### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# math 165, 2-3 week notes Math 165

ISU

### View Full Document

## About this Document

## 41

## 2

1 review

## Popular in Calculus 1

## 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.

## 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*

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "I bought an awesome study guide, which helped me get an A in my Math 34B class this quarter!"

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.