MAT271-272 Notes.pdf - 1 Calculus 8th Edition Notes Table...

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Unformatted text preview: 1 Calculus 8th Edition Notes Table of Contents Chapter 1: Functions and Limits ………………………………………………………………………………………………………………4 1.1: Four Ways to Represent a Function …………………………………………………………………………………….4 1.2: Mathematical Models: A Catalog of Essential Functions ……………………………………………………..8 1.3: New Functions from Old Functions ……………………………………………………………………………………11 1.4: The Tangent and Velocity Problems …………………………………………………………………………………..13 1.5: The Limit of a Function ………………………………………………………………………………………………………14 1.6: Calculating Limits Using the Limit Laws ……………………………………………………………………………..18 1.7: The Precise Definition of a Limit ………………………………………………………………………………………..21 1.8: Continuity ………………………………………………………………………………………………………………………….29 Chapter 2: Derivatives …………………………………………………………………………………………………………………………..36 2.1: Derivatives and Rates of Change ……………………………………………………………………………………….36 2.2: The Derivative as a Function ……………………………………………………………………………………………..39 2.3: Differentiation Formulas ……………………………………………………………………………………………………43 2.4: Derivatives of Trigonometric Functions ……………………………………………………………………………..46 2.5: The Chain Rule …………………………………………………………………………………………………………………..49 2.6: Implicit Differentiation ………………………………………………………………………………………………………50 2.7: Rates of Change in the Natural and Social Sciences …………………………………………………………..51 2.8: Related Rates …………………………………………………………………………………………………………………….51 2.9: Linear Approximations and Differentials ……………………………………………………………………………52 Chapter 3: Applications of Differentiation …………………………………………………………………………………………….53 3.1: Maximum and Minimum Values ………………………………………………………………………………………..53 3.2: The Mean Value Theorem …………………………………………………………………………………………………55 3.3: How Derivatives Affect the Shape of a Graph ……………………………………………………………………58 3.4: Limits at Infinity; Horizontal Asymptotes …………………………………………………………………………..61 3.5: Summary of Curve Sketching …………………………………………………………………………………………….64 3.7: Optimization Problems ……………………………………………………………………………………………………..64 3.8: Newton’s Method ……………………………………………………………………………………………………………..66 3.9: Antiderivatives ………………………………………………………………………………………………………………….67 Chapter 4: Integrals ………………………………………………………………………………………………………………………………68 4.1: Areas and Distances ………………………………………………………………………………………………………….68 4.2: The Definite Integral ………………………………………………………………………………………………………….70 4.3: The Fundamental Theorem of Calculus ……………………………………………………………………………..78 4.4: Indefinite Integrals and the Net Change Theorem ……………………………………………………………..80 4.5: The Substitution Rule ………………………………………………………………………………………………………..82 Top of Document 2 Chapter 5: Applications of Integration ………………………………………………………………………………………………….86 5.1: Areas Between Curves ………………………………………………………………………………………………………86 5.2: Volumes …………………………………………………………………………………………………………………………….88 5.3: Volumes by Cylindrical Shells …………………………………………………………………………………………….90 5.4: Work …………………………………………………………………………………………………………………………………92 5.5: Average Value of a Function ………………………………………………………………………………………………93 Chapter 6: Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions …………….95 6.1: Inverse Functions ………………………………………………………………………………………………………………95 6.2*: The Natural Logarithmic Function ……………………………………………………………………………………98 6.3*: The Natural Exponential Function ………………………………………………………………………………….104 6.4*: General Logarithmic and Exponential Functions ……………………………………………………………107 6.5: Exponential Growth and Decay ……………………………………………………………………………………….111 6.6: Inverse Trigonometric Functions ……………………………………………………………………………………..112 6.7: Hyperbolic Functions ……………………………………………………………………………………………………….117 6.8: Indeterminate Forms and l’Hospital’s Rule ………………………………………………………………………125 Chapter 7: Techniques of Integration ………………………………………………………………………………………………….127 7.1: Integration by Parts …………………………………………………………………………………………………………127 7.2: Trigonometric Integrals …………………………………………………………………………………………………..129 7.3: Trigonometric Substitution ……………………………………………………………………………………………..130 7.4: Integration of Rational Functions by Partial Fractions ……………………………………………………..131 7.5: Strategy for Integration …………………………………………………………………………………………………..132 7.6: Integration Using Tables and Computer Algebra Systems ………………………………………………..133 7.7: Approximate Integration ………………………………………………………………………………………………134 7.8: Improper Integrals …………………………………………………………………………………………………………..137 Chapter 8: Further Applications of Integration ……………………………………………………………………………………140 8.1: Arc Length ……………………………………………………………………………………………………………………….140 8.2: Area of a Surface of Revolution ……………………………………………………………………………………….142 Chapter 9: Differential Equations ………………………………………………………………………………………………………..145 9.1: Modeling with Differential Equations ………………………………………………………………………………145 9.2: Direction Fields and Euler’s Method ………………………………………………………………………………..148 9.3: Separable Equations ………………………………………………………………………………………………………..149 9.4: Models for Population Growth ………………………………………………………………………………………..150 9.5: Linear Equations ………………………………………………………………………………………………………………153 9.6: Predatory-Prey Systems …………………………………………………………………………………………………..154 Chapter 10: Parametric Equations and Polar Coordinates …………………………………………………………………..156 10.1: Curves Defined by Parametric Equations ……………………………………………………………………….156 10.2: Calculus with Parametric Curves ……………………………………………………………………………………157 10.3: Polar Coordinates ………………………………………………………………………………………………………….159 10.4: Areas and Lengths in Polar Coordinates …………………………………………………………………………162 10.5: Conic Sections ……………………………………………………………………………………………………………….163 10.6: Conic Sections in Polar Coordinates ………………………………………………………………………………169 Top of Document 3 Chapter 11: Infinite Sequences and Series ………………………………………………………………………………………….174 11.1: Sequences ……………………………………………………………………………………………………………………..174 11.2: Series …………………………………………………………………………………………………………………………….178 11.3: The Integral Test and Estimates of Sums ……………………………………………………………………….182 11.4: The Comparison Tests ……………………………………………………………………………………………………186 11.5: Alternating Series ………………………………………………………………………………………………………….188 11.6: Absolute Convergence and the Ratio and Root Tests …………………………………………………….189 11.7: Strategy for Testing Series …………………………………………………………………………………………….194 11.8: Power Series ………………………………………………………………………………………………………………….195 11.9: Representations of Functions as Power Series ………………………………………………………………197 11.10: Taylor and Maclaurin Series ………………………………………………………………………………………..198 11.11: Applications of Taylor Polynomials ………………………………………………………………………………205 Top of Document 4 Chapter 1: Functions and Limits Section 1.1: Four Ways to Represent a Function In general, a function exists whenever one quantity depends on another. Calculus is the study of the behaviors of functions and how a change in one quantity causes a change in the other quantity. Definition 1.1: A function is a rule that assigns to each element in a set exactly one element, called (), in a set . Note: In the general sense, sets and can be sets of real numbers, complex numbers, or some other number system entirely. However, we will stay within the real number system for the course. For the function , the set is the domain, and the set is the range. While the domain is the set of all real numbers for which () is defined, the range is the set of all function values () as varies throughout the domain. The notation () is read as “ of ” and is the value of the function at . For every function , there is an associated equation = (), which conveys the following information. • • • is the independent variable since it is free to vary throughout the function’s domain. is the dependent variable since its value depends on the value of . is the name of the function. Visually, functions can be displayed using arrow diagrams or a graph. Examples of each are given below. For the arrow diagram, arrows are drawn from representative values in the domain to their corresponding function values in the range. The rule that connects to (), to (), and so on, is the function . The graph of a function with domain is a set of ordered pairs �, ()� that correspond to points on the Real Cartesian Plane. In other words, we have that Top of Document 5 Graph() = ��, ()�� ∈ � With increasing occurring to the right along the − and increasing occurring upward along the − , points on a graph are determined by their heights above (or below) the point . Additionally, the domain of a function can be determined by reading the graph from left to right, and the range can be determined by reading the graph from bottom to top. For a function , the expression ( + ℎ) − () , ℎ ℎ≠0 is called the difference quotient for . To say that this will be used frequently in this course is an understatement… In general, there are, as the title of the section suggests, four different ways to represent a function. 1. 2. 3. 4. Verbally – by a description in words. Numerically – by a table of values. Visually – by a graph. Analytically – by an explicit formula. Each way has its benefits and deficits, so we will always strive to find a balance among them when representing and analyzing functions. Notes: • • For algebraic functions (functions that are defined solely with the operations of addition, subtraction, multiplication, division, powers, and roots), domain restrictions occur whenever division by zero is possible or when the even root of a negative number is possible. For transcendental functions (exponential, logarithmic, trigonometric, and inverse trigonometric functions), they exhibit their own unique domain restrictions. If the “exactly one” expression is removed from the definition of a function, then a more general concept of a relation arises. Both relations and functions can be graphed on the Real Cartesian Plane, and so because we will frequently want to use relations that are not functions, it is helpful to be able to determine whether a relation is a function. Graphically, this can be done using the Vertical Line Test. Theorem 1.2 (The Vertical Line Test): A curve in the -plane is the graph of a function of if and only if no vertical line intersects the curve more than once. Note: The phrase “if and only if” is what’s called a biconditional since it implies two conditional statements. 