Four Ways to Represent a FunctionMathematical Models: A Catalog of Essential FunctionsNew Functions from Old FunctionsGraphing Calculators and ComputersExponential FunctionsInverse Functions and LogarithmsLimits and DerivativesThe Tangent and Velocity ProblemsThe Limit of a FunctionCalculating Limits Using the Limit LawsThe Precise Definition of a LimitContinuityLimits at Infinity; Horizontal AsymptotesDerivatives and Rates of ChangeThe Derivative as a FunctionDifferentiation RulesDerivatives of Polynomials and Exponential FunctionsThe Product and Quotient RulesDerivatives of Trigonometric FunctionsThe Chain RuleImplicit DifferentiationDerivatives of Logarithmic FunctionsRates of Change i the Natural and Social SciencesExponential Growth and DecayRelated RatesLinear Approximations and DifferentialsHyperbolic FunctionsApplications of DifferentiationMaximum and Minimum ValuesThe Mean Value TheoremHow Derivatives Affect the Shape of a GraphIndeterminate Forms and l'Hopital's RulesSummary of Curve SketchingGraphing with Calculus and CalculatorsOptimization ProblemsNewton's MethodAntiderivativesIntegralsAreas and DistancesThe Definite IntegralThe Fundamental Theorem of CalculusIndefinite Integrals and the Net Change TheoremThe Substitution RuleApplications of Integration[Red]Areas Between Curves[Red]Volumes[Red]Volumes by Cylindrical Shells[Red]WorkAverage Value of a FunctionTechniques of Integration[Red]Integration by Parts[Red]Trigonometric Integrals[Red]Trigonometric Substitution[Red]Integration of Rational Functions by Partial FractionsStrategy for IntegrationIntegration Using Tables and Computer Algebra Systems[Red]Approximate Integration[Red]Improper IntegralsFurther Applications of Integration[Red]Arc Length[Red]Area of a Surface of RevolutionApplications to Physics and EngineeringApplications to Economics and BiologyProbabilityDifferential Equations[Green]Direction Fields and Euler's Method[Green]Separable EquationsModels for Population GrowthLinear EquationsPredator-Prey SystemsParametric Equations and Polar Coordinates[Yellow]Curves Defined by Parametric Equations[Yellow]Calculus with Parametric Curves[Yellow]Polar Coordinates[Yellow]Areas and Lengths in Polar CoordinatesConic SectionsConic Sections in Polar CoordinatesInfinite Sequences and Series[Yellow]Sequences[Yellow]Series[Yellow]The Integral Test and Estimates of Sums[Yellow]The Comparison Tests[Yellow]Alternating Series[Yellow]Absolute Convergence and the Ration and Root TestsStrategy for Testing Series[Yellow]Power Series[Yellow]Representations of Functions as Power Series[Yellow]Taylor and Maclaurin SeriesCalculus 1, 2, and 3The AuthorCalculus 11 Functions and Models1.1 Four Ways to Represent a Function1. Function: A function f is a rule that assigns to each element x is a set D exactly one element, called f (x),in a set E.2. Domain: In the definition of a function, D is called the domain.3. Range: In the definition of a function, the range of f is the set of all possible values of f (x) as x variesthroughout the domain.4. The Vertical Line Test: A curve in the xy-plane is the graph of a function of x if and only if no verticalline intersects the curve more than once5. Piecewise Defined Functions: Functions that are defined by different formulas in different parts of theirdomains are called piecewise defined functions6. Step-functions: Piecewise functions that jump from one value to the next and look like steps7. Even-function: If a function f satisfies f (−x) = f(x) for every number x in its domain, then f is called aneven function.8. Odd-function: If a function f satisfies f(−x) = −f (x) for every number x in its domain, then f is calledan odd function.9. Increasing function: A function f is called increasing on an interval I if f (x1) < f(x2) whenever x1< x2in I.10. Decreasing function: A function f is called decreasing on an interval I if f(x1) > f(x2) whenever x1< x2in I.1.2 Mathematical Models: A Catalog of Essential Functions1. Linear Function: y is a linear function of x if we can write a formula for the function as y = f(x) = mx + bwhere m is the slope of the line and b is the y-intercept.2. Polynomial: A function P is called a polynomial if P (x) = anxn+ an−1xn−1+ . . . + a2x2+ a1x + a0wheren > 0 is an integer and all aiare constant coefficients of the polynomial.3. Quadratic Functions: A polynomial of degree 2 is of the form P (x) = ax2+bx +c and is called a quadraticfunction.4. Cubic Functions: A polynomial of degree 3 is of the form P (x) = ax3+ bx2+ cx + d and is called a cubicfunction.5. Power Functions: A function of the form f (x) = xawhere a is a constant is called a power function.16. Rational Functions: A rational function f is the ratio of two polynomials: f(x) =P (x)Q(x)where P and Qare polynomials. The domain of a rational function is all values of x such that Q(x) 6= 0.7. Algebraic Functions: A function f is called an algebraic function if it can be constructed using algebraicoperations starting with polynomials. All rational functions are algebraic functions.8. Trigonometric Functions: Here are some properties of trigonometric functions:• |sin(x)| ≤ 1• |cos(x)| ≤ 1• sin(x + 2π) = sin(x)• cos(x + 2π) = cos(x)• tan(x) =sin(x)cos(x)• tan(x + π) = tan(x)9. Exponential Functions: The exponential functions are the functions of the form f(x) = ax, where a is apositive constant.10. Logarithmic Functions: The logarithmic functions f(x) = loga(x), where a is a positive constant, are theinverse functions of the exponential functions.1.3 New Functions from Old Functions1. Vertical and Horizontal Shifts (Translations): Suppose c > 0. To obtain the graph of:(a) y = f (x) + c, shift the graph of y = f (x) a distance of c units upward.(b) y = f (x) −c, shift the graph of y = f (x) a distance of c units downward.(c) y = f (x −c), shift the graph of y = f (x) a distance of c units to the right.(d) y = f (x + c), shift the graph of y = f (x) a distance of c units to the left.2. Vertical and Horizontal Stretching and Reflecting: Suppose c > 1. To obtain the graph of:(a) y = c · f(x), stretch the graph of y = f(x) vertically by a factor of c.(b) y =1c· f(x), shrink the graph of y = f(x) vertically by a factor of c.(c) y = f (cx), shrink the graph of y = f (x) horizontally by a factor of c.(d) y = fxc, stretch the graph of y = f (x) horizontally by a factor of c.(e) y = −f (x), reflect the graph of y = f (x) about the x-axis.(f) y = f (−x), reflect the graph of y =