calc 1 midterm

Function Notation for Difference Quotient

f(x) = (x+h) - f(x) / h

monotonic

the function does not change direction

monotonically increasing

the function for all x, such that x,1<x,2

Definition of an Exponential Function

“P” is an exponential function of “T” with a base of ‘a’

exponential growth or decay function

𝑷 =𝑷𝟎𝒂𝒕

what does Pnot mean

initial population/amount

what does time equal

time = 0

exponential growth (for a )

a>1

exponential decay (a )

0<a <1

what is the equation for growth/decay rate that includes e + k

𝑃(t)=𝑃0𝑒^𝑘𝑡

if k is greater than 0 is there growth or decay

if k is greater than 0, there is growth! if less than zero, there is decay

Half Life

how much time it takes for something to decay to half → to cut pnot by half

half life equation

P0/2=p0(a^t)

double time

time to double initial value Pnot

double time equation

2P0 = P0(a^t)

Transformation

moving function about the coordinate grid to maintain shape

even functions

there is symmetry about the y axis

odd functions

there is symmetry about the origin

prove an even function algebraically

f(-x)=f(x)

prove an odd function algebraically

f(-x)=-f(x)

Inverse Function

if something is invertible, it must pass both the vertical and horizontal line tests. A function has an inverse If and only if it’s graph intersects any horizontal line at most once.

Properties of Logs: log(AB)

log(A) + log(B)

properties of Logs: Log(A /B)

Log(A)-log(B)

Properties of Logs: log(A^P)

P x log(A)

properties of logs: Log1

0

what is needed for a two sided limit to exist

need both left and right imits to be equal for a two-sided limit to exist

what is the algebraic test for limits

divide all terms by largest value of X in the denominator, rthen apply the limit

concavity of an even power

when K>0 —> concave up

when K<0, concave down

concavity of an odd power

K>0 —> low to high

k<0—> high to low

definition of a limit

the value that a function approaches as the input gets closer and closer to a specific target number

Written as lim x—> a} f(x) = L, which means "the limit of f(x) —> a as x approaches a “L"

if ‘b’ is a constant (Properties of Limits)

then lim b(f(x) = b (lim f(x))

properties of limits: (f(x) x g(x))

limf(x) x limg(x)

properties of limits: lim f(x)/g(x)

limf(x)/limg(x)

intermediate value theorem definition

suppose f(x) is continuous on a closed interval [a .b]. if k is an output value between f(a) and f(b), then there is at least ONE number (c) between [a ,b] such that f(c) is equal to k