Calculus: Early Transcendentals, Books A La Carte Edition (3rd Edition)
Calculus: Early Transcendentals, Books A La Carte Edition (3rd Edition)
3rd Edition
ISBN: 9780134770512
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Concept explainers

Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
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Chapter 4, Problem 18RE

Use f′ and f″ to complete parts (a) and (b).

a. Find the intervals on which f is increasing and the intervals on which it is decreasing.

b. Find the intervals on which f is concave up and the intervals on which it is concave down.

18. f ( x ) = e x 1 + e x

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Chapter 4 Solutions

Calculus: Early Transcendentals, Books A La Carte Edition (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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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY