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## math 165, 2-3 week notes

1 review
by: shreyash Notetaker

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# math 165, 2-3 week notes Math 165

Marketplace > Iowa State University > Mathematics (M) > Math 165 > math 165 2 3 week notes
shreyash Notetaker
ISU

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these notes cover limits from 2.4-2.6 and also has a little bit of chapter 3 ( tangent and derivative of points)
COURSE
Calculus 1
PROF.
Professor Miriam Castillo-Gil
TYPE
Bundle
PAGES
28
WORDS
CONCEPTS
math165, calculus-1, Limits
KARMA
75 ?

## 2

1 review
"Yes YES!! Thank you for these. I'm such a bad notetaker :/ will definitely be looking forward to these"
Elton Kiehn

## Popular in Mathematics (M)

This 28 page Bundle was uploaded by shreyash Notetaker on Thursday January 28, 2016. The Bundle belongs to Math 165 at Iowa State University taught by Professor Miriam Castillo-Gil in Fall 2015. Since its upload, it has received 41 views. For similar materials see Calculus 1 in Mathematics (M) at Iowa State University.

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## Reviews for math 165, 2-3 week notes

Yes YES!! Thank you for these. I'm such a bad notetaker :/ will definitely be looking forward to these

-Elton Kiehn

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Date Created: 01/28/16
2.4 One-Sided Limits Deﬁnitions: (Side limits) Left-hand side limit; we write This is the value that f(x) approaches, if it exists, when x approaches c from the left, that is we consider values Similarly, right- hand side limit; we write This is the value f(x) approaches, if it exists, when x approaches c from the right, that is we consider values Example: Let f(x) = . Its domain is For a function f(x) to have a limit L as xc, f(x) must be deﬁned on both sides of c and the one sided limits must both approach L as x c and x c. Theorem. Claim. We will ﬁnd the limit of as ( in radians) In fact we will ﬁnd the right-sided limit, the left-sided is found similarly. We will need the formula for the area of a sector of a circle or radius r that sweeps radians. Since By squeeze theorem 2.5 Continuity A continuious function has an "unbroken" graph (one you can draw without lifting your pen) We ﬁrst deﬁne continuity at a point. Deﬁnition: A function y=f(x) is continuous at a point c of its domain (not an endpoint) if When the domain includes endpoints of an interval we use one-sided limits to deﬁne continiuty f is continuous at a left endpoint if f is continuous at a right endpoint if The function g(x) is continuous at the points: And discontinuous at A function f(x) is continuous at an interior Continuity Test point x=c of its domain if and only if: 1. f(x) is 2. 3. We have three kinds of discontinuities: Discontiniuty Discontiniuty Discontiniuty A function is continuous if it is continuous at every point of its domain. Combinations of continuous functions are continuous If f(x) and g(x) are continuous functions at x=c, then the following functions are also continuous at x=c. 1. Sums and Differences 6. Compositions 2. Constant Multiples 3. Products 4. Quotients 5. Powers and Roots Examples Polynomials are continuous functions Sine and Cosine are continuous at zero, from example 11 in 2.2 we know: In general all trigonometric functions are continuous ( Theorem 10 If g(x) is a continuous function at x=b and Lim f(x)=b , then Example Find Lim cos(2x + sin ( + x)) Example Find Lim ( x+1 ) ( e ) , given that continuous. Example The inverse of a continuous function is continuous. Continuous Extension at a Point We can redeﬁne functions with removable discontinuities to obtain continuous functions. The function f(x)= is discontinuous at x=0 Since the limit as x approaches zero exists, the discontinuity is removable. We know Lim For the function to be continuous at zero we need to deﬁne f(0) we make f(0)= and redeﬁne the function: F(x)= In general we can redeﬁne a function f with removable discontinuity at x= c by F(x)= We say F is the continuous extension of f to x=c. Example Find a continuous extension of to x=2 Intermediate Value Theorem If f is a continuous function on a closed interval [a,b], and if y is any value between f(a) and f(b), then y = for some value c such that a < c < b Example Show that there is a root of the equation between 1 and 2. Example Use the Intermediate Value Theorem (I.V.Thm) to prove

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