 4.10PE: Extreme Valuesa. Give an example of a differentiable function ƒ who...
 4.135PE: Applications and ExamplesGraph Explain what you see.
 4.10QGY: What is an inflection point? Give an example. What physical signifi...
 4.27PE: Theory and ExamplesAssume that the brakes of an automobile produce ...
 4.38PE: Graphs and GraphingGraph the curves
 4.28AAE: Theory and ExamplesLet ƒ(x), g(x) be two continuously differentiabl...
 4.28PE: Graphs and GraphingGraph the curves
 4.39PE: Graphs and GraphingGraph the curves
 4.70PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.88PE: OptimizationA customer has asked you to design an opentop rectangu...
 4.89PE: OptimizationFind the height and radius of the largest right circula...
 4.71PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.90PE: OptimizationThe figure here shows two right circular cones, one ups...
 4.72PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.123PE: Initial Value Solve the initial value problems.
 4.15PE: The Mean Value Theorema. Show that decreases on every interval in i...
 4.15AAE: OptimizationHole in a water tank You want to bore a hole in the sid...
 4.15QGY: List the steps you would take to graph a rational function. Illustr...
 4.35AAE: Theory and ExamplesThe best branching angles for blood vessels and ...
 4.35PE: Graphs and GraphingGraph the curves
 4.36PE: Graphs and GraphingGraph the curves
 4.1AAE: Functions and DerivativesWhat can you say about a function whose ma...
 4.1PE: Extreme ValuesDoes have any local maximum or minimum values? Give r...
 4.85PE: OptimizationThe sum of two nonnegative numbers is 36. Find the numb...
 4.1QGY: What can be said about the extreme values of a function that is con...
 4.86PE: OptimizationThe sum of two nonnegative numbers is 20. Find the numb...
 4.87PE: OptimizationAn isosceles triangle has its vertex at the origin and ...
 4.18QGY: How can you sometimes handle limits that lead to indeterminate form...
 4.136PE: Applications and ExamplesA round underwater transmission cable cons...
 4.43PE: Graphs and GraphingEach of exercise gives the first derivative of a...
 4.18AAE: OptimizationMinimizing a parameter Find the smallest value of the p...
 4.18PE: The Mean Value Theorema. Show that increases on every interval in i...
 4.44PE: Graphs and GraphingEach of exercise gives the first derivative of a...
 4.45PE: Graphs and GraphingEach of exercise gives the first derivative of a...
 4.95PE: Newton’s MethodLet Show that the equation ƒ(x) = 4 has a solution ...
 4.94PE: OptimizationThe ladder problem What is the approximate length (in f...
 4.16AAE: OptimizationKicking a field goal An American football player wants ...
 4.96PE: Newton’s MethodLet Show that the equation ƒ(x) = 75 has a solution ...
 4.21AAE: Theory and ExamplesSuppose that it costs a company y = a + bx dolla...
 4.16QGY: Outline a general strategy for solving maxmin problems. Give examp...
 4.37PE: Graphs and GraphingGraph the curves
 4.16PE: The Mean Value Theorema. Show that increases on every interval in i...
 4.9AAE: The derivative of but ƒ is not a constant function. Doesn’t this co...
 4.9PE: Extreme ValuesThe greatest integer function defined for all values ...
 4.9QGY: How do you test a twicedifferentiable function to determine where ...
 4.26AAE: Theory and ExamplesGroup blood testing During World War II it was n...
 4.7QGY: How can you sometimes identify a function ƒ(x) by knowing and knowi...
 4.24QGY: What is an initial value problem? How do you solve one? Give an exa...
 4.27AAE: Theory and ExamplesAssume that the brakes of an automobile produce ...
 4.13AAE: OptimizationLargest inscribed triangle Points A and B lie at the en...
 4.13PE: Extreme ValuesA graph that is large enough to show a function’s glo...
 4.13QGY: List the steps you would take to graph a polynomial function. Illus...
 4.32AAE: Theory and ExamplesA particle moves with acceleration Assuming that...
 4.32PE: Graphs and GraphingGraph the curves
 4.3AAE: Functions and DerivativesCan you conclude anything about the extrem...
 4.33AAE: Theory and ExamplesGiven By considering the minimum, prove that for...
 4.3PE: Extreme ValuesDoes have an absolute minimum value? An absolute maxi...
 4.3QGY: How do you find the absolute extrema of a continuous function on a ...
 4.20AAE: LimitsL’Hôpital’s Rule does not help with the following limits. Fin...
 4.79PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.20PE: The Mean Value TheoremThe formula gives a different function for ea...
 4.80PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.6PE: Extreme ValuesDoes have an absolute minimum value? An absolute maxi...
 4.20QGY: Describe Newton’s method for solving equations. Give an example. Wh...
 4.81PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.23AAE: Theory and ExamplesTo find we apply Newton’s method to Here we assu...
 4.130PE: Applications and ExamplesFind the absolute maximum and minimum valu...
 4.49PE: Graphs and GraphingGraph each function. Then use the function’s fir...
 4.131PE: Applications and ExamplesFind the absolute maxima and minima of the...
 4.50PE: Graphs and GraphingGraph each function. Then use the function’s fir...
 4.132PE: Applications and ExamplesFind the absolute maxima and minima of the...
 4.51PE: Graphs and GraphingGraph each function. Then use the function’s fir...
 4.11AAE: In either case, if the answer is yes, give a proof. If the answer i...
 4.100PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.101PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.102PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.17AAE: OptimizationA maxmin problem with a variable answer Sometimes the ...
 4.17PE: The Mean Value Theorema. Show that the equation has exactly one sol...
 4.17QGY: Describe l’Hôpital’s Rule. How do you know when to use the rule and...
 4.40PE: Graphs and GraphingGraph the curves
 4.41PE: Graphs and GraphingGraph the curves
 4.91PE: OptimizationManufacturing tires Your company can manufacture x hund...
 4.42PE: Graphs and GraphingGraph the curves
 4.92PE: OptimizationParticle motion The positions of two particles on the s...
 4.93PE: OptimizationOpentop box An opentop rectangular box is constructed...
 4.5AAE: Local extremaa. Suppose that the first derivative of is At what poi...
 4.5PE: Extreme ValuesDoes have an absolute minimum value? An absolute maxi...
 4.5QGY: What are the hypotheses and conclusion of the Mean Value Theorem? W...
 4.22AAE: Theory and ExamplesEstimating reciprocals without division You can ...
 4.22PE: The Mean Value TheoremCalculate the first derivatives of and What c...
 4.2AAE: Functions and DerivativesIs it true that a discontinuous function c...
 4.22QGY: What is an indefinite integral? How do you evaluate one? What gener...
 4.55PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.2PE: Extreme ValuesDoes have any local maximum values? Give reasons for ...
 4.2QGY: What does it mean for a function to have a local extreme value on i...
 4.56PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.57PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.19AAE: LimitsEvaluate the following limits.
