Topics
Math 140: Calculus I 


Topics 
Last modified: October 27, 2009 
 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
 Onesided 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 nondifferentiability
 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
Copyright 2014,
Catalin Zara.
This work is licensed under a
Creative Commons License
Cite/attribute Resource.
administrator. (Jul 14, 2010). Topics. Retrieved Nov 06, 2014, from UMass Boston OpenCourseware Web site: http://ocw.umb.edu/mathematics/math140calculusifall2009/Topics.