AP CALCULUS 1003 Limits pt.3 Limits at Infinity and End Behavior.

Slides:



Advertisements
Similar presentations
9.3 Rational Functions and Their Graphs
Advertisements

1.2 Functions & their properties
Horizontal Vertical Slant and Holes
5.2 Rational Functions and Asymptotes
Graphing Rational Functions
Discussion X-intercepts.
Section 5.2 – Properties of Rational Functions
Rational Functions. 5 values to consider 1)Domain 2)Horizontal Asymptotes 3)Vertical Asymptotes 4)Holes 5)Zeros.
3208 Unit 2 Limits and Continuity
Applications of Differentiation Curve Sketching. Why do we need this? The analysis of graphs involves looking at “interesting” points and intervals and.
WARM UP: Factor each completely
Chapter 3: The Nature of Graphs Section 3-1: Symmetry Point Symmetry: Two distinct points P and P’ are symmetric with respect to point M if M is the midpoint.
Finding Limits Graphically and Numerically An Introduction to Limits Limits that Fail to Exist A Formal Definition of a Limit.
 A continuous function has no breaks, holes, or gaps  You can trace a continuous function without lifting your pencil.
AP Calculus 1004 Continuity (2.3). C CONVERSATION: Voice level 0. No talking! H HELP: Raise your hand and wait to be called on. A ACTIVITY: Whole class.
Chapter 1 Limit and their Properties. Section 1.2 Finding Limits Graphically and Numerically I. Different Approaches A. Numerical Approach 1. Construct.
Sec 5: Vertical Asymptotes & the Intermediate Value Theorem
Pre Calculus Functions and Graphs. Functions A function is a relation where each element of the domain is paired with exactly one element of the range.
Continuity 2.4.
AP Calculus 1005 Continuity (2.3). General Idea: General Idea: ________________________________________ We already know the continuity of many functions:
9.3 Rational Functions and Their Graphs Rational Function – A function that is written as, where P(x) and Q(x) are polynomial functions. The domain of.
Infinite Limits Lesson 1.5.
Limits at Infinity Horizontal Asymptotes Calculus 3.5.
NPR1 Section 3.5 Limits at Infinity NPR2 Discuss “end behavior” of a function on an interval Discuss “end behavior” of a function on an interval Graph:
2.7 Limits involving infinity Wed Sept 16
RATIONAL FUNCTIONS A rational function is a function of the form:
Copyright © Cengage Learning. All rights reserved.
Rational Functions and Their Graphs. Example Find the Domain of this Function. Solution: The domain of this function is the set of all real numbers not.
Asymptotes.
2.2 Limits Involving Infinity Goals: Use a table to find limits to infinity, use the sandwich theorem, use graphs to determine limits to infinity, find.
Limits. What is a Limit? Limits are the spine that holds the rest of the Calculus skeleton upright. Basically, a limit is a value that tells you what.
Review Limits When you see the words… This is what you think of doing…  f is continuous at x = a  Test each of the following 1.
1.4 Continuity  f is continuous at a if 1. is defined. 2. exists. 3.
Graphing Rational Functions. What is a rational function? or.
Jonathon Vanderhorst & Callum Gilchrist Period 1.
Date: 1.2 Functions And Their Properties A relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain.
AP CALCULUS 1003 Limits pt.2 One Sided Limits and Infinite Limits.
Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: If you look at your speedometer during this trip, it might read 65 mph.
AP CALCULUS 1004 Limits pt.3 Limits at Infinity and End Behavior.
Analyzing and sketching the graph of a rational function Rational Functions.
Limits Involving Infinity Section 1.4. Infinite Limits A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite.
GRAPHS OF RATIONAL FUNCTIONS F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of.
1.4 Continuity and One-Sided Limits Main Ideas Determine continuity at a point and continuity on an open interval. Determine one-sided limits and continuity.
1.4 Continuity Calculus.
The foundation of calculus
Limits and Continuity Definition Evaluation of Limits Continuity
Ch. 2 – Limits and Continuity
1.5 Infinite Limits Main Ideas
Graphing Rational Functions Part 2
Horizontal Asymptotes
25. Rational Functions Analyzing and sketching the graph of a rational function.
Unit 4: Graphing Rational Equations

Ch. 2 – Limits and Continuity
2.2 Limits Involving Infinity, p. 70
The Sky is the Limit! Or is it?
3. Limits and infinity.
Warm UP! Factor the following:.
AP Calculus September 6, 2016 Mrs. Agnew
Important Values for Continuous functions
Graphing Rational Functions
Sec 4: Limits at Infinity
2.2 Limits Involving Infinity
Section 5.2 – Properties of Rational Functions
26 – Limits and Continuity II – Day 1 No Calculator
Domain, Range, Vertical Asymptotes and Horizontal Asymptotes
1.4 Continuity and One-Sided Limits This will test the “Limits”
Solving and Graphing Rational Functions
Find the zeros of each function.
Asymptotes, End Behavior, and Infinite Limits
Presentation transcript:

