Chapter 2

Question 1

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

Answer 1

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)

Question 2

What is average velocity?

Answer 2

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.

Question 3

What is the Intuitive Definition of a Limit?

Answer 3

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.

Question 4

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

Answer 4

No. Ex: a hole at (a)

Question 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))

Answer 5

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)

Question 6

What does LOGaN=x mean?

Answer 6

(a)^x = N

Question 7

What is a natural log? (ln)

Answer 7

a logarithm with the base of e

Question 8

What are even and odd functions?

Answer 8

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.

Question 9

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

Answer 9

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

Question 10

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

Answer 10

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)

Question 11

What is the intuitive definition of an infinite limit?

Answer 11

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.

Question 12

What is the definition of a vertical asymptote using limits?

Answer 12

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)

Question 13

How do you find vertical asympotes?

Answer 13

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)

Question 14

What are the five arithmetic Limit Laws?

Answer 14

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(cf(x))= climf(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)

Question 15

What is the power Limit Law?

Answer 15

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

Question 16

Limit Theorem 1

Answer 16

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

Question 17

Squeeze Theorem (Limit Theorem 3)

Answer 17

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))

Question 18

Precise definition of a limit

Answer 18

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

Question 19

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

Answer 19

Open interval does not include endpoints (x,y)

Closed interval does include endpoints [x,y]

Question 20

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

Answer 20

With lim(x^2-x+5) x->2=7 
Given E>0. Let d=\_\_. If |x-2|

Question 21

Precise definition of left hand limits

Answer 21

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

Question 22

Precise definition of right hand limits

Answer 22

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

Question 23

How do you change the method of Verifying/Proving a limit using the Precise definition if you end up with
|x-a||x+n|

Answer 23

set |x-a|

Question 24

What is the precise definition of an infinite limit?

Answer 24

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Question 25

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

Answer 25

``` 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)) ```

Question 26

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

Answer 26

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)

Question 27

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

Answer 27

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)

Question 28

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

Answer 28

1: f+g 2: f-g 3: cf 4: fg 5: f/g (if g does not =0)

Question 29

Where are polynomials continuous? Where are rational functions continuous?

Answer 29

Polynomials are continuous everywhere (R=(-infinity,infinity) A rational function is continuous wherever it is defined

Question 30

What does (fog)(x) mean? When is (fog) (x) continuous at x=a?

Answer 30

f(g(x)) It is continuous if g is continuous at x=a AND f is continuous at g(a) Ex: show that h(x)=sinx^2 is continuous because x^2 is continuous for all real numbers and so is sinx

Question 31

What does the Intermediate Value Theorem state? How do you use it to prove solutions?

Answer 31

Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) does not equal f(b). Then there exists a number c in (a,b) such that f(c) =N Use it to prove a specific solution of a function by showing that there are solutions above and below the specific one asked for (as long as the function is continuous)

Question 32

Intuitive Definition of a limit at Infinity

Answer 32

Let f be a function defined on some interval (a,infinity) ((-infinity,a)) Then limf(x) x-> (-)infinity =L means that the values of f(x) can be made arbitrarily close to L by requiring x to be sufficiently large (negative)

Question 33

What is the definition of a horizontal asymptote?

Answer 33

The line y=L is called a horizontal asymptote of the curve y=f(x) if either limf(x) x->infinity = L or limf(x) x->-infinity =L

Question 34

How do you evaluate a limit at infinity?

Answer 34

Since any function C(constant)/x^n(rational number) when x goes to infinity eventually reaches 0, divide both the numerator and the denominator by the highest power of x that occurs in the denominator. After that, set all values with a denominator x^n to zero, and simplify

Question 35

What should you remember when evaluating a limit at | -infinity? How does this work for finding horizontal asymptotes?

Answer 35

For x>0, the square root of x^2 = |x| = -x. In finding a horizontal asymptote of a function, you need to find out what the limit of that function is when x-> both infinity and -infinity

Question 36

How do you find a vertical asymptote of a function using limits?

Answer 36

You set the function as a limit, with x-> whatever value makes the denominator 0

Question 37

What is the precise definition of a limit at infinity?

Answer 37

Let f be a function defined on some interval (a,infinity) ((infinity,a)). Then limf(x) x->infinity=L means that for every Epsilon >0 there is a corresponding number N such that if x>N then |f(x)-L|

Question 38

What is the precise definition for an infinite limit at infinity?

Answer 38

Let f be a function defined on some interval (a,infinity). Then limf(x) x->infinity = infinty means for every positive number M there is a corresponding positive number N such that if x>N then f(x) >M

Question 39

What the equation for the (Derivative)/(Slope of the tangent line)/(Instantaneous Rate of Change) of f(x) at a?

Answer 39

``` f prime(f^1)(a)= lim x->a (f(x)-f(a))/(x-a) OR f^1(a)=lim h->0 (f(a+h)-f(a))/h ((IF H=X-A)) ```

Question 40

What is the equation for instantaneous velocity?

Answer 40

v(a)=lim h->0 (f(a+h)-f(a))/h

Question 41

What does it mean to say a function is differentiable? When is a function differentiable on an open limit?

Answer 41

A function is differentiable at a if fprime(a) exists. It is differentiable on an open interval (a,b) [or (a,infinity) or (-infinity,a) or (-infinity,infinity)] if it is differentiable at every number in the interval

Question 42

How do you graph f^1?

Answer 42

since f prime is the slope, graph the slope

Question 43

How do you find acceleration?

Answer 43

The instantaneous rate of change of velocity with respect to time, the s^1(derivative of the velocity) (s^1 of the s^1 (s^2)

Question 44

How do you find velocity?

Answer 44

The derivative of the displacement with respect to time, substitute t in for x using the derivative equation