CALCULUS 1

by Peter J. Ponzo

TABLE OF CONTENTS

EXAMPLE PROBLEMS
ASSORTED PROBLEMS
LECTURE 0
    SOME BASICS
    NUMBERS ... and INFINITY
            5/0 is NOT a number
            Infinity is NOT a number
    INEQUALITIES
    FUNCTIONS
            vertical line test
            functions and their domain
    ABSOLUTE VALUES
            To plot y = | f(x) |
    SOME TRIG IDENTITIES
            the RADIAN measure of an angle
    SOME TRIG GRAPHS
    SOME GEOMETRY
    LOGARITHMS and EXPONENTIALS
            exponential functions
            logarithmic function
    ODDS 'n' ENDS
            geometric series
    SIGMA NOTATION
    MAPLE
LECTURE 1
    LIMITS
    LIMIT RULES
    ONE SIDED LIMITS
LECTURE 2
    INFINITE LIMITS
    ASYMPTOTES
    CONTINUOUS FUNCTIONS
LECTURE 3
    TECHNIQUES FOR EVALUATING LIMITS WHEN THE "RULES" DON'T APPLY
            The form infinity/infinity
            The form infinity - infinity
            The form ??
            Reduce the given limit to one you know
            to make tea
            the SQUEEZE THEOREM
            the graph of y = f(x) sin x
LECTURE 4
    the DERIVATIVE
    DIFFERENTIATION RULES
    the CHAIN RULE
    HIGHER DERIVATIVES
            concave up
            concave down
            the Logistic Equation
            velocity
            acceleration
LECTURE 5
    IMPLICIT DIFFERENTIATION & TRANSCENDENTAL FUNCTIONS
    IMPLICIT DIFFERENTIATION
            the slope at a point (x,y)
            greatest integer function
            trig, exponential and log functions
            weird limits
    the TRIG FUNCTIONS and their derivatives
    the EXPONENTIAL and LOG functions
            ln x
LECTURE 6
    INVERSE FUNCTIONS
            horizontal line test
    TESTING TO SEE IF A FUNCTION HAS AN INVERSE
            Examples of Inverses
            the Derivative of an Exponential Function
    LOGARITHMIC DIFFERENTIATION
            About Exponential Growth
            About the number e
    ODDS 'n' ENDS ON CURVE SKETCHING
            Even and Odd Functions
            Quick&Dirty Curve Sketching
LECTURE 7
    MORE ON INVERSE FUNCTIONS
            the INVERSE TANGENT
            the INVERSE SINE
            restricting the domain
            Check the dimensions
            the limit of ??
            y = sin x with x in DEGREES
LECTURE 8
    ABSOLUTE MAXIMUM AND MINIMUM
            closed interval
            critical point
    RELATIVE MAXIMA and MINIMA
            First Derivative Test
            give it a name and use it!
            Snell's law
LECTURE 9
    RELATED RATE PROBLEMS
LECTURE 10
    The TANGENT LINE APPROXIMATION
            Rule of 72
    POLYNOMIAL APPROXIMATIONS
            quadratic approximation
            cubic approximation
            quartic approximation
            "best" linear approximation
LECTURE 11
    NEWTON'S METHOD for finding roots
            a computer algebra system
            The error goes to zero!
            What is the annual rate of return from this mutual fund?
            a computer spreadsheet
    DIFFICULTIES WITH NEWTON'S METHOD
            Pick a reasonable value for x1
LECTURE 12
    L'HÔPITAL'S RULE
            the form 0/0
            the infinity/infinity form
            Interpretation of a Limiting Value
LECTURE 13
    POLAR COORDINATES
            a distance and a direction
            polar curves
            y² = f(x)
    INTERSECTION OF POLAR CURVES

End of part 1

LECTURE 14
            the AREA UNDER A CURVE
            the SUM of rectangles
    the DEFINITE INTEGRAL
            a Riemann SUM
    PROPERTIES of the DEFINITE INTEGRAL
    THE FUNDAMENTAL THEOREM
            the "area function"
    an ANTIDERIVATIVE
            constant of integration
LECTURE 15
    DEFINITE INTEGRATION
            "negative" areas
            elemental areas
            47,000,000 elemental rectangles
            The error in area
LECTURE 16
    AREAS IN POLAR COORDINATES
    AREA SWEPT OUT BY THE RADIUS
            this "swept out" business. Sounds like a broom
            check it for reasonableness
LECTURE 17
    TECHNIQUES OF INTEGRATION
    THE METHOD OF SUBSTITUTION
            "next to dx"
            integration is an ART
            Heaviside calculus
            Shift Theorem
    INTEGRATION BY PARTS
            who's u and who's v
            the Ponzo function
            the lower limit
            the upper limit
LECTURE 18
    VOLUMES
            cut the solid into many very thin slices
            Volume of a cylinder
            Volume of a cone
    VOLUMES OF SOLIDS OF REVOLUTION
            the volume of a sphere
            the volume of a torus
            make a reasonable diagram
            the centre of area
    THE THEOREM OF PAPPUS
            the CENTROID
            The centroid of a triangle
LECTURE 19
            Volumes of solids of revolution using horizontal rectangles
            a cylindrical shell
            the volume of a "disc"
            guess who's the student?
            distance travelled
            whatzits per doodle
            The total work
            digging a well
            cost of manufacturing
            a reasonable approximation
    AVERAGE VALUE OF A FUNCTION
            an average temperature
            the "average height"
            average velocity
    A PARADOX
LECTURE 20
    IMPROPER INTEGRALS
    DEFINITION of an IMPROPER INTEGRAL
            f(x) must approach zero very rapidly
            Another Kind of Improper Integral
            f(x) must get small enough fast enough
    SOLUTIONS TO "ASSORTED PROBLEMS"

End of Calculus 1