AP CALCULUS 1003 Limits pt.3 Limits at Infinity and End Behavior

REVIEW: ALGEBRA is a ________________________ machine that ___________________ a function ___________ a point. CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point Function Evaluates at Limit Describes the behavior of near

END BEHAVIOR

LIMITS AT INFINITY Part 2: End Behavior GENERAL IDEA: The behavior of a function as x gets very large ( in a positive or negative direction)

END Behavior: Limit Layman’s Description: Notation: Horizontal Asymptotes: Note: m Closer to L than ε If it has a limit = L then the HA y=L

GNAW: Graphing EX: If you cover the middle what happens?

GNAW: Algebraic Method: DIRECT SUBSTITUTION gives a second INDETERMINANT FORM Theorem: Method:Divide by largest degree in denominator

End Behavior Models EX: (with Theorem) = End behavior HA y=

End Behavior Models Summary: ________________________________________ A). B). C). Leading term test (reduce leading term) If degree on bottom is largest limit = 0 If the degrees are the same then the limit = reduced fraction If the degrees on top is larger limit DNE but EB acts like reduced power function E.B. HA y=0

Continuity

General Idea: General Idea: ________________________________________ We already know the continuity of many functions: Polynomial (Power), Rational, Radical, Exponential, Trigonometric, and Logarithmic functions DEFN: A function is continuous on an interval if it is continuous at each point in the interval. DEFN: A function is continuous at a point IFF a) b) c) Can you draw without picking up your pencil Has a point f(a) exists Has a limit Limit = value

Continuity Theorems

Continuity on a CLOSED INTERVAL. Theorem: A function is Continuous on a closed interval if it is continuous at every point in the open interval and continuous from one side at the end points. Example : The graph over the closed interval [-2,4] is given. From the right From the left

Discontinuity No value f(a) DNE hole jump Limit does not equal value Limit ≠ value Vertical asymptote a) c) b)

Discontinuity: cont. Method: (a). (b). (c). Removable or Essential Discontinuities Test the value =Look for f(a) = Test the limit Holes and hiccups are removable Jumps and Vertical Asymptotes are essential Test f(a) =f(a) = Lim DNE Jump = cont. ≠ hiccup

Examples: EX: Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? removable or essential? = x≠ 4 Hole discontinuous because f(x) has no value It is removable

Examples: cont. Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? removable or essential? x≠3 VA discontinuous because no value It is essential

Examples: cont. Identify the x-values (if any) at which f(x)is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? Step 1: Value must look at 4 equation f(1) = 4 Step 2: Limit It is a jump discontinuity(essential) because limit does not exist

Graph: Determine the continuity at each point. Give the reason and the type of discontinuity. x = -3 x = -2 x = 0 x =1 x = 2 x = 3 Hole discont. No value VA discont. Because no value no limit Hiccup discont. Because limit ≠ value Continuous limit = value VA discont. No limit Jump discont. Because limit DNE

Algebraic Method a. b. c. Value:f(2) = 8 Look at function with equal Limit: Limit = value: 8=8

Algebraic Method At x=1 a. b. c. At x=3 a. b. Limit c. Value: Limit: Jump discontinuity because limit DNE, essential Value: Hole discontinuity because no value, removable No further test necessary f(1) = -1

Rules for Finding Horizontal Asymptotes 1. If degree of numerator < degree of denominator, horizontal asymptote is the line y=0 (x axis) 2. If degree of numerator = degree of denominator, horizontal asymptote is the line y = ratio of leading coefficients. 3. If degree of numerator > degree of denominator, there is no horizontal asymptote, but possibly has an oblique or slant asymptote.

Consequences of Continuity: A. INTERMEDIATE VALUE THEOREM ** Existence Theorem EX: Verify the I.V.T. for f(c) Then find c. on If f© is between f(a) and f(b) there exists a c between a and b c ab f(a) f(c) f(b) f(1) =1 f(2) = 4 Since 3 is between 1 and 4. There exists a c between 1 and 2 such that f(c) =3 x 2 =3 x=±1.732

Consequences: cont. EX: Show that the function has a ZERO on the interval [0,1]. I.V.T - Zero Locator Corollary CALCULUS AND THE CALCULATOR: The calculator looks for a SIGN CHANGE between Left Bound and Right Bound f(0) = -1 f(1) = 2 Since 0 is between -1 and 2 there exists a c between 0 and 1 such that f(c) = c Intermediate Value Theorem

Consequences: cont. EX: I.V.T - Sign on an Interval - Corollary (Number Line Analysis) We know where the zeroes are located Choose a point between them to determine whether the graph is positive or negative

Consequences of Continuity: B. EXTREME VALUE THEOREM On every closed interval there exists an absolute maximum value and minimum value.