 4.106PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.19PE: The Mean Value TheoremWater in a reservoir As a result of a heavy r...
 4.19QGY: How can you sometimes handle limits that lead to indeterminate form...
 4.107PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.46PE: Graphs and GraphingEach of exercise gives the first derivative of a...
 4.12AAE: In either case, if the answer is yes, give a proof. If the answer i...
 4.108PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.47PE: Graphs and GraphingEach of exercise gives the first derivative of a...
 4.12PE: Extreme ValuesWhat are the maximum and minimum values of the functi...
 4.12QGY: What do the derivatives of a function tell you about the shape of i...
 4.48PE: Graphs and GraphingEach of exercise gives the first derivative of a...
 4.30PE: Graphs and GraphingGraph the curves
 4.14AAE: OptimizationProving the second derivative test The Second Derivativ...
 4.97PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.14PE: Extreme Values(Continuation of Exercise 13.)a.Graph over the interv...
 4.31AAE: Theory and ExamplesA particle moves along the xaxis. Its accelerat...
 4.14QGY: What is a cusp? Give examples.
 4.98PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.99PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.31PE: Graphs and GraphingGraph the curves
 4.33PE: Graphs and GraphingGraph the curves
 4.76PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.77PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.34AAE: Theory and ExamplesSchwarz’s inequalitya. In Exercise 33, let and d...
 4.78PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.127PE: Applications and ExamplesThe rectangle shown here has one side on t...
 4.82PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.83PE: Checking Antiderivative FormulasRight, or wrong? Say which for each...
 4.8QGY: What is the First Derivative Test for Local Extreme Values? Give ex...
 4.128PE: Applications and ExamplesThe rectangle shown here has one side on t...
 4.34PE: Graphs and GraphingGraph the curves
 4.8PE: Extreme ValuesFind the absolute maximum and absolute minimum values...
 4.129PE: Applications and ExamplesFind the absolute maximum and minimum valu...
 4.25PE: Analyzing GraphsEach of the graphs in Exercises 25 and 26 is the gr...
 4.25AAE: Theory and ExamplesFree fall in the fourteenth century In the middl...
 4.84PE: Checking Antiderivative FormulasRight, or wrong? Say which for each...
 4.133PE: Applications and ExamplesGraph the following functions and use what...
 4.25QGY: If you know the acceleration of a body moving along a coordinate li...
 4.10AAE: Extrema and inflection points Let h = ƒg be the product of two diff...
 4.134PE: Applications and ExamplesGraph Does the function appear to have an ...
 4.4AAE: Functions and DerivativesLocal extrema Use the sign pattern for the...
 4.4PE: Extreme ValuesFind values of a and b such that the function has a l...
 4.4QGY: What are the hypotheses and conclusion of Rolle’s Theorem? Are the ...
 4.6AAE: If for all x, what is the most the values of ƒ can increase on [0, ...
 4.6QGY: State the Mean Value Theorem’s three corollaries.
 4.7AAE: Bounding a function Suppose that ƒ is continuous on [a, b] and that...
 4.7PE: Extreme ValuesFind the absolute maximum and absolute minimum values...
 4.8AAE: An inequalitya. Show that for every value of x.b. Suppose that ƒ is...
 4.11PE: Extreme ValuesThe function does not take on either a maximum or a m...
 4.11QGY: What is the Second Derivative Test for Local Extreme Values? Give e...
 4.21PE: The Mean Value TheoremShow that even though Doesn’t this contradict...
 4.21QGY: Can a function have more than one antiderivative? If so, how are th...
 4.23PE: Analyzing GraphsUse the graph to answer the questions.Identify any ...
 4.23QGY: How can you sometimes solve a differential equation of the form
 4.24AAE: Theory and ExamplesThe family of straight lines y = ax + b (a, b ar...
 4.24PE: Analyzing GraphsEstimate the intervals on which the function isa. i...
 4.26PE: Analyzing GraphsEach of the graphs in Exercises 25 and 26 is the gr...
 4.29AAE: Theory and ExamplesCan there be a curve satisfying the following co...
 4.29PE: Graphs and GraphingGraph the curves
 4.30AAE: Theory and ExamplesFind the equation for the curve in the xyplane ...
 4.52PE: Graphs and GraphingGraph each function. Then use the function’s fir...
 4.53PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.54PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.58PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.59PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.60PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.61PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.62PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.63PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.64PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.65PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.66PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.67PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.68PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.69PE: Finding Indefinite IntegralsFind the most general antiderivative or...
 4.73PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.74PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.75PE: Checking Antiderivative FormulasVerify the formulas by differentiation
 4.103PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.104PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.105PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.109PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.110PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.111PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.112PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.113PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.114PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.115PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.116PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.117PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.118PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.119PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.120PE: Finding Indefinite IntegralsFind the indefinite integrals (most gen...
 4.121PE: Initial Value Solve the initial value problems.
 4.122PE: Initial Value Solve the initial value problems.
 4.124PE: Initial Value Solve the initial value problems.
 4.125PE: Applications and ExamplesCan the integrations in (a) and (b) both b...
 4.126PE: Applications and ExamplesCan the integrations in (a) and (b) both b...
Solutions for Chapter 4: Thomas' Calculus: Early Transcendentals 13th Edition
Full solutions for Thomas' Calculus: Early Transcendentals  13th Edition
ISBN: 9780321884077
Solutions for Chapter 4
Get Full SolutionsThomas' Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321884077. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4 includes 196 full stepbystep solutions. Since 196 problems in chapter 4 have been answered, more than 43690 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Thomas' Calculus: Early Transcendentals , edition: 13th.

