Calculus midterm study terms

what is calculus?

the mathematics of continuous change

Newtons question

if an apple is dropped from the height of x ft, its height h at time t is given by h(t)=-xt² +x

informal definition of a limit

if f(x) becomes arbitrarily close to a single number L as x approaches c from both sides, then the limit of f(x) as x approaches c is L. (lim f(x) x→ c = L)

arbitrarily close

a variable approaches a particular point, a function approaches a specific value.

formal definition of a limit

the statement lim f(x) x→c =L means that for each (epsilon) ε>0, there exists a (delta) δ>0 such that if 0<|x-c|<δ, then |f(x)-L|<ε.

what does c represent in lim f(x) x→ c = L

c is an x-value at which the limit is evaluated.

what does L represnt in lim f(x) x→ c = L

L is a y-value that the function approaches as x approaches c.

introduction to proofs step 1

1. write the whole statement out using the definition

introduction to proofs step 2

2. start with what you want to prove (the “then” part)

   |f(x)-L| <ε

introduction to proofs step 3

3. work backwards from the “then” part to the “if” part

introduction to proofs step 4

4. write a proof that goes from “if” to “then”

0<|x-c|<δ

special trig limits

sinax/bx= a/b,

sinax/x=a

tanx/x=sinx/cosx,

only works when x→0

squeeze theorem

if h(x)≤ f(x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself.

As x equals c, h(x) and g(x) have the same limit, then if f(x) exists, it is equal to L.

must have the same limit

continuity and one-sided limits (informally)

a function f is continuous at a if the graph of does not have a hole or break at a.

point of discontinuity

not continuous at a

continuity, basic definition

if a graph continues without holes or breaks

continuity at a point

1. f(c ) is defined

2. lim x→c f(x) exists

3. lim x→f(x) =f(c )

continuity on an open interval

a function is continuous on an open interval (a,b) when the function is continuous at each point on the interval.

A function that is continuous on the entire real number line (-∞,∞) everywhere is continuous

Right sided limits

x approaches c from values greater than c

Left sided limits

x approaches c from values less than c

continuity on a closed interval

f is continuous on the closed interval when f is continuous on the open interval

greatest integer function

f(x)=[[x]] is defined as less than or equal to x.

tan x

sinx/cosx

sec x

1/cosx

sinx