Chapter 1

Functions and Graphs

Read chapter 1 in the book

Summary

The power function f ( x ) = x n is an even function if n is even and n ≠ 0 , and it is an odd function if n is odd Even functions are symmetric about the y -axis whereas odd functions are symmetric about the origin The root function f ( x ) = x 1 / n has the domain [ 0 , ∞ ) if n is even and the domain ( -∞ , ∞ ) if n is odd. If no domain is stated for a function y = f ( x ) , the domain is considered to be the set of all real numbers x for which the function is defined

Key terms

function

a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output

graph of a function

the set of points ( x , y ) such that x is in the domain of f and y = f ( x )

odd function

a function is odd if f ( - x ) = - f ( x ) for all x in the domain of f

even function

a function is even if f ( - x ) = f ( x ) for all x in the domain of f

power function

a function of the form f ( x ) = x n for any positive integer n ≥ 1

root function

a function of the form f ( x ) = x 1 / n for any integer n ≥ 2

range

the set of outputs for a function

domain

the set of inputs for a function

Chapter 2

Limits

Read chapter 2 in the book

Summary

Differential calculus arose from trying to solve the problem of determining the slope of a line tangent to a curve at a point. Integral calculus arose from trying to solve the problem of finding the area of a region between the graph of a function and the x -axis. The integral is also calculated by finding a limit and, in fact, is related to the derivative of a function The slope of the tangent line indicates the rate of change of the function, also called the derivative .

Key terms

limit

the process of letting x or t approach a in an expression; the limit of a function f ( x ) as x approaches a is the value that f ( x ) approaches as x approaches a

tangent

A tangent line to the graph of a function at a point ( a , f ( a ) is the line that secant lines through ( a , f ( a ) approach as they are taken through points on the function…

integral

also calculated by finding a limit and, in fact, is related to the derivative of a function

differential calculus

the field of calculus concerned with the study of derivatives and their applications

multivariable calculus

the study of the calculus of functions of two or more variables

integral calculus

the study of integrals and their applications

average velocity

the change in an object’s position divided by the length of a time period; the average velocity of an object over a time interval [ t , a ] (if t < a or [ a , t ] if t > a ) …

continuity at a point

A function f ( x ) is continuous at a point a if and only if the following three conditions are satisfied: (1) f ( a ) is defined, (2) lim x → a f ( x ) exists, and (3) lim x → a…

Chapter 3

Derivatives

Read chapter 3 in the book

Summary

The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment h The derivative of a function f ( x ) at a value a is found using either of the definitions for the slope of the tangent line As such, the velocity v ( t ) at time t is the derivative of the position s ( t ) at time t .

Key terms

derivative

the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative

instantaneous rate of change

the rate of change of a function at any point along the function a , also called f ′ ( a ) , or the derivative of the function at a

constant multiple rule

the derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative: d x ( c f ( x ) = c f ′ ( x )

derivative function

gives the derivative of a function at each point in the domain of the original function for which the derivative is defined

difference rule

the derivative of the difference of a function f and a function g is the same as the difference of the derivative of f and the derivative of g : d x ( f ( x ) - g ( x ) = f ′ ( x…

implicit differentiation

is a technique for computing d y d x for a function defined by an equation, accomplished by differentiating both sides of the equation (remembering to treat the variable y as a…

logarithmic differentiation

is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the…

Chapter 4

Applications of Derivatives

Read chapter 4 in the book

Summary

To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time The actual change in y is Δ y = f ( a + d x ) - f ( a ) In terms of the quantities, state the information given and the rate to be found Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates

Key terms

related rates

are rates of change associated with two or more related quantities that are changing over time

actual change in y

Δ y = f ( a + d x ) - f ( a )

concavity test

suppose f is twice differentiable over an interval I ; if f ″ > 0 over I , then f is concave up over I ; if f ″ < 0 over I , then f is concave down over I

differential

the differential d x is an independent variable that can be assigned any nonzero real number; the differential d y is defined to be d y = f ′ ( x ) d x

differential form

given a differentiable function y = f ′ ( x ) , the equation d y = f ′ ( x ) d x is the differential form of the derivative of y with respect to x

first derivative test

let f be a continuous function over an interval I containing a critical point c such that f is differentiable over I except possibly at c ; if f ′ changes sign from positive to…

indefinite integral

the most general antiderivative of f ( x ) is the indefinite integral of f ; we use the notation ∫ f ( x ) d x to denote the indefinite integral of f

indeterminate forms

when evaluating a limit, the forms 0 , ∞ / ∞ , 0 · ∞ , ∞ - ∞ , 0 , ∞ 0 , and 1 ∞ are considered indeterminate because further analysis is required to determine whether the limit…

Chapter 5

Integration

Read chapter 5 in the book

Summary

Riemann sums allow for much flexibility in choosing the set of points { x i * } at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum Left- and right-endpoint approximations are special kinds of Riemann sums where the values of { x i * } are chosen to be the left or right endpoints of the subintervals, respectively The definite integral can be used to calculate net signed area, which is the area above the x -axis less the area below the x -axis. The width of each rectangle is Δ x = b - a n

Key terms

definite integral

a primary operation of calculus; the area between the curve and the x -axis over a given interval is a definite integral

net signed area

the area between a function and the x -axis such that the area below the x -axis is subtracted from the area above the x -axis; the result is the same as the definite integral of…

right-endpoint approximation

the right-endpoint approximation is an approximation of the area of the rectangles under a curve using the right endpoint of each subinterval to construct the vertical sides of…

riemann sum

an estimate of the area under the curve of the form A ≈ ∑ i = 1 n f ( x i * ) Δ x

lower sum

a sum obtained by using the minimum value of f ( x ) on each subinterval

upper sum

a sum obtained by using the maximum value of f ( x ) on each subinterval

integrable function

a function is integrable if the limit defining the integral exists; in other words, if the limit of the Riemann sums as n goes to infinity exists

Chapter 6

Applications of Integration

Read chapter 6 in the book

Summary

Using the slicing method, we can find a volume by integrating the cross-sectional area If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals and add the results to find the area of the region The principles are the same regardless of which variable is used as the variable of integration

Key terms

principles

the same regardless of which variable is used as the variable of integration

slicing method

a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an…

region

complex, use the absolute value of the difference of the functions

cross-section

the intersection of a plane and a solid object

catenary

a curve in the shape of the function y = a cosh ( x / a ) is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary

centroid

the centroid of a region is the geometric center of the region; laminas are often represented by regions in the plane; if the lamina has a constant density, the center of mass of…

density function

a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in…

doubling time

if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by ( ln 2 ) / k