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Limits, Asymptotes, and Continuity Ex.
Def. A horizontal asymptote of f (x) occurs at y = L if or Def. A vertical asymptote of f (x) occurs at values of x where f (x) is undefined (sort of). and are examples of graphs that have a hole.
Ex. Find all asymptotes of, then sketch the graph.
means that x approaches 2 from the right (larger than 2) means that x approaches 2 from the left (smaller than 2) One-Sided Limits Ex.
Thm. The limit exists if both sides agree.
Ex. For the function given, find: a. b. c.
Ex. Find if
Def. (loose) A function is continuous on an interval if the graph has no gaps, jumps, or breaks on the interval. Ex. Is continuous on [0,5]?
Def. (tight) A function f (x) is continuous on an interval if, for all points c on the interval: i. exists ii. exists iii.
Ex. Let Find a value of B so that f (x) is continuous at x = 0.
The test on Chapter 1 will be on Monday. Next class we will review and Ill pass out a Sample Test so you know what types of questions I can ask.
AP Calculus Review First Semester Differentiation to the Edges of Integration Sections , 3.9, (7.7)
2.3 Continuity When you plot function values generated in a laboratory or collected in a field, you can connect the plotted points with an unbroken curve.
LIMITS What is Calculus? What are Limits? Evaluating Limits –Graphically –Numerically –Analytically What is Continuity? Infinite Limits This presentation.
ASYMPTOTES TUTORIAL Horizontal Vertical Slant and Holes.
Page 0 Introduction to Limits. Page 1 Definition of the limit of f(x) as x approaches a: We write and say the limit of f(x), as x approaches a, equals.
Page 56 We have done limits at infinity (i.e., lim x f(x) and lim x – f(x) where f(x) is a rational function), what happen if f(x) is not a rational function?
Rational Functions. Definition: A Rational Function is a function in the form: f(x) = where p(x) and q(x) are polynomial functions and q(x) 0. In this.
Linear & Quadratic Functions PPT What is a function? In order for a relation to be a function, for every input value, there can only be one output.
INFINITE LIMITS. LIMITS What is Calculus? What are Limits? Evaluating Limits –Graphically –Numerically –Analytically What is Continuity? Infinite Limits.
3.5 Continuity & End Behavior. Discontinuous – you cannot trace the graph of the function without lifting your pencil. (step and piecewise functions)
The first chapter is primarily a review of topics that youve seen in Pre-Calculus and other classes. If you are having trouble with any of these topics,
3.1 Derivative of a Function What youll learn Definition of a derivative Notation Relationships between the graphs of f and f Graphing the derivative from.
Horizontal Lines Vertical Lines Lines, Lines, Lines!!! ~
Finding x- and y-intercepts algebraically. y-intercepts For every point along the y-axis, the value of x will be zero (x=0) We can use this fact to find.
EXAMPLE 1 Write an equation of a line from a graph SOLUTION m 4 – (– 2) 0 – 3 = 6 – 3 = = – 2 STEP 2 Find the y -intercept. The line intersects the y -axis.
A set of ordered pairs is called a __________.. A set of ordered pairs is called a relation.
Chapter 3: Applications of Differentiation L3.5 Limits at Infinity.
1 Graph Sketching: Asymptotes and Rational Functions OBJECTIVES Find limits involving infinity. Determine the asymptotes of a function’s graph.
9.3 Rational Functions and Their Graphs. If the graph is not continuous at x = a then the function has a point of discontinuity at x = a.
Graphing Using Tables (continued). Graph 2x + y = 4 using a table.
Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs OBJECTIVES Find relative extrema of a continuous function using the First-Derivative.
Weve looked at linear and quadratic functions, polynomial functions and rational functions. We are now going to study a new function called exponential.
3.7 Graphs of Rational Functions. A rational function is a quotient of two polynomial functions. A rational function is a quotient of two polynomial functions.
Pre-Algebra 11-2 Slope of a Line 11-2 Slope of a Line Pre-Algebra Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation.
3.1 Derivatives. Derivative A derivative of a function is the instantaneous rate of change of the function at any point in its domain. We say this is.
2.2 Limits Involving Infinity Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.
Section c Let h be a function defined for all such that h(4) = -3 and the derivative of h is given by for. Write an equation for the line tangent.
Inverses of Functions Part 2 Lesson 2.9. Reminder from yesterday.
Y – Intercept of a Line The y – intercept of a line is the point where the line intersects or “cuts through” the y – axis.
2.2 Limits Involving Infinity Finite Limits as – The symbol for infinity does not represent a real number. – We use infinity to describe the behavior of.
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