Arctangent function
See Inverse tangent function.

Cube root
nth root, where n = 3 (see Principal nth root),

Directed distance
See Polar coordinates.

Elimination method
A method of solving a system of linear equations

Equilibrium point
A point where the supply curve and demand curve intersect. The corresponding price is the equilibrium price.

Focal length of a parabola
The directed distance from the vertex to the focus.

Focal width of a parabola
The length of the chord through the focus and perpendicular to the axis.

Inverse cosine function
The function y = cos1 x

Irrational zeros
Zeros of a function that are irrational numbers.

Linear equation in x
An equation that can be written in the form ax + b = 0, where a and b are real numbers and a Z 0

Logarithmic regression
See Natural logarithmic regression

nset
A set of n objects.

Origin
The number zero on a number line, or the point where the x and yaxes cross in the Cartesian coordinate system, or the point where the x, y, and zaxes cross in Cartesian threedimensional space

Phase shift
See Sinusoid.

Positive association
A relationship between two variables in which higher values of one variable are generally associated with higher values of the other variable, p. 717.

Quadratic function
A function that can be written in the form ƒ(x) = ax 2 + bx + c, where a, b, and c are real numbers, and a ? 0.

Randomization
The principle of experimental design that makes it possible to use the laws of probability when making inferences.

Upper bound for ƒ
Any number B for which ƒ(x) ? B for all x in the domain of ƒ.

Vertex form for a quadratic function
ƒ(x) = a(x  h)2 + k

yaxis
Usually the vertical coordinate line in a Cartesian coordinate system with positive direction up, pp. 12, 629.