Chapter 2 Flashcards
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)
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.
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.
In the Definition of a Limit, does f(x) ever have to be defined at (a)?
No. Ex: a hole at (a)
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)
What does LOGaN=x mean?
(a)^x = N
What is a natural log? (ln)
a logarithm with the base of e
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.
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
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)
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.
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)
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)
What are the five arithmetic Limit Laws?
- A limit of a sum is the sum of the limits
Ex: lim(f(x)+g(x))= limf(x)+limg(x) - A limit of a difference is the difference of the limits
Ex: lim(f(x)-g(x))= limf(x)-limg(x) - The limit of a constant times a function is the constant times the limit of the function
Ex: lim(cf(x))= climf(x) - The limit of a product is the product of the limits
Ex: lim(f(x)g(x))= limf(x)*limg(x) - 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)
What is the power Limit Law?
lim(f(x))^n= (limf(x))^n. Same for square roots
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+
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))
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
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]
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|
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
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
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|
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 asd]as]dfa]sdf asd]f]asdf]asd]fas ds]fa]sdf]as]df asdf]asd fa sdf asdf as df asd f asdf as df asd f asdf as df asd
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))
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)
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)
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)
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
What does (fog)(x) mean? When is (fog) (x) continuous at x=a?
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
What does the Intermediate Value Theorem state? How do you use it to prove solutions?
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)
Intuitive Definition of a limit at Infinity
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)
What is the definition of a horizontal asymptote?
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
How do you evaluate a limit at infinity?
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
What should you remember when evaluating a limit at
-infinity? How does this work for finding horizontal asymptotes?
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
How do you find a vertical asymptote of a function using limits?
You set the function as a limit, with x-> whatever value makes the denominator 0
What is the precise definition of a limit at infinity?
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|
What is the precise definition for an infinite limit at infinity?
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
What the equation for the (Derivative)/(Slope of the tangent line)/(Instantaneous Rate of Change) of f(x) at a?
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))
What is the equation for instantaneous velocity?
v(a)=lim h->0 (f(a+h)-f(a))/h
What does it mean to say a function is differentiable? When is a function differentiable on an open limit?
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
How do you graph f^1?
since f prime is the slope, graph the slope
How do you find acceleration?
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)
How do you find velocity?
The derivative of the displacement with respect to time, substitute t in for x using the derivative equation
