1(1/x),forx0.Ch. 4.2 - Prob. 52ECh. 4.2 - Suppose f(x)2, for allx2, and f(2) = 7. Show that...Ch. 4.2 - Suppose f(x)1, for all x 0, and f(0) = 0. Show...Ch. 4.2 - Use the Mean Value Theorem to prove that 1+a21+a...Ch. 4.2 - Prove the following statements. a.|sinasinb||ab|,...Ch. 4.2 - Generalized Mean Value Theorem Suppose the...Ch. 4.2 - Prob. 58ECh. 4.3 - Explain why a positive derivative on an interval...Ch. 4.3 - Sketch a function f that is differentiable on (−∞,...Ch. 4.3 - Explain how the First Derivative Test determines...Ch. 4.3 - Verify that the function f(x) = x4 is concave up...Ch. 4.3 - Prob. 5QCCh. 4.3 - Explain how the first derivative of a function...Ch. 4.3 - Explain how to apply the First Derivative Test.Ch. 4.3 - Suppose the derivative of f isf(x)=x3. a.Find the...Ch. 4.3 - Suppose the derivative of f isf(x)=(x1)(x2)....Ch. 4.3 - Sketch the graph of a function that has neither a...Ch. 4.3 - The following graph of the derivative g' has...Ch. 4.3 - Functions from derivatives The following figures...Ch. 4.3 - Functions from derivatives The following figures...Ch. 4.3 - Sketches from properties Sketch a graph of a...Ch. 4.3 - f(x) 0 on (, 2); f(x) 0 on (2, 5); f(x) 0 on...Ch. 4.3 - Sketches from properties Sketch a graph of a...Ch. 4.3 - Sketches from properties Sketch a graph of a...Ch. 4.3 - Supposeg(x)=2x. a.On what intervals is g concave...Ch. 4.3 - The following graph of g has exactly three...Ch. 4.3 - Is it possible for a function to satisfy f(x) 0,...Ch. 4.3 - Sketch a function that changes from concave up to...Ch. 4.3 - Give a function that does not have an inflection...Ch. 4.3 - Suppose f is continuous on an interval containing...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - Increasing and decreasing functions Find the...Ch. 4.3 - First Derivative Test a. Locale the critical...Ch. 4.3 - First Derivative Test a. Locale the critical...Ch. 4.3 - First Derivative Test a. Locale the critical...Ch. 4.3 - First Derivative Test a. Locale the critical...Ch. 4.3 - First Derivative Test a. Locale the critical...Ch. 4.3 - First Derivative Test a. Locale the critical...Ch. 4.3 - First Derivative Test a. Locale the critical...Ch. 4.3 - First Derivative Test a. Locale the critical...Ch. 4.3 - First Derivative Test a. Locale the critical...Ch. 4.3 - First Derivative Test a.Locate the critical points...Ch. 4.3 - Absolute extreme values Verify that the following...Ch. 4.3 - Absolute extreme values Verify that the following...Ch. 4.3 - Absolute extreme values Verify that the following...Ch. 4.3 - Absolute extreme values Verify that the following...Ch. 4.3 - Prob. 59ECh. 4.3 - Prob. 60ECh. 4.3 - Prob. 61ECh. 4.3 - Prob. 62ECh. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Concavity Determine the intervals on which the...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Prob. 82ECh. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Second Derivative Test Locate the critical points...Ch. 4.3 - Explain why or why not Determine whether the...Ch. 4.3 - Is it possible? Determine whether the following...Ch. 4.3 - Matching derivatives and functions The following...Ch. 4.3 - Prob. 100ECh. 4.3 - Prob. 101ECh. 4.3 - Designer functions Sketch the graph of a function...Ch. 4.3 - Prob. 103ECh. 4.3 - Designer functions Sketch the graph of a function...Ch. 4.3 - Designer functions Sketch the graph of a function...Ch. 4.3 - Graph carefully Graph the function f(x) = 60x5 ...Ch. 4.3 - Interpreting the derivative The graph of f on the...Ch. 4.3 - Prob. 108ECh. 4.3 - Prob. 109ECh. 4.3 - Prob. 110ECh. 4.3 - Population models The population of a species is...Ch. 4.3 - Tangent lines and concavity Give an argument to...Ch. 4.3 - General quartic Show that