Flashcards in Chapter 2 Deck (44)

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1

## How do you find the tangent line of a slope when given one point? Section 2.1 (Not usable on test)

### Find the slope of multiple secant lines on either side of the tangent line using a point on the line (that follows the equation) and the point given. Look at what the slope approximates as it gets closer to the limit. (as the chosen point gets closer to the point given)

2

## What is average velocity?

### Change in position/Time elapsed (dealing with a single instance of time, uses the tangent line of a slope equation) Ex: Calculating the instantaneous velocity of a ball after 5 seconds dropped by looking at its change of position from 5 seconds to 5.00...1 and dividing that change in position by 5.00...1-5.

3

## What is the Intuitive Definition of a Limit?

### Supposing f(x) is defined when x nears a, you can say "the limit of f(x), as x approaches a, equals L" if we can make the values of f(x) arbitrarily close to L by making that value of x sufficiently close to a, but not equal to a.

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## In the Definition of a Limit, does f(x) ever have to be defined at (a)?

### No. Ex: a hole at (a)

5

## How do you find the value of a limit if the limit is not continuous? (If the limit of f(x) as x approaches a is NOT f(a))

### Either guess by plugging in x values close to a but not equal to (a) and seeing what they approximate, or solve by simplifying until you can plug in (a)

6

## What does LOGaN=x mean?

### (a)^x = N

7

## What is a natural log? (ln)

### a logarithm with the base of e

8

## What are even and odd functions?

### Even: f(-x)=f(x) and is symmetrical to the y-axis. Odd: symmetrical to the x-axis. The function f(x)=x^n is even if n is even, and odd if n is odd.

9

## How do you multiply powers? Ex: (x^1/3)*(x^2/5)

### Add them. Ex: (x^1/3)*(x^2/5)= x^(1/3+2/5)=x^11/15

10

## What is the intuitive definition of a one-sided limit?

### The left (right) hand limit of f(x) as x approaches (a) is equal to L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to (a) with x less (greater) than (a)

11

## What is the intuitive definition of an infinite limit?

### Let f be a function defined on both sides of a, except possibly a. The values of f(x) can be made arbitrarily large (negative) by taking x sufficiently close to a, but not equal to a.

12

## What is the definition of a vertical asymptote using limits?

### The vertical line x=a is called a vertical asymptote if one of the following is true: limf(x) as x approaches a^(-+) is (-) infinity. (If any way x approaches a creates either positive or negative infinity)

13

## How do you find vertical asympotes?

###
Create a LIMIT where the numerator is fixed and the denominator approaches 0.

Ex: Vert Asymptotes of f(x)=tanx are where cosx=0, which is pi/2 (The limit of tanx as x approaches pi/2 is infinity)

14

## What are the five arithmetic Limit Laws?

###
1. A limit of a sum is the sum of the limits

Ex: lim(f(x)+g(x))= limf(x)+limg(x)

2. A limit of a difference is the difference of the limits

Ex: lim(f(x)-g(x))= limf(x)-limg(x)

3. The limit of a constant times a function is the constant times the limit of the function

Ex: lim(c*f(x))= c*limf(x)

4. The limit of a product is the product of the limits

Ex: lim(f(x)g(x))= limf(x)*limg(x)

5. The limit of a quotient is the quotient of the limits (Provided the limit of the denominator does not equal 0)

Ex: lim(f(x)/g(x))= limf(x)/limg(x)

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## What is the power Limit Law?

### lim(f(x))^n= (limf(x))^n. Same for square roots

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## Limit Theorem 1

### Thm 1: limf(x) as x->a is L if and only if limf(x) x->a- = L = limf(x) x->a+

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## Squeeze Theorem (Limit Theorem 3)

### If limf(x)a x->a and limh(x) x->a = L, the limg(x) x->a =L (If limf(x) is less than or equal to limg(x) which is less than or equal to limh(x))

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## Precise definition of a limit

### Let f(x) be defined...The limit of f(x) as x approaches a is L, and write limf(x) x->a =L, if for every Epsilon>0 there is a Delta>0 such that if Ix-aI

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## What does defined at an open interval, closed interval mean?

###
Open interval does not include endpoints (x,y)

Closed interval does include endpoints [x,y]

20

## Method for Proving/Verifying a limit using the Precise definition. With lim (x^2 -x+5) x->2=7

###
With lim(x^2-x+5) x->2=7

Given E>0. Let d=__. If |x-2|

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## Precise definition of left hand limits

### limf(x) x->a- =L if for every Epsilon >0 there is a number Delta >0 such that if a-delta

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## Precise definition of right hand limits

### limf(x) x->a+ if for every Epsilon >0 there is a number Delta>0 such that if a

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##
How do you change the method of Verifying/Proving a limit using the Precise definition if you end up with

|x-a||x+n|

### set |x-a|

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## What is the precise definition of an infinite limit?

###
Let f be a function defined on some open interval that contains the number a, except possibly a itself. Then the limf(x) x->a = (-)infinity means that for every positive number M there is a positive number Delta such that if |x-a|(wirgiuqwegh ds

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25

## What is the definition of a continuous function? What does this implicitly require?

###
A function f is continuous at a number a if

limf(x) x->a =f(a)

This implicitly requires that

f(a) is defined (a is in the domain of f,Ex: f(a) is not a hole)

The limf(x) x->a exists Ex: It's not a jump discontinuity

The limf(x) x->a =f(a) Ex: If there is an open hole for the limit of f(x) and a closed dot floating above it, the closed dot is f(a) but the limit of f(x) is the open dot

((To show that a function is continuous, must show that its limit follows these rules))

26

## What is the definition of continuous from the right or from the left function?

###
A function f is continuous from the right at a number a if

limf(x) x->a+ =f(a)

f is continuous from the left at a if limf(x) x->a- =f(a)

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## What is the definition of continuous at an interval? How do you show this with a given function?

###
A function f is continuous on an interval if it is continuous at every number in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the left or continuous from the right)

You show this by showing that the given functions limit as x->a is f(a)

28

## If f and g are continuous at a and c is a constant, what five functions are also continuous at a?

###
1:f+g

2:f-g

3:cf

4: fg

5: f/g (if g does not =0)

29

## Where are polynomials continuous? Where are rational functions continuous?

###
Polynomials are continuous everywhere (R=(-infinity,infinity)

A rational function is continuous wherever it is defined

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