 3.20AAE: Even differentiable functions Is there anything special about the d...
 3.1AAE: An equation like sin2 ? + cos2?= 1 is called an identity because it...
 3.1PE: Derivatives of FunctionsFind the derivatives of the functions
 3.1QGY: What is the derivative of a function ƒ? How is its domain related t...
 3.2AAE: If the identity sin (x + a) = sin x cos a + cos x sin ais different...
 3.2PE: Derivatives of FunctionsFind the derivatives of the functions
 3.2QGY: What role does the derivative play in defining slopes, tangents, an...
 3.3AAE: a. Find values for the constants a, b, and c that will make satisfy...
 3.3PE: Derivatives of FunctionsFind the derivatives of the functions
 3.3QGY: How can you sometimes graph the derivative of a function when all y...
 3.4AAE: Solutions to differential equationsa. Show that y = sin x, y = cos ...
 3.4PE: Derivatives of FunctionsFind the derivatives of the functions
 3.4QGY: What does it mean for a function to be differentiable on an open in...
 3.5AAE: An osculating circle Find the values of h, k, and a that make the c...
 3.5PE: Derivatives of FunctionsFind the derivatives of the functions
 3.5QGY: How are derivatives and onesided derivatives related?
 3.6AAE: Marginal revenue A bus will hold 60 people. The number x of people ...
 3.6PE: Derivatives of FunctionsFind the derivatives of the functions
 3.6QGY: Describe geometrically when a function typically does not have a de...
 3.7AAE: Industrial productiona. Economists often use the expression “rate o...
 3.7PE: Derivatives of FunctionsFind the derivatives of the functions
 3.7QGY: How is a function’s differentiability at a point related to its con...
 3.8AAE: Designing a gondola The designer of a 30ftdiameter spherical hot ...
 3.8PE: Derivatives of FunctionsFind the derivatives of the functions
 3.8QGY: What rules do you know for calculating derivatives? Give some examp...
 3.9AAE: Pisa by parachute On August 5, 1988, Mike McCarthy of London jumped...
 3.9PE: Derivatives of FunctionsFind the derivatives of the functions
 3.9QGY: Explain how the three formulas enable us to differentiate any polyn...
 3.10AAE: Motion of a particle The position at time t ? 0of a particle moving...
 3.10PE: Derivatives of FunctionsFind the derivatives of the functions
 3.10QGY: What formula do we need, in addition to the three listed in Questio...
 3.11AAE: Shooting a paper clip On Earth, you can easily shoot a paper clip 6...
 3.11PE: Derivatives of FunctionsFind the derivatives of the functions
 3.11QGY: What is a second derivative? A third derivative? How many derivativ...
 3.12AAE: Velocities of two particles At time t sec, the positions of two par...
 3.12PE: Derivatives of FunctionsFind the derivatives of the functions
 3.12QGY: What is the derivative of the exponential function ex? How does the...
 3.13AAE: Velocity of a particle A particle of constant mass m moves along th...
 3.13PE: Derivatives of FunctionsFind the derivatives of the functions
 3.13QGY: What is the relationship between a function’s average and instantan...
 3.14AAE: Average and instantaneous velocitya. Show that if the position x of...
 3.14PE: Derivatives of FunctionsFind the derivatives of the functions
 3.14QGY: How do derivatives arise in the study of motion? What can you learn...
 3.15AAE: Find all values of the constants m and b for which the function isa...
 3.15PE: Derivatives of FunctionsFind the derivatives of the functions
 3.15QGY: How can derivatives arise in economics?
 3.16AAE: Does the function have a derivative at x = 0? Explain.
 3.16PE: Derivatives of FunctionsFind the derivatives of the functions
 3.16QGY: Give examples of still other applications of derivatives
 3.17AAE: a. For what values of a and b will be differentiable for all values...
 3.17PE: Derivatives of FunctionsFind the derivatives of the functions
 3.17QGY: What do the limits have to do with the derivatives of the sine and ...
 3.18AAE: a. For what values of a and b will be differentiable for all values...
 3.18PE: Derivatives of FunctionsFind the derivatives of the functions
 3.18QGY: Once you know the derivatives of sin x and cos x, how can you find ...
 3.19AAE: Odd differentiable functions Is there anything special about the de...
 3.19PE: Derivatives of FunctionsFind the derivatives of the functions
 3.19QGY: At what points are the six basic trigonometric functions continuous...
 3.20PE: Derivatives of FunctionsFind the derivatives of the functions
 3.20QGY: What is the rule for calculating the derivative of a composite of t...
 3.21AAE: Suppose that the functions ƒ and g are defined throughout an open i...
 3.21PE: Derivatives of FunctionsFind the derivatives of the functions
 3.21QGY: If u is a differentiable function of x, how do you find if n is an ...
 3.22AAE: Use the result of Exercise 21 to show that the following functions ...
 3.22PE: Derivatives of FunctionsFind the derivatives of the functions
 3.22QGY: What is implicit differentiation? When do you need it? Give examples.
 3.23AAE: Is the derivative of Give reasons for your answers.
 3.23PE: Derivatives of FunctionsFind the derivatives of the functions
 3.23QGY: What is the derivative of the natural logarithm function ln x? How ...
 3.24AAE: Suppose that a function ƒ satisfies the following conditions for al...
 3.24PE: Derivatives of FunctionsFind the derivatives of the functions
 3.24QGY: What is the derivative of the exponential function and ? What is th...