the general quartic...Ch. 4.3 - Properties of cubics Consider the general cubic...Ch. 4.3 - Concavity of parabolas Consider the general...Ch. 4.4 - Graph f(x) = x3/3 - 400x using various windows on...Ch. 4.4 - Explain why the functions f and f + C, where C is...Ch. 4.4 - Prob. 3QCCh. 4.4 - Why is it important to determine the domain of f...Ch. 4.4 - Prob. 2ECh. 4.4 - Prob. 3ECh. 4.4 - Where are the vertical asymptotes of a rational...Ch. 4.4 - How do you find the absolute maximum and minimum...Ch. 4.4 - Describe the possible end behavior of a...Ch. 4.4 - Shape of the curve Sketch a curve with the...Ch. 4.4 - Shape of the curve Sketch a curve with the...Ch. 4.4 - Designer functions Sketch a continuous function f...Ch. 4.4 - Designer functions Sketch a continuous function f...Ch. 4.4 - Designer functions Sketch a continuous function f...Ch. 4.4 - Designer functions Sketch a continuous function f...Ch. 4.4 - Let f(x)=(x3)(x+3)2. a.Verify that f(x)=3(x1)(x+3)...Ch. 4.4 - If , it can be shown that and . Use these...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing rational functions Use the guidelines of...Ch. 4.4 - Graphing rational functions Use the guidelines of...Ch. 4.4 - Graphing rational functions Use the guidelines of...Ch. 4.4 - Graphing rational functions Use the guidelines of...Ch. 4.4 - Graphing rational functions Use the guidelines of...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Prob. 38ECh. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Prob. 40ECh. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Graphing functions Use the guidelines of this...Ch. 4.4 - Functions from graphs Use the graphs of f and f to...Ch. 4.4 - Functions from graphs Use the graphs of f and f to...Ch. 4.4 - Graphing with technology Make a complete graph of...Ch. 4.4 - Graphing with technology Make a complete graph of...Ch. 4.4 - Graphing with technology Make a complete graph of...Ch. 4.4 - Graphing with technology Make a complete graph of...Ch. 4.4 - Graphing with technology Make a complete graph of...Ch. 4.4 - Graphing with technology Make a complete graph of...Ch. 4.4 - Explain why or why not Determine whether the...Ch. 4.4 - Functions from derivatives Use the derivative f to...Ch. 4.4 - Functions from derivatives Use the derivative f to...Ch. 4.4 - Functions from derivatives Use the derivative f to...Ch. 4.4 - Functions from derivatives Use the derivative f to...Ch. 4.4 - e e Prove that e e by first finding the maximum...Ch. 4.4 - Oscillations Consider the function f(x) = cos (ln...Ch. 4.4 - Local max/min of x1/x Use analytical methods to...Ch. 4.4 - Local max/min of xx Use analytical methods to find...Ch. 4.4 - Derivative information Suppose a continuous...Ch. 4.4 - Prob. 66ECh. 4.4 - Combining technology with analytical methods Use a...Ch. 4.4 - Combining technology with analytical methods Use a...Ch. 4.4 - Combining technology with analytical methods Use a...Ch. 4.4 - Special curves The following classical curves have...Ch. 4.4 - Special curves The following classical curves have...Ch. 4.4 - Special curves The following classical curves have...Ch. 4.4 - Special curves The following classical curves have...Ch. 4.4 - Special curves The following classical curves have...Ch. 4.4 - Prob. 78ECh. 4.4 - Prob. 79ECh. 4.5 - Verify that in the example to the right, the same...Ch. 4.5 - Find the objective function in Example 1 (in terms...Ch. 4.5 - Find the objective function in Example 2 (in terms...Ch. 4.5 - Fill in the blanks: The goal of an optimization...Ch. 4.5 - Prob. 2ECh. 4.5 - Suppose the objective function is Q = x2y and you...Ch. 4.5 - Suppose you wish to minimize a continuous...Ch. 4.5 - Suppose the objective function P = xy is subject...Ch. 