1. If a curve on the -plane is a function, then no vertical line intersects the curve more than once. 2. If no vertical line intersects a curve on the -plane more than once, then the curve represents a function. Therefore, to prove this theorem, both statements must separately be proven. Biconditionals are frequently used in calculus as a means for showing equivalence between two (or more) statements. Top of Document 6 Proof: Both statements will be proven by using logical contrapositives. For a conditional statement of the logical contrapositive will be if A, then B if not B, then not A A conditional statement and its logical contrapositive have the same truth value, so determining the truth value of one statement automatically determines the truth value of the other statement. 1. Suppose there exists a vertical line = that intersects a curve on the -plane at two points, (, 1 ) and (, 2 ), where 1 and 2 are the two distinct -values at which the intersections take place. However, this implies that for the input , there are two distinct corresponding outputs 1 ≠ 2 . This contradicts Definition 1.1, so the relation is not a function. 2. Suppose a curve on the -plane represents a relation that is not a function. Therefore, there exists at least one in the domain of the relation for which (, 1 ) and (, 2 ), with 1 ≠ 2 , are distinct ordered pairs, and therefore distinct points. So, the line = will intersect the curve defining the relation at these two points. Not every function is defined with a single explicit formula. Functions that are defined with multiple different explicit formulas (where the domains for each are disjoint (nonintersecting) sets) are called piecewise-defined functions. Two canonical examples are the absolute value function and the class of step functions. Definition 1.3 (The Absolute Value Function): Denoted by ||, the absolute value function determines the distance is from 0. Therefore, || ≥ 0. More generally, we have the piecewise-defined function given below. () = || = � , if ≥ 0 −, if < 0 A step function is a piecewise-defined function, where each of the “pieces” are constant functions. They form the basis for our discussion of functional Riemann integration, discussed in Chapter 4. An example of a step function is the greatest integer function, denoted by () = �[]�. For any real , it’s defined as the largest integer less than or equal to . A graph of it is given below. Top of Document 7 While the graph of a relation or function can have any of infinitely many types of symmetry, we will focus mostly on three: symmetry about each coordinate axis and the origin. The graph of a relation can exhibit all three types of symmetry, but the only function that has − symmetry is () = 0. • • • A relation that has − symmetry implies that for every point (, ) on the graph, the point (, −) also exists on the graph. Therefore, − symmetry can be determined analytically by interchanging with − in the equation. If, after simplifying, the equation is unchanged, then − symmetry exists. A function/relation that has − symmetry implies that for every point (, ) on the graph, the point (−, ) also exists on the graph. Therefore, − symmetry can be determined analytically by finding (−). If (−) = (), ∀ ∈ , then − symmetry exists, and the function/relation is called even. A function/relation that has origin symmetry implies that for every point (, ) on the graph, the point (−, −) also exists on the graph. Therefore, origin symmetry can be determined analytically by finding (−). If (−) = −(), ∀ ∈ , then origin symmetry exists, and the function/relation is called odd. Definition 1.4: • • A function is increasing on an (open) interval if (1 ) < (2 ), ∀1 < 2 ∈ (1 ) > (2 ), ∀1 < 2 ∈ A function is decreasing on an (open) interval if Note: Recall that for open intervals, the endpoints of the interval are not included. The general open intervals are (−∞, ∞), (, ∞), (−∞, ), and (, ) for < . Later, we will discuss increasing and decreasing at a single point. We’ll then be able to use that to describe a function increasing or decreasing on the non-open intervals [, ∞), (−∞, ], [, ], [, ), and (, ] for < . Top of Document 8 Section 1.2: Mathematical Models: A Catalog of Essential Functions One of the main goals of functions is to use them to represent mathematical models, or descriptions of real-world phenomena. The model can then be used to better understand the phenomena as well as to, perhaps, make predictions about future behavior. For a real-world problem, we will want to follow a general procedure of (1) formulating a mathematical model, (2) solving the model to develop conclusions, (3) interpreting the conclusions in the context of the problem, (4) making predictions using the model, and finally (5) testing the predictions on new real data. Below are some of the many types of functions we will use to construct mathematical models. 