 3.25AAE: The generalized product rule Use mathematical induction to prove th...
 3.25PE: Derivatives of FunctionsFind the derivatives of the functions
 3.25QGY: What is the derivative of Are there any restrictions on a?
 3.26AAE: Leibniz’s rule for higherorder derivatives of products Leibniz’s r...
 3.26PE: Derivatives of FunctionsFind the derivatives of the functions
 3.26QGY: What is logarithmic differentiation? Give an example.
 3.27AAE: The period of a clock pendulum The period T of a clock pendulum (ti...
 3.27PE: Derivatives of FunctionsFind the derivatives of the functions
 3.27QGY: How can you write any real power of x as a power of e? Are there an...
 3.28AAE: The melting ice cube Assume that an ice cube retains its cubical sh...
 3.28PE: Derivatives of FunctionsFind the derivatives of the functions
 3.28QGY: What is one way of expressing the special number e as a limit? What...
 3.29PE: Derivatives of FunctionsFind the derivatives of the functions
 3.29QGY: What are the derivatives of the inverse trigonometric functions? Ho...
 3.30PE: Derivatives of FunctionsFind the derivatives of the functions
 3.30QGY: How do related rates problems arise? Give examples.
 3.31PE: Derivatives of FunctionsFind the derivatives of the functions
 3.31QGY: Outline a strategy for solving related rates problems. Illustrate w...
 3.32PE: Derivatives of FunctionsFind the derivatives of the functions
 3.32QGY: What is the linearization L(x) of a function ƒ(x) at a point x = a?...
 3.33PE: Derivatives of FunctionsFind the derivatives of the functions
 3.33QGY: If x moves from a to a nearby value a + dx how do you estimate the ...
 3.34PE: Derivatives of FunctionsFind the derivatives of the functions
 3.35PE: Derivatives of FunctionsFind the derivatives of the functions
 3.36PE: Derivatives of FunctionsFind the derivatives of the functions
 3.37PE: Derivatives of FunctionsFind the derivatives of the functions
 3.38PE: Derivatives of FunctionsFind the derivatives of the functions
 3.39PE: Derivatives of FunctionsFind the derivatives of the functions
 3.40PE: Derivatives of FunctionsFind the derivatives of the functions
 3.41PE: Derivatives of FunctionsFind the derivatives of the functions
 3.42PE: Derivatives of FunctionsFind the derivatives of the functions
 3.43PE: Derivatives of FunctionsFind the derivatives of the functions
 3.44PE: Derivatives of FunctionsFind the derivatives of the functions
 3.45PE: Derivatives of FunctionsFind the derivatives of the functions
 3.46PE: Derivatives of FunctionsFind the derivatives of the functions
 3.47PE: Derivatives of FunctionsFind the derivatives of the functions
 3.48PE: Derivatives of FunctionsFind the derivatives of the functions
 3.49PE: Derivatives of FunctionsFind the derivatives of the functions
 3.50PE: Derivatives of FunctionsFind the derivatives of the functions
 3.51PE: Derivatives of FunctionsFind the derivatives of the functions
 3.52PE: Derivatives of FunctionsFind the derivatives of the functions
 3.53PE: Derivatives of FunctionsFind the derivatives of the functions
 3.54PE: Derivatives of FunctionsFind the derivatives of the functions
 3.55PE: Derivatives of FunctionsFind the derivatives of the functions
 3.56PE: Derivatives of FunctionsFind the derivatives of the functions
 3.57PE: Derivatives of FunctionsFind the derivatives of the functions
 3.58PE: Derivatives of FunctionsFind the derivatives of the functions
 3.59PE: Derivatives of FunctionsFind the derivatives of the functions
 3.60PE: Derivatives of FunctionsFind the derivatives of the functions
 3.61PE: Derivatives of FunctionsFind the derivatives of the functions
 3.62PE: Derivatives of FunctionsFind the derivatives of the functions
 3.63PE: Derivatives of FunctionsFind the derivatives of the functions
 3.64PE: Derivatives of FunctionsFind the derivatives of the functions
 3.65PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.66PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.67PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.68PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.69PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.70PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.71PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.72PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.73PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.74PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.75PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.76PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.77PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.78PE: Implicit DifferentiationFind dy/dx by implicit differentiation.
 3.79PE: Implicit DifferentiationFind dp/dq.
 3.80PE: Implicit DifferentiationFind dp/dq.
 3.81PE: Implicit DifferentiationFind dr/ds.
 3.82PE: Implicit DifferentiationFind dr/ds.
 3.83PE: Implicit DifferentiationFind by implicit differentiation:
 3.84PE: Implicit Differentiationa. By differentiating implicitly, show that...
 3.85PE: Suppose that functions ƒ(x) and g(x) and their first derivatives ha...
 3.86PE: Suppose that functions ƒ(x) and g(x) and their first derivatives ha...
 3.87PE: Find the value of
 3.88PE: Find the value of
 3.89PE: Find the value of
 3.90PE: Find the value of at if and .
 3.91PE: If find the value of at the point (0, 1).
 3.92PE: If at the point (8, 8).
 3.93PE: Applying the Derivative DefinitionFind the derivative using the def...
 3.94PE: Applying the Derivative DefinitionFind the derivative using the def...