4.5 - Suppose S=x+2y is an objective function subject to...Ch. 4.5 - Maximum product What two nonnegative real numbers...Ch. 4.5 - Sum of squares What two nonnegative real numbers a...Ch. 4.5 - Minimum sum What two positive real numbers whose...Ch. 4.5 - Maximum product Find numbers x and y satisfying...Ch. 4.5 - Maximum area rectangles Of all rectangles with a...Ch. 4.5 - Maximum area rectangles Of all rectangles with a...Ch. 4.5 - Minimum perimeter rectangles Of all rectangles of...Ch. 4.5 - Minimum perimeter rectangles Of all rectangles...Ch. 4.5 - Minimum sum Find positive numbers x and y...Ch. 4.5 - Pen problems a. A rectangular pen is built with...Ch. 4.5 - Rectangles beneath a semicircle A rectangle is...Ch. 4.5 - Rectangles beneath a parabola A rectangle is...Ch. 4.5 - Minimum-surface-area box Of all boxes with a...Ch. 4.5 - Maximum-volume box Suppose an airline policy...Ch. 4.5 - Shipping crates A square-based, box-shaped...Ch. 4.5 - Closest point on a line What point on the line y =...Ch. 4.5 - Closest point on a curve What point on the...Ch. 4.5 - Minimum distance Find the point P on the curve y =...Ch. 4.5 - Minimum distance Find the point P on the line y =...Ch. 4.5 - Walking and rowing A boat on the ocean is 4 mi...Ch. 4.5 - Laying cable An island is 3.5 mi from the nearest...Ch. 4.5 - Laying cable again Solve the problem in Exercise...Ch. 4.5 - Shortest ladder A 10-ft-tall fence runs parallel...Ch. 4.5 - Shortest laddermore realistic An 8-ft-tall fence...Ch. 4.5 - Circle and square A piece of wire of length 60 is...Ch. 4.5 - Maximum-volume cone A cone is constructed by...Ch. 4.5 - Prob. 34ECh. 4.5 - Optimal soda can a. Classical problem Find the...Ch. 4.5 - Covering a marble Imagine a flat-bottomed...Ch. 4.5 - Optimal garden A rectangular flower garden with an...Ch. 4.5 - Rectangles beneath a line a. A rectangle is...Ch. 4.5 - Designing a box Two squares of length x are cut...Ch. 4.5 - Folded boxes a. Squares with sides of length x are...Ch. 4.5 - A window consists of rectangular pane of glass...Ch. 4.5 - Light transmission A window consists of a...Ch. 4.5 - Keplers wine barrel Several mathematical stories...Ch. 4.5 - Blood testing Suppose a blood test for a disease...Ch. 4.5 - Maximum-volume cylinder in a sphere Find the...Ch. 4.5 - Maximizing profit Suppose you own a tour bus and...Ch. 4.5 - Cone in a cone A right circular cone is inscribed...Ch. 4.5 - Cylinder in a sphere Find the height h, radius r,...Ch. 4.5 - Travel costs A simple model for travel costs...Ch. 4.5 - Do dogs know calculus? A mathematician stands on a...Ch. 4.5 - Viewing angles An auditorium with a flat floor has...Ch. 4.5 - Suspension system A load must be suspended 6 m...Ch. 4.5 - Light sources The intensity of a light source at a...Ch. 4.5 - Basketball shot A basketball is shot with an...Ch. 4.5 - Fermats Principle a. Two poles of heights m and n...Ch. 4.5 - Prob. 56ECh. 4.5 - Making silos A grain silo consists of a...Ch. 4.5 - Prob. 58ECh. 4.5 - Minimizing related functions Complete each of the...Ch. 4.5 - Searchlight problemnarrow beam A searchlight is...Ch. 4.5 - Metal rain gutters A rain gutter is made from...Ch. 4.5 - Gliding mammals Many species of small mammals...Ch. 4.5 - Watching a Ferris wheel An observer stands 20 m...Ch. 4.5 - Crease-length problem A rectangular sheet of paper...Ch. 4.5 - Crankshaft A crank of radius r rotates with an...Ch. 4.5 - Maximum angle Find the value of x that maximizes ...Ch. 4.5 - Sum of isosceles distances a. An isosceles...Ch. 4.5 - Cylinder and cones (Putnam Exam 1938) Right...Ch. 4.5 - Slowest shortcut Suppose you are standing in a...Ch. 4.5 - Rectangles in triangles Find the dimensions and...Ch. 4.5 - Prob. 71ECh. 4.5 - Another pen problem A rancher is building a horse...Ch. 4.5 - Minimum-length roads A house is located at each...Ch. 4.5 - The arbelos An arbelos is the region enclosed by...Ch. 4.5 - Prob. 75ECh. 4.5 - Turning a corner with a pole a. What is the length...Ch. 4.5 - Tree notch (Putnam Exam 1938, rephrased) A notch...Ch. 4.5 - Prob. 78ECh. 4.5 - A challenging pen problem A farmer uses 200 meters...Ch. 4.5 - Prob. 80ECh. 4.6 - Sketch the graph of a function f that is concave...Ch. 4.6 - In Example 1, suppose you travel one mile in 75...Ch. 4.6 - Prob. 3QCCh. 4.6 - Prob. 4QCCh. 4.6 - Prob. 5QCCh. 4.6 - Sketch the graph of a smooth function f and label...Ch. 4.6 - Suppose you find the linear approximation to a...Ch. 4.6 - How is linear approximation used to approximate...Ch. 4.6 - How can linear approximation be used to...Ch. 4.6 - Suppose f is differentiable on (,),f(1)=2, and...Ch. 4.6 - Suppose f is differentiable on (,) and the...Ch. 4.6 - Linear approximation Estimate f(3.85) given that...Ch. 4.6 - Linear approximation Estimate f(5.1) given that...Ch. 4.6 - Given a function f that is differentiable on its...Ch. 4.6 - Does the differential dy represent the change in f...Ch. 4.6 - Suppose f is differentiable on (,),...Ch. 4.6 - Suppose f is differentiable on (,), f(5.99)=7 and...Ch. 4.6 - Estimating speed Use the linear approximation...Ch. 4.6 - Prob. 14ECh. 4.6 - Estimating time Suppose you want to travel D miles...Ch. 4.6 - Prob. 16ECh. 4.6 - Estimating time Suppose you want to travel D miles...Ch. 4.6 - Estimating time Suppose you want to travel D miles...Ch. 4.6 - Linear approximation Find the linear approximation...Ch. 4.6 - Linear approximation Find the linear approximation...Ch. 4.6 - Linear approximation Find the linear approximation...Ch. 4.6 - Linear approximation Find the linear approximation...Ch. 4.6 - Linear approximation Find the linear approximation...Ch. 4.6 - Linear approximation Find the linear approximation...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Linear approximation a.Write the equation of the...Ch. 4.6 - Estimations with linear approximation Use linear...Ch. 4.6 - Estimations with linear approximation Use linear...Ch. 4.6 - Estimations with linear approximation Use linear...Ch. 4.6 - Estimations with linear approximation Use linear...Ch. 4.6 - Estimations with linear approximation Use linear...Ch. 4.6 - Estimations with linear approximation Use linear...Ch. 4.6 - Estimations with linear approximation Use linear...Ch. 4.6 - Estimations with linear approximation Use linear...Ch. 4.6 - Estimations with linear approximation Use linear...Ch. 4.6 - Prob. 46ECh. 4.6 - Linear approximation and concavity Carry out the...Ch. 4.6 - Linear approximation and concavity Carry out the...Ch. 4.6 - Prob. 49ECh. 4.6 - Linear approximation and concavity Carry out the...Ch. 4.6 - Prob. 51ECh. 4.6 - Ideal Gas Law The pressure P, temperature T, and...Ch. 4.6 - Explain why or why not Determine whether the...Ch. 4.6 - Prob. 54ECh. 4.6 - Approximating changes 35. Approximate the change...Ch. 4.6 - Approximating changes 36. Approximate the change...Ch. 4.6 - Approximating changes 37. Approximate the change...Ch. 4.6 - Approximating changes 38. Approximate the change...Ch. 4.6 - Approximating changes 39. Approximate the change...Ch. 4.6 - Approximating changes 40. Approximate the change...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Differentials Consider the following functions and...Ch. 4.6 - Prob. 71ECh. 4.6 - Errors in approximations Suppose f(x) = 1/(1 + x)...Ch. 4.6 - Prob. 73ECh. 4.7 - Which of the following functions lead to an...Ch. 4.7 - Prob. 2QCCh. 4.7 - What is the form of the limit limx/2(x/2)(tanx)?...Ch. 4.7 - Explain why a limit of the form 0 is not an...Ch. 4.7 - Before proceeding, use your intuition and rank...Ch. 4.7 - Compare the growth rates of f(x)=x2 and g(x)=x3 as...