1. Linear Model is a linear function of if the graph of the function relating the variables is a line. Its mathematical model is given by the slope-intercept form = () = + where is the slope and is the -intercept. Linear models have the unique characteristic of exhibiting a constant rate of change of with respect to . In other words, as changes, always changes by the same constant factor. When no physical law or principle is present in a real-world problem, an empirical model can be constructed solely out of observed/measured data. As a result, there are typically numerous methods for constructing accurate models. For instance, for a linear model, a linear function can be formed by finding the line that contains the first and last points in the ordered data, but typically a more sophisticated (and accurate) linear model is achieved through linear regression by finding the least squares regression line, or the line formed by minimizing the sum of the squares of the vertical distances between the data points and the line. Luckily, a calculator can be used to construct a least squares regression line rather easily. Simply input the -values into one list, the -values into another list, and then run the LinReg command. The calculator will then produce the linear function’s slope and -intercept. Note: Once a model is determined, such as a linear model, estimating a value between observed values is called interpolation, and estimating a value outside the range of observed values is called extrapolation. Top of Document 9 2. Polynomial Model A function is called a polynomial if () = + −1 −1 + ⋯ + 2 2 + 1 + 0 , ≠ 0 where is a nonnegative integer and the numbers , ∀: 1 → , are constants called coefficients. The domain of any polynomial is ℝ = (−∞, ∞). The degree of the polynomial is , the constant is called the leading coefficient, and the term is called the leading term. • • • A polynomial of degree 1 is the linear function 1 () = + . A polynomial of degree 2 is the quadratic function 2 () = 2 + + . o The graph is a parabola that opens upward if > 0 and downward if < 0. o Just like for a linear model, a quadratic model can be determined on the calculator using the least squares method by the QuadReg command. A polynomial of degree 3 is the cubic function 3 () = 3 + 2 + + . o Just like for linear and quadratic models, a cubic model can be determined on the calculator using the least squares method by the CubicReg command. 3. Power Model A function of the form () = , where is a constant, is called a power function. • Let be a positive integer. Except for when = 1 (which gives the line = ), the parity of ultimately determines the graph’s shape. Note that as increases, the graph flattens out more and more on the interval (−1, 1), but becomes steeper on the intervals (−∞, −1) and (1, ∞). • Let = , where is a positive integer. For ≥ 2, the function () = = √ becomes a 1 1 root function. If is even, then the domain of √ is [0, ∞), whereas odd makes the domain ℝ. Top of Document 10 • 1 Let = −1. The graph of () = −1 = is the reciprocal function. Its graph is a hyperbola with asymptotes along each coordinate axis. o Reciprocal functions are used whenever there exists an inverse proportionality between variables. More specifically, they’re used when a change in one variable causes an alternate type of change in the other variable (for example, an increase in causes a decrease in ). 4. Rational Function Model A rational function is a ratio of two polynomials: () = () () where and are polynomials. The domain of a rational function is given by the set {|() ≠ 0}. Note: All the functions discussed thus far are examples of algebraic functions since their explicit formulas only use the operations of addition, subtraction, multiplication, division, powers, and roots. The next set of functions are among the class of transcendentals. 5. Trigonometric Models A trigonometric function is any one of the six functions () = sin() () = csc() () = cos() () = sec() () = tan() () = cot() The Sine and Cosine functions have domain ℝ and range [−1, 1]. They are both 2-periodic. In other words, we have that sin( + 2) = sin() , cos( + 2) = cos() , The Tangent function is the ratio of the Sine function to the Cosine function. tan() = Top of Document sin() cos() ∀ ∈ ℝ 11 The Tangent function is undefined when cos() = 0, or the set �� ≠ (2+1) 2 , ∈ ℤ�. Its range is ℝ. Unlike the Sine and Cosine functions, it is -periodic. In other words, we have that tan( + ) = tan() , ∀ ∈ ℝ The Cosecant, Secant, and Cotangent functions are the reciprocals of the Sine, Cosine, and Tangent functions, respectively. Note: Unless stated otherwise, always assume the independent variable of a trigonometric function is in radian measure. 6. Exponential Model An exponential function is of the form () = , ∈ ℝ+ \{1}. Note: The expressio...
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