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##### Document Actions UMass Boston, Fall 2009
Math 140: Calculus I Course Info Calendar Syllabus Assignments Topics

• 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
• 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
• 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
• 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
• 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
• 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 