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 - Give examples of each of the following. a.A limit...Ch. 4.7 - Give examples of each of the following. a. A limit...Ch. 4.7 - Which of the following limits can be evaluated...Ch. 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 - Prob. 9ECh. 4.7 - Evaluate limx2x33x2+2xx2 using lHpitals Rule and...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 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...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. 23ECh. 4.7 - / form Evaluate the following limits. 38....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 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Prob. 29ECh. 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 - / form Evaluate the following limits. 39....Ch. 4.7 - Prob. 38ECh. 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. 41ECh. 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 - / form Evaluate the following limits. 41....Ch. 4.7 - / form Evaluate the following limits. 42....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. 51ECh. 4.7 - Miscellaneous limits by any means Use analytical...Ch. 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 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - form Evaluate the following limits. 51....Ch. 4.7 - form Evaluate the following limits. 53....Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Prob. 64ECh. 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 Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...Ch. 4.7 - Limits Evaluate the following limits. Use lHpitals...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 - Two methods Evaluate the following limits in two...Ch. 4.7 - Two methods Evaluate the following limits in two...Ch. 4.7 - More limits Evaluate the following limits. 88....Ch. 4.7 - More limits Evaluate the following limits. 89....Ch. 4.7 - More limits Evaluate the following limits. 90....Ch. 4.7 - 88-94. More limits Evaluate the following...Ch. 4.7 - More limits Evaluate the following limits. 92....Ch. 4.7 - More limits Evaluate the following limits. 93....Ch. 4.7 - More limits Evaluate the following limits. 94....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 - Comparing growth rates Use limit methods to...Ch. 4.7 - Prob. 102ECh. 4.7 - Comparing growth rates Use limit methods to...Ch. 4.7 - Prob. 104ECh. 4.7 - Explain why or why not Determine whether the...Ch. 4.7 - Graphing functions Make a complete graph of the...Ch. 4.7 - Graphing functions Make a complete graph of the...Ch. 4.7 - Graphing functions Make a complete graph of the...Ch. 4.7 - Graphing functions Make a complete graph of the...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 - Prob. 118ECh. 4.7 - Exponentials vs. super exponentials Show that xx...Ch. 4.7 - Exponential growth rates a. For what values of b ...Ch. 4.8 - Verity that setting y = 0 in the equation...Ch. 4.8 - What happens if you apply Newtons method to the...Ch. 4.8 - Give a geometric explanation of Newtons method.Ch. 4.8 - Prob. 2ECh. 4.8 - A graph of f and the lines tangent to f at x = 1,...Ch. 4.8 - A graph of f and the lines tangent to f at x = 3,...Ch. 4.8 - Let f(x)=2x36x2+4x. Use Newtons method to find x1...Ch. 4.8 - The function f(x)=4xx2+4 is differentiable and has...Ch. 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 For the given...Ch. 4.8 - Finding roots with Newtons method For the given...Ch. 4.8 - Finding roots with Newtons method For the given...Ch. 4.8 - Finding roots with Newtons method For the given...Ch. 4.8 - Finding roots with Newtons method For the given...Ch. 4.8 - Finding roots with Newtons method For the given...Ch. 4.8 - Finding roots with Newtons method For the given...