LIMITS & AN INTRODUCTION TO CALCULUS

• Introduction to Limits  and  Properties of Limits
• What is a limit?  (slide show)
• Introduction to Limits problems, solutions, and explanations
  
• Limits from a graphical point of view
• Evaluating limits graphically and numerically
• Informal view of limits
  
• Slides for showing different ways to have the same limit -- click on "The limit"
  
• Limits from a numerical point of view
• Another numerical view of limits
• Generating tables to demonstrate limits applet
• Tabular view of limits
  
• Limits from a symbolic point of view
• Another algebraic approach to limits
• Formal definition of limits -- Cauchy definition of limits with examples; also uses with one-sided limits
• Formal definition of limits
• Precise epsilon-delta definition of a limit
• Problems on the formal definition
• Problems, solutions, and explanations on formal definitions
• Proof of Various Limit Properties  
  
• Formal definition of limits applet
  
• Limits of functions as x approaches a number
• Proof of two well-used trigonometric limit formulas
• Problems about limits with hints, answers, and details
• More problems on limits, but need to be able to find first derivative
• One-sided limits problems, solutions, & explanations
• One and two sided limits and when limits fail to exist
  
•  Limits and Properties of Limits
• Properties of Limits problems, solutions, and explanations
  
• When does a limit exist?  (slide show)
• Techniques for Evaluating Limits
• More on the Sandwich Theorem and the Special Trigonometric Limit
• Limits of Functions using the Squeeze Theorem
• Slides to use in working with sin x / x (Click on "A trigonomeric limit")
• Explanation and applet dealing with sin x /x
• Proofs of some trigonometric limits
• How do you evaluate a limit?  (slide show)
• Evaluating limits analytically
  
• Computing limits problems, solutions, and explanations
  
• Infinite Limits and Limits at Infinity
• Types of infinity  
  
• Limits of functions as x approaches + or - infinity
• More on Infinite Limits
• Problems to work
• Problems, solutions, and explanations
• Limits at Infinity, Part 1, problems, solutions, and explanations
• Limits at Infinity, Part 1I, problems, solutions, and explanations
• Infinite Limits
• Limits at Infinity - horizontal asymptotes, etc 
  
• Horizontal asymptotes
• Vertical asymptotes
• Limits and Infinity (slide show)
• Limits at infinity applet
  
• Review problems for finding asymptotes -- Go to Calculus Book I, then Limits and Continuity, then Asymptotes, More Asymptotes, or Asymptotes of Oscillating Functions
• Limits of Sequences
• Some special limits 
  
• More challenging limits 
  
• Continuity
• More on Continuity and the Intermediate Value Theorem
• Intermediate Value Theorem
• Intermediate Value Theorem (slide show)
• Intermediate Value Theorem applet
  
• Definition of continuous functions
• What makes a function continuous? (slide show)
• Continuity applet
• An informal look at continuity
• Definition of continuity using limits
  
• Continuity problems, solutions, and explanations
• More on continuity  
  
• Problems on the importance of continuity
• Continuity of functions in one variable
• Discussion and visualization of continuity
• A function continuous at all irrationals, discontinuous at all rationals.
• A function that is continuous at only one point (salt & pepper functions)
• Definition of the Derivative and the Tangent Line Problem
• Review of linear equations  
  
• Limit definition of the derivative
• More on the definition of the derivative
• The difference quotient  (slide show)
• Problems, solutions, and explanations on the definition of the derivative
  
• Geometric interpretation of the derivative
• Animated View of zooming in on a tangent line -- linearization of a curve 
• Animated view of a secant line approaching a tangent line (Click on "Secants and tangents")
• Secant graphs of functions applet
• Graphs a tangent to a function applet
• Problems
• A function that is differentiable at one point and discontinous every where else.
• A function with a derivative defined for all x, but whose derivative is discontinuous.
• A continuous, nowhere differentiable function.
• Symmetric Difference Quotient
• Velocity, Acceleration, & Other Rates of Change
• Average velocity and speed applet
• Instantaneous velocity aplet
  
• Summaries and Review problems
• Review problems for calculating limits and deciding continuity--Go to Calculus Book I, then Limits and Continuity, then anything in that section (the problems on asymptotes and epsilon & deltas may not have been covered in c
• Summary of introduction to the derivative
• Problems

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