theorems Flashcards
function
A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. The
set of inputs is called the domain of the function. The set of outputs is called the range of the function.
vertical line test
Given a function f , every vertical line that may be drawn intersects the graph of f no more than once. If any vertical
line intersects a set of points more than once, the set of points does not represent a function.
when we have a graph that’s symmetric with respect to the y axis, we would call it
even
x -intercept
let y be zero and solve equation for x
y-intercept
let x be zero and solve equation for y
symmetry of a graph
There three types of symmetry
- symmetric with respect to the y-axis. this mean the portion of the graph in the y-axis is the mirror image of the portion of the graph in the y-axis
- symmetric with respect to the x-axis this mean the portion of the graph in the x-axis is the mirror image of the portion of the graph in the x axis
- symmetric with respect to the origin
Symmetry theorem
- The graph of an equation in x and y is symmetric with respect to the y-axis when replacing x by -x yields an equivalent equation
- The graph of an equation in x and y is symmetric with respect to the x-axis when replacing y by -y yields an equivalent equation
- The graph of an equation in x and y is symmetric with respect to the origin when replacing x by -y and y by -y yields and equivalent eqation .
symmetric to the y test - just change the x to -x
symmetric to the x test - just change the y to -y, which makes the whole function negative
The graph of a polynomial has symmetry with respect to the y-axis when each term has
even exponent (constant) y=2x^4-x^2+2
the graph of a polynomia has a symmetry with respect to the origin when each term has an
odd exponent
even function test
f(-x)=f(x), the graph of cos and sec
odd function test
f(-x)=-f(x) the graph of sin, tan, cot, and csc
inverse function test
f(g(x))= g(f(x)) = x, they both had yeild the same value
vertical line test
Given a function f , every vertical line that may be drawn intersects the graph of f no more than once. If any vertical
line intersects a set of points more than once, the set of points does not represent a function.
if a graph does not pass a vertical line test, then that graph is not a function
to find the zeros of a function
just make f(x) equal to zero and solve for the x
for example:
find the zeros of f(x)=x^2+2
x=(x)^1/2
three scenarios limit does not exist
- Asymptotes
- break and jump
- biesce wise function
if a function is differentiabile
continuous
if a function is not differentiable
not continuous and you can not find the derivative
three scenarios that derivative, does not exist
vertical line, for example: (x)^1/3
sharp point: |x|
squeeze theorem
If f, g, and h are functions defined on some open interval containing a such that g(x)<=f(x),=h(x) for all x in the interval except possibly at a itself, and lim g(x) = lim h(x) = L then the lim f(x) = L x--> a x-->a x--->
infinite limits
if f is a function defined at every number in some open interval containing a, except possible at a itself, then
if a function approaches to the positive infinity, we’re going to say f(x) increases without bound as x approaches a.
if a function approaches to the negative infinity, we’re going to say f(x) decreases without bound as a x approaches a.
limit theorems
- If n is a positive integer, then
the limit of a function as x approaches to infinity, the power of the numerator is less than the power of the denominator, so the limit is
0
if denominator is equal with numerator, the limit to infinity is equal to
coefficient
if the degree of denominator is less than numerator, just evaluate the limit by
reducing the variable and find for the real limit
the limit of a constant
constant
finding the aymptote using limit
just make it approach to infinity then u’re going to find horizontal asymptote
just make it approach to the denominator and u’re going to find vertical asymptote
a function said to be continuous
f(a) exists lim f(x) = exists x---> a lim f(x) = f(a) x---> a
continuous
a graph with no hole or break
continuity theorem:
1. A polynomial function is continuous every where
2. A rational function is continuous, except at a point results the denominator to zero
3. If a function f and g are continuous at a, then the functions f+g, f-g, f*g, f/g, and g(a)not equal to 0, are also continuous at a.
Intermediate value theorem
If a function f is continuous on a closed interval [a, b] and k is a number with f(a) <=k<=f(b), then there exists a number c in [a, b] such that f(c) =k
differentiability
given a function f, if f’(x) exists at x=a, then the function f is said to be differentiable at x=a. If a function is differentiable at x=a, the f is continuous at x=a.
If a function f is continuous at x=a, then f may or may not be differentiable at x=a.
cases for differentiability
cusp = not differentiable
vertical tangent = not differential
has corner= not differential
to know if a function has an inverse
just find the derivative of the function and compare with zero or check if it is greater than zero and the function would be increasing and it would pass the horizontal line test.
if a function passes a horizontal line test and that function has an inverse
higher order derivatives
if the derivative f’ of a function f is differentiable, then the derivative of f’ is the second derivative of f represented by f’’ (reads as f double prime). You can continue to differentiate f as long as there is differentiability.
Rolle’s Theorem
if f is continuous on a closed interval [a, b], f is differentiable on the open interval (a,b) and f(a)=f(b)=0, then there exists a number c where f’(c)=0
Mean value Theorem
if a f is continuous on a closed interval [a, b] and f is differentiable on the open interval (a, b) then there exists a number c in (a, b) such that f’(c) =f(b)-f(a)/b-a
