SMC Calculus 1

Definition of a function

a function f is a rule that assigns each element x in a set D exactly one element, called f(x) in a set R

4 different representations of functions

verbally, numerically (table of values), visually (graph), algebraically (explicit formula)

domain

the set of all inputs for which the formula makes sense and gives a real-number output

Which rules define functions?

vertical line test

piecewise defined functions

even and odd functions

vertical line test

a visual way to determine if a relation of x if and only if no vertical line intersects the curve more than once

even and odd functions

• even function if f satisfies f(-x) = f(x) for every number x in its domain (symmetric w.r.t the y-axis)

• odd function if f satisfies f(-x) = -f(x) for every number x in its doman (symmetric about the origin i.e. rotating 180 degrees about the origin)

def of increasing

if f(x1) < f(x2) whenever x1 < x2 in interval K

def of decreasing

if f(x1) > f(x2) whenever x1 < x2 in interval K

mathematical model

a mathematical description (often by means of a function of an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product, the life expectancy of a person at birth, or the cost of emissions reductions

process of mathematical model

given a real-world purpose, understand and make predictions

1. identify and name independent and dependent variables

2. apply mathematics

3. interpret conclusions

   1. test prediction by checking against new real data

linear function (slope-intercept form)

y = mx + b

polynomial def

where n is nonnegative integer and a1…. are constants and called coefficients, leading coefficient an ≠ 0 then the degree of the polynomial is n

degree 2 poly/ quadratic

ax² + bx + c, a≠0

degree 3 poly/ cubic

ax³+ bx² + cx + d, a≠0

power function

f(x) = x^a where a is a constant

rational function

f is a ration of 2 polynomials, f(x) = P(x)/ Q(x)

• where P and Q are polynomials

• domain is all x such that Q(x) ≠ 0

algebraic function

f if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials,

• note: any rational function is an algebraic function because it constructed using basic operations

  • functions not algebraic are transcendental

trigonometric functions

sine, cos, tan, sec, csc, cot

periodic functions

sin(x + 2π) = sin x , cos (x+2π) = cosx

exponential functions

f(x) = b^x

• where base b is a positive constant

logarithmic functions

f(x) = logb(x)

• where base b is a positive contant

transformations of functions

translations

stretching or compressing/shrinking
reflection

combinations of cuntions

given 2 functions and f and g, the sum, difference, product, and quotient functions:

composite functions

given 2 functions f and g, the composite function f of g defined by f(g(x))

precise definition of a limit

def of one sided limits

describes the specific value a function approaches as the input (x) gets closer to a target number from only one direction—either from the left or from the righ

def of limits and one sided limits

A limit describes the value a function approaches as its input gets closer to a specific point. A one-sided limit restricts this approach to only one direction—either from the left (values smaller than the target) or from the right (values larger than the target)

how can limits fail to exist?

infinite limit

occurs when a function's output values grow or shrink without bound -∞ or ∞ as x approaches a specific value.

vertical symptote

the vertical line x=a is called a vertical symptote of the curve y = f(x) if at least one of the follow statements is true

1. limit as x approaches a is infinity

2. limit as x approaches a is neg infinity

3. left limit as x approaches a is infinity

4. left limit as x approaches a is neg infinity

5. right limit as x approaches a is infinity

6. right limit as x approaches a is neg infinity

limit laws

• Sum Rule: To find the limit of the sum of two functions, you can simply find the limit of each function individually and add the results together. [1, 2]

• Difference Rule: To find the limit of the difference between two functions, find the limit of each function individually and subtract the second limit from the first. [1, 2]

• Constant Multiple Rule: If you are finding the limit of a function multiplied by a fixed number, you can find the limit of the function first and then multiply the result by that number. [1]

• Product Rule: To find the limit of the product of two functions, find the limit of each function individually and multiply those two limits together. [1, 2]

• Quotient Rule: To find the limit of the division of two functions, find the limit of the top function and divide it by the limit of the bottom function. (This only works if the limit of the bottom function is not zero.) [1, 2]

power law

root law

constant law

identity law

direct substitution property

if f is apolynomial or a rational function and a is in the domain of f the the limit as x approaches a f(x) = f(a)

• when f(a) is not defined, then use algebraic manipulation like cancellation and rationalization techniques bc of 0/0 so know how to factor

thm 1: using one-sided limits

the limit as x approaches a f(x) = L if the limit as x approaches a from the right and left f(x) = L

theorem 2

if f(x) ≤ g(x) when x is near a (except posisble at a) and the limits of f and g both exist as x approaches a, then the limit as x approachesa f(x) ≤ limit as x approaches a of g(x)

squeeze thm

if f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a)

and limit as x approaches a of f(x) and h(x) = L,

then the limit as x approaches a of g(x) is L