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 - Finding intersection points Use Newtons method to...Ch. 4.8 - Prob. 28ECh. 4.8 - Finding intersection points Use Newtons method to...Ch. 4.8 - Prob. 30ECh. 4.8 - Finding intersection points Use Newtons method to...Ch. 4.8 - Prob. 32ECh. 4.8 - Prob. 33ECh. 4.8 - Prob. 34ECh. 4.8 - Prob. 35ECh. 4.8 - Investment problem A one-time investment of 2500...Ch. 4.8 - Applications 45. A damped oscillator The...Ch. 4.8 - The sinc function The sinc function, sinc(x)=sinxx...Ch. 4.8 - Estimating roots The values of various roots can...Ch. 4.8 - Prob. 40ECh. 4.8 - Prob. 41ECh. 4.8 - Prob. 42ECh. 4.8 - Prob. 43ECh. 4.8 - Newtons method and curve sketching Use Newtons...Ch. 4.8 - Newtons method and curve sketching Use Newtons...Ch. 4.8 - Prob. 46ECh. 4.8 - Prob. 47ECh. 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 - Pitfalls of Newtons method Let f(x)=x1+x2, which...Ch. 4.8 - Prob. 53ECh. 4.8 - Approximating square roots Let a 0 be given and...Ch. 4.8 - Prob. 55ECh. 4.8 - Prob. 56ECh. 4.8 - An eigenvalue problem A certain kind of...Ch. 4.8 - Prob. 58ECh. 4.8 - Prob. 59ECh. 4.8 - Prob. 60ECh. 4.9 - Verify by differentiation that x4 is an...Ch. 4.9 - Find the family of antiderivatives for each of...Ch. 4.9 - Use differentiation to verify result 6 in Table...Ch. 4.9 - Prob. 4QCCh. 4.9 - Position is an antiderivative of velocity. But...Ch. 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 - Give the antiderivatives of a/1x2, where a is a...Ch. 4.9 - Give the antiderivatives of 1/x.Ch. 4.9 - Evaluate acosxdxand asinxdx, where a is a...Ch. 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 - 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 - 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 - Finding antiderivatives Find all the...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 - 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 involving trigonometric...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals involving trigonometric...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 - Other indefinite integrate 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 - Other indefinite integrate Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Other indefinite integrate Determine the following...Ch. 4.9 - Other indefinite integrate Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Miscellaneous indefinite integrals Determine the...Ch. 4.9 - Miscellaneous indefinite integrals Determine the...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Indefinite integrals Determine the following...Ch. 4.9 - Miscellaneous indefinite integrals Determine the...Ch. 4.9 - Particular antiderivatives For the following...Ch. 4.9 - Particular antiderivatives For the following...Ch. 4.9 - Particular antiderivatives For the following...Ch. 4.9 - Particular antiderivatives For the following...Ch. 4.9 - Particular antiderivatives For the following...Ch. 4.9 - Particular antiderivatives For the following...Ch. 4.9 - Particular antiderivatives For the following...Ch. 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 - 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 - 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 - Solving initial value problems Find the solution...Ch. 4.9 - Graphing general solutions Graph several functions...Ch. 4.9 - Prob. 88ECh. 4.9 - Graphing general solutions Graph several functions...Ch. 4.9 - Graphing general solutions Graph several functions...Ch. 4.9 - Velocity to position Given the following velocity...Ch. 4.9 - Velocity to position Given the following velocity...Ch. 4.9 - Velocity to position Given the following velocity...Ch. 4.9 - Velocity to position Given the following velocity...Ch. 4.9 - Velocity to position Given the following velocity...Ch. 4.9 - Velocity to position Given the following velocity...