 3.95PE: Applying the Derivative Definitiona. Graph the function b. Is ƒ con...
 3.96PE: Applying the Derivative Definitiona. Graph the function b. Is ƒ con...
 3.97PE: Applying the Derivative Definitiona. Graph the function b. Is ƒ con...
 3.98PE: Applying the Derivative DefinitionFor what value or values of the c...
 3.99PE: Slopes, Tangents, and NormalsTangents with specified slope Are ther...
 3.100PE: Slopes, Tangents, and NormalsTangents with specified slope Are ther...
 3.101PE: Slopes, Tangents, and NormalsHorizontal tangents Find the points on...
 3.102PE: Slopes, Tangents, and NormalsTangent intercepts Find the x and yi...
 3.103PE: Slopes, Tangents, and NormalsTangents perpendicular or parallel to ...
 3.104PE: Slopes, Tangents, and NormalsIntersecting tangents Show that the ta...
 3.105PE: Slopes, Tangents, and NormalsNormals parallel to a line Find the po...
 3.106PE: Slopes, Tangents, and NormalsTangent and normal lines Find equation...
 3.107PE: Slopes, Tangents, and NormalsTangent parabola The parabola is to be...
 3.108PE: Slopes, Tangents, and NormalsSlope of tangent Show that the tangent...
 3.109PE: Slopes, Tangents, and NormalsTangent curve For what value of c is t...
 3.110PE: Slopes, Tangents, and NormalsNormal to a circle Show that the norma...
 3.111PE: Slopes, Tangents, and NormalsFind equations for the lines that are ...
 3.112PE: Slopes, Tangents, and NormalsFind equations for the lines that are ...
 3.113PE: Slopes, Tangents, and NormalsFind equations for the lines that are ...
 3.114PE: Slopes, Tangents, and NormalsFind equations for the lines that are ...
 3.115PE: Slopes, Tangents, and NormalsFind equations for the lines that are ...
 3.116PE: Slopes, Tangents, and NormalsFind equations for the lines that are ...
 3.117PE: Slopes, Tangents, and NormalsFind the slope of the curve at the poi...
 3.118PE: Slopes, Tangents, and NormalsThe graph shown suggests that the curv...
 3.119PE: Analyzing GraphsEach of the figures in shows two graphs, the graph ...
 3.120PE: Analyzing GraphsEach of the figures in shows two graphs, the graph ...
 3.121PE: Analyzing GraphsRepeat Exercise 121, supposing that the graph start...
 3.122PE: Analyzing Graphs Repeat Exercise 121, supposing that the graph star...
 3.123PE: Analyzing GraphsIt is about the accompanying graphs. The graphs in ...
 3.124PE: Analyzing GraphsThe graphs in part (a) show the numbers of rabbits ...
 3.125PE: Trigonometric LimitsFind the limits
 3.126PE: Trigonometric LimitsFind the limits
 3.127PE: Trigonometric LimitsFind the limits
 3.128PE: Trigonometric LimitsFind the limits
 3.129PE: Trigonometric LimitsFind the limits
 3.130PE: Trigonometric LimitsFind the limits
 3.131PE: Trigonometric LimitsFind the limits
 3.132PE: Trigonometric LimitsFind the limits
 3.133PE: Trigonometric LimitsShow how to extend the functions to be continuo...
 3.134PE: Trigonometric LimitsShow how to extend the functions to be continuo...
 3.135PE: Logarithmic DifferentiationUse logarithmic differentiation to find ...
 3.136PE: Logarithmic DifferentiationUse logarithmic differentiation to find ...
 3.137PE: Logarithmic DifferentiationUse logarithmic differentiation to find ...
 3.138PE: Logarithmic DifferentiationUse logarithmic differentiation to find ...
 3.139PE: Logarithmic DifferentiationUse logarithmic differentiation to find ...
 3.140PE: Logarithmic DifferentiationUse logarithmic differentiation to find ...
 3.141PE: Related RatesRight circular cylinder The total surface area S of a ...
 3.142PE: Related RatesRight circular cone The lateral surface area S of a ri...
 3.143PE: Related RatesCircle’s changing area The radius of a circle is chang...
 3.144PE: Related RatesCube’s changing edges The volume of a cube is increasi...
 3.145PE: Related RatesResistors connected in parallel If two resistors of R1...
 3.146PE: Related RatesImpedance in a series circuit The impedance Z (ohms) i...
 3.147PE: Related RatesSpeed of moving particle The coordinates of a particle...
 3.148PE: Related RatesMotion of a particle A particle moves along the curve ...
 3.149PE: Related RatesDraining a tank Water drains from the conical tank sho...
 3.150PE: Related RatesRotating spool As television cable is pulled from a la...
 3.151PE: Related RatesMoving searchlight beam The figure shows a boat 1 km o...
 3.152PE: Related RatesPoints moving on coordinate axes Points A and B move a...
 3.153PE: Find the linearizations ofa. Graph the curves and linearizations to...
 3.154PE: We can obtain a useful linear approximation of the function by comb...
 3.155PE: Find the linearization of
 3.156PE: Find the linearization of at x = 0.
 3.157PE: Surface area of a cone Write a formula that estimates the change th...
 3.158PE: Controlling errora. How accurately should you measure the edge of a...
 3.159PE: Differential Estimates of ChangeCompounding error The circumference...
 3.160PE: Differential Estimates of ChangeFinding height To find the height o...
Solutions for Chapter 3: Thomas' Calculus: Early Transcendentals 13th Edition
Full solutions for Thomas' Calculus: Early Transcendentals  13th Edition
ISBN: 9780321884077
Solutions for Chapter 3
Get Full SolutionsSince 221 problems in chapter 3 have been answered, more than 41353 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Thomas' Calculus: Early Transcendentals , edition: 13th. Chapter 3 includes 221 full stepbystep solutions. Thomas' Calculus: Early Transcendentals was written by Sieva Kozinsky and is associated to the ISBN: 9780321884077.

