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Photo of Catalin Zara UMass Boston, Fall 2009
Math 140: Calculus I
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Course Info Calendar Syllabus Assignments Topics
  Topics Last modified:
October 27, 2009
Functions
  • Four ways of representing a function
    • Verbal
    • Numerical
    • Graphical
    • Analytical
  • A Library of functions
    • Power functions
    • Polynomial functions
    • Rational functions
    • Exponential functions
    • Logarithmic functions
    • Trigonometric functions
  • Operations with functions
    • Algebraic operations
    • Composition
    • Inverse functions
Limits and Continuity
  • Intuitive numerical and graphical approach
    • Limit of a function at a point; numerical and graphical approach
    • One-sided limits; numerical and graphical approach
    • Infinite limits; numerical and graphical approach
    • Vertical asymptotes
  • Computation of limits: Limit laws
    • Sum, difference, constant multiple
    • Product
    • Quotient
    • Powers
    • Direct substitution property
    • Composition law
    • One sided limits and limits
    • The squeeze theorem
  • Continuity
    • Continuity of a function at a point
    • Point of discontinuity
    • Types of discontinuity: removable, infinite, jump
    • One sided continuity of a function at a point
    • Continuity on an interval
    • Algebraic operations with continuous functions
    • Composition of continuous functions
    • Classes of continuous functions
    • The Intermediate Value Theorem
Derivatives
  • Motivation: Rates of change
    • Secants and tangents
    • Instantaneous velocity
  • Derivatives
    • Definition of derivative of a function at a point
    • Graphical interpretation of derivative
    • Derivatives as rates of change
  • The derivative as a function
    • The derivative function
    • From graph of function to graph of derivative
    • Differentiable functions
    • Differentiability and continuity
    • Points of non-differentiability
  • Computation of derivatives from the definition
    • Constant functions
    • Power functions; power rule
  • Differentiation rules: Algebraic operations
    • Constant multiple rule
    • Sum and difference rule
    • Product rule
    • Quotient rule
  • Derivatives of trigonometric functions
    • Derivatives of sin and cos
    • Derivatives of tan and cot
    • Derivatives of sec and csc
    • Connecting formulas for tan and sec
  • Derivatives of exponential functions
  • Differentiation rules: Composition of functions
    • The chain rule
    • General rules
    • Inverse functions and their derivatives
    • Logarithmic functions and their derivatives
  • Other differentiation rules
    • Logarithmic differentiation
    • Implicit differentiation
      • Implicit differentiation
      • Orthogonal trajectories
  • Higher order derivatives
    • Second derivative and acceleration
    • Third derivative and jerk
    • Higher order derivatives of implicit functions
    • Inductive formulas
Applications of Derivatives
  • Related rates problems
  • Approximations
    • Tangent line and linearization
    • Linear approximation
    • Differentials
    • Errors
  • Minimum and maximum values
    • The language of local/global, minimum/maximum
    • The extreme value theorem
    • Fermat's theorem
    • Critical numbers
    • The closed interval method
  • The Mean Value Theorem
    • Rolle's theorem
    • The mean value theorem
    • Estimate of function from estimate of derivative
    • Functions with zero derivative
  • Derivatives and the shape of the graph
    • First derivative: increasing/decreasing test
    • The first derivative test for extreme points
    • Concavity
    • Second derivative and the concavity test
    • The second derivative test for extreme points
    • Curve sketching
  • Optimization problems
  • Newton's method
Integrals
  • Motivation: Areas and distances
    • Estimates of area using rectangles
    • Area of a region: definition
    • Sigma notation for sum
    • Distance for variable velocity
  • The definite integral
    • Riemann sums
    • Left, right, and midpoint Riemann sums
    • The definite integral for continuous functions
    • The midpoint rule for approximations
    • Properties of definite integrals
      • Linearity of integrand
      • Additivity of interval
      • Comparison properties
  • Antiderivatives
    • Definition of antiderivative
    • Set of antiderivatives on an interval
    • Indefinite integrals
    • The geometry of antiderivatives: direction fields
    • Rectilinear motion
  • The Fundamental Theorem of Calculus
    • Functions defined as integrals
    • The Fundamental Theorem of Calculus
    • Derivative of a function defined as an integral
    • The net change theorem
    • Net change and total change
  • The substitution rule
    • The substitution rule for indefinite integrals
    • The substitution rule for definite integrals
    • Symmetry: even and odd functions
Applications of definite integrals
  • Area between curves
  • Volumes
    • As integral of area of a section
    • Solids of revolution, by washers
  • Work done by a variable force
  • Average value of a function
Copyright 2014, Catalin Zara. This work is licensed under a Creative Commons License