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 - Acceleration to position Given the following...Ch. 4.9 - Acceleration to position Given the following...Ch. 4.9 - Prob. 102ECh. 4.9 - A car starting at rest accelerates at 16 ft/s2-for...Ch. 4.9 - Prob. 104ECh. 4.9 - Races The velocity function and initial position...Ch. 4.9 - Prob. 106ECh. 4.9 - Motion with gravity Consider the following...Ch. 4.9 - Motion with gravity Consider the following...Ch. 4.9 - Motion with gravity Consider the following...Ch. 4.9 - Motion with gravity Consider the following...Ch. 4.9 - Explain why or why not Determine whether the...Ch. 4.9 - Prob. 112ECh. 4.9 - Functions from higher derivatives Find the...Ch. 4.9 - Functions from higher derivatives Find the...Ch. 4.9 - Prob. 115ECh. 4.9 - Prob. 116ECh. 4.9 - How rate A large tank is filled with water when an...Ch. 4.9 - Prob. 118ECh. 4.9 - Verifying indefinite integrals Verify the...Ch. 4.9 - Prob. 120ECh. 4.9 - Prob. 121ECh. 4.9 - Prob. 122ECh. 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 - Use the graphs of f and f to complete the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Critical points Find the critical points of the...Ch. 4 - Absolute values Consider the function f(x) = |x ...Ch. 4 - Use f and f to complete parts (a) and (b). a. Find...Ch. 4 - Use f and f to complete parts (a) and (b). a.Find...Ch. 4 - Use f and f to complete parts (a) and (b). a.Find...Ch. 4 - Inflection points Does f(x) = 2x5 10x4 + 20x3 + x...Ch. 4 - Does f(x)=x62+5x4415x2 have any inflection points?...Ch. 4 - Identify the critical points and the inflection...Ch. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Prob. 32RECh. 4 - Curve sketching Use the guidelines given in...Ch. 4 - Prob. 34RECh. 4 - Optimal popcorn box A small popcorn box is created...Ch. 4 - Minimizing time Hannah is standing on the edge of...Ch. 4 - Minimizing sound intensity Two sound speakers are...Ch. 4 - Hockey problem A hockey player skates on a line...Ch. 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 For the function f(x)=10x and...Ch. 4 - Mean Value Theorem Explain why the Mean Value...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. 56RECh. 4 - Newtons method Use Newtons method to approximate...Ch. 4 - Two methods Evaluate the following limits in two...Ch. 4 - Two methods Evaluate the following limits in two...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. 68RECh. 4 - Prob. 69RECh. 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. 73RECh. 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. 78RECh. 4 - Prob. 79RECh. 4 - Prob. 80RECh. 4 - Limits Evaluate the following limits. Use lHpitals...Ch. 4 - Comparing growth rates Determine which of the two...Ch. 4 - Prob. 83RECh. 4 - Prob. 84RECh. 4 - Comparing growth rates Determine which of the two...Ch. 4 - Comparing growth rates Determine which of the two...Ch. 4 - Prob. 87RECh. 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 - Indefinite integrals Determine the following...Ch. 4 - Indefinite integrals Determine the following...Ch. 4 - Indefinite integrals Determine the following...Ch. 4 - Indefinite integrals Determine the following...Ch. 4 - Indefinite integrals Determine the following...Ch. 4 - Prob. 97RECh. 4 - Indefinite integrals Determine the following...Ch. 4 - Prob. 99RECh. 4 - Indefinite integrals Determine the following...Ch. 4 - Prob. 101RECh. 4 - Prob. 102RECh. 4 - Prob. 103RECh. 4 - Prob. 104RECh. 4 - Prob. 105RECh. 4 - Functions from derivatives Find the function f...Ch. 4 - Functions from derivatives Find the function f...Ch. 4 - Prob. 108RECh. 4 - Prob. 109RECh. 4 - Distance traveled A car starting at rest...Ch. 4 - Prob. 111RECh. 4 - Logs of logs Compare the growth rates of ln x, ln...Ch. 4 - Prob. 113RECh. 4 - Prob. 114RECh. 4 - Prob. 115RECh. 4 - Prob. 116RECh. 4 - Prob. 117RECh. 4 - A family of super-exponential functions Let f(x) =...