Base
See Exponential function, Logarithmic function, nth power of a.

Damping factor
The factor Aea in an equation such as y = Aeat cos bt

Decreasing on an interval
A function f is decreasing on an interval I if, for any two points in I, a positive change in x results in a negative change in ƒ(x)

Factored form
The left side of u(v + w) = uv + uw.

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

Instantaneous rate of change
See Derivative at x = a.

Linear combination of vectors u and v
An expression au + bv , where a and b are real numbers

Linear correlation
A scatter plot with points clustered along a line. Correlation is positive if the slope is positive and negative if the slope is negative

Logarithmic reexpression of data
Transformation of a data set involving the natural logarithm: exponential regression, natural logarithmic regression, power regression

Major axis
The line segment through the foci of an ellipse with endpoints on the ellipse

nset
A set of n objects.

Parametric equations
Equations of the form x = ƒ(t) and y = g(t) for all t in an interval I. The variable t is the parameter and I is the parameter interval.

Partial fractions
The process of expanding a fraction into a sum of fractions. The sum is called the partial fraction decomposition of the original fraction.

Principal nth root
If bn = a, then b is an nth root of a. If bn = a and a and b have the same sign, b is the principal nth root of a (see Radical), p. 508.

Row operations
See Elementary row operations.

Sequence
See Finite sequence, Infinite sequence.

Solve a triangle
To find one or more unknown sides or angles of a triangle

Symmetric about the origin
A graph in which (x, y) is on the the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ? + ?) is on the graph whenever (r, ?) is

Terminal side of an angle
See Angle.

Unit circle
A circle with radius 1 centered at the origin.