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- Provethat a) prove that for any irrational numbers there exists? asequence of rational numbers Xn converg to S. b) let S: RR be a sunctions-t. f(x)=(x-1) arc tan (x), xe Q 3(x-1) 1+x² x&Q Show that lim f(x)= 0 14x C) For any set A define the set -A=yarrow_forwardQ2: Find the interval and radius of convergence for the following series: Σ n=1 (-1)η-1 xn narrow_forward8. Evaluate arctan x dx a) xartanx 2 2 In(1 + x²) + C b) xartanx + 1½-3ln(1 + x²) + C c) xartanx + In(1 + x²) + C d) (arctanx)² + C 2 9) Evaluate Inx³ dx 3 a) +C b) ln x² + C c)¾½ (lnx)² d) 3x(lnx − 1) + C - x 10) Determine which integral is obtained when the substitution x = So¹² √1 - x²dx sine is made in the integral πT π π a) √ sin cos e de b) √ cos² de c) c Ꮎ Ꮎ cos² 0 de c) cos e de d) for cos² e de πT 11. Evaluate tan³xdx 1 a) b) c) [1 - In 2] 2 2 c) [1 − In2] d)½½[1+ In 2]arrow_forward12. Evaluate ſ √9-x2 -dx. x2 a) C 9-x2 √9-x2 - x2 b) C - x x arcsin ½-½ c) C + √9 - x² + arcsin x d) C + √9-x2 x2 13. Find the indefinite integral S cos³30 √sin 30 dᎾ . 2√√sin 30 (5+sin²30) √sin 30 (3+sin²30) a) C+ √sin 30(5-sin²30) b) C + c) C + 5 5 5 10 d) C + 2√√sin 30 (3-sin²30) 2√√sin 30 (5-sin²30) e) C + 5 15 14. Find the indefinite integral ( sin³ 4xcos 44xdx. a) C+ (7-5cos24x)cos54x b) C (7-5cos24x)cos54x (7-5cos24x)cos54x - 140 c) C - 120 140 d) C+ (7-5cos24x)cos54x e) C (7-5cos24x)cos54x 4 4 15. Find the indefinite integral S 2x2 dx. ex - a) C+ (x²+2x+2)ex b) C (x² + 2x + 2)e-* d) C2(x²+2x+2)e¯* e) C + 2(x² + 2x + 2)e¯* - c) C2x(x²+2x+2)e¯*arrow_forward4. Which substitution would you use to simplify the following integrand? S a) x = sin b) x = 2 tan 0 c) x = 2 sec 3√√3 3 x3 5. After making the substitution x = = tan 0, the definite integral 2 2 3 a) ៖ ស្លឺ sin s π - dᎾ 16 0 cos20 b) 2/4 10 cos 20 π sin30 6 - dᎾ c) Π 1 cos³0 3 · de 16 0 sin20 1 x²√x²+4 3 (4x²+9)2 π d) cos²8 16 0 sin³0 dx d) x = tan 0 dx simplifies to: de 6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy? a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor 7. Evaluate 3x x+4 - dx 1 a) 3x+41nx + 4 + C b) 31n|x + 4 + C c) 3 ln x + 4+ C d) 3x - 12 In|x + 4| + C x+4arrow_forward1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.arrow_forward2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?arrow_forwardQuestion 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.arrow_forwardPlease find all values of x.arrow_forward3. Consider the initial value problem 9y" +12y' + 4y = 0, y(0) = a>0: y′(0) = −1. Solve the problem and find the value of a such that the solution of the initial value problem is always positive.arrow_forward5. Euler's equation. Determine the values of a for which all solutions of the equation 5 x²y" + axy' + y = 0 that have the form (A + B log x) x* or Ax¹¹ + Bä” tend to zero as a approaches 0.arrow_forward4. Problem on variable change. The purpose of this problem is to perform an appropriate change of variables in order to reduce the problem to a second-order equation with constant coefficients. ty" + (t² − 1)y'′ + t³y = 0, 0arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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