45 terms

continuity

we can draw its graph without lifting a pencil from paper, has no holes, breaks, or jumps on interval

f(c) exists

lim x->c f(x) exists

lim x->c f(x)=f(c)

f(c) exists

lim x->c f(x) exists

lim x->c f(x)=f(c)

jump discontinuity

when the left and right hand limits exist, but are different

removable discontinuity

lim x->c f(x) not equal to f(2)

infinite discontinuity

when it has a vertical asymptote

extreme value theorem

if f is continuous on the closed interval [a,b], then f attains a minimum and maximum value somewhere in the interval

the intermediate value theorem

if f is continuous on the closed interval [a,b], and M is a number such that f(a)<M<f(b), then there exists a number c, the the interval

instantaneous rate of change

limit of the difference quotient

memorize derivatives

refer to page 114

Mean Value Theorem

if the function f(x) is continuous at each point on the closed interval a<x<b and has a derivative at each point on the open interval a<x<b, then there is at least one number c, a<c<b, such that f(b)-f(a)/b-a=fprimec

Rolle's Theorem

addition to hypotheses of the MVT, given that f(b)=f(a)=k then there is a number c, between a and b such that fprimec=0

intermediate forms

0/0, or infinity/infinity, 0xinfinity, infinity-infinity, o^0, 1^infinity, infinity^0

L'Hopital's rule

take the derivative of the functions int he numerator and the denominator if it results in intermediate form

instantaneous rate of change

limit of the average rate of change as h->0

instantaneous rate of change vs average rate of change

instantaneous change is defined as the derivative of the function evaluated at a single point. The average change is the same as always (pick two points, and divide the difference in the function values by the difference in the values).

to determine increasing and decreasing functions

derivatives with discontinuities

derivatives with discontinuities

solve for critical values and use number line, if the derivative is negative then f(x) is decreasing, and vice versa

derivatives with discontinuities

when the derivative is not defined

To determine minimums, maximums, and points of inflection

cusps

cusps

find derivative/second derivative, find critical points, if second derivative is greater than 0 then c yields a local minimum and vice versa

cusps

don't have minimums or maximums, but can change in concavity

optimization problems

find equations, find equation you need to minimize, substitute original equation into equation for minimization, take the derivative, set equal to 0 and solve

relating a function and its derivatives graphically

refer to page 174

to construct graphs of f, f', f''

use number lines and know when they change signs, minimums, maximums, points of inflections

motion rules

if velocity is greater than 0 then the particles is moving to the right and vv, if acceleration if greater than 0 then velocity is increasing and vv, if acceleration and velocity are both positive then they are accelerating and if they have different signs then they are decelerating, if the position of the particle is continuous then the particle reverses its direction whenever velocity is 0 and the acceleration is different from 0

local linear approximation

local linear approximation

local linear approximation

if f'(a) exists, then the local linear approximation of f(x) at a is f(a) + f'(a)(x-a)

related rates

related rates

related rates

refer to page 185

anti derivative formulas

refer to page 211-212

differential equations: motion problems

refer to page 223

Fundamental Theorem of Calculus

if f is continuous on the closed interval [a,b] and F' = f, then according to the FTC, F(b)-F(a)

properties of definite integrals

refer to page 242

definite integrals for F'(x)

use FTC and if it is a constant then you can ignore, just multiply by derivative

definition of definite integral as the limit of Riemann sum

refer to page 246

Riemann sum

start by taking (b-a)/n in interval to get width (where n is the number of sub intervals needed)

left: use left most x point and substitute into f(x), then multiply by width calculated initially, then do this process for the number of sub intervals needed while adding the width each time you move on to a new point

right: use right most x point out of first interval and substitute into f(x), then multiply by width calculated initially, then do this process for the number of sub intervals needed while adding the width each time you move on to a new point

midpoint: use left and right most x point on the first interval and divide by 2, then substitute into f(x), then multiply by width calculated initially, then do this process for the number of sub intervals needed while adding the width each time you move on to a new point

left: use left most x point and substitute into f(x), then multiply by width calculated initially, then do this process for the number of sub intervals needed while adding the width each time you move on to a new point

right: use right most x point out of first interval and substitute into f(x), then multiply by width calculated initially, then do this process for the number of sub intervals needed while adding the width each time you move on to a new point

midpoint: use left and right most x point on the first interval and divide by 2, then substitute into f(x), then multiply by width calculated initially, then do this process for the number of sub intervals needed while adding the width each time you move on to a new point

trapezoidal rule

T(n)= [(y0+y1)/2] ** h1 + [(y1+y2)/2] ** h2 + [(y2+y3)/2] ** h3 + [(y3+y4)/2] ** h4 +..........

average value

(intergal)/b-a

to find are between curves

find intersection points, then put into integral with top curve minus bottom curve, or right curve minus left curve

to find volume between curves

same process as area

finding volume with disks

same process but use (pi)r^2(dx)

finding volume using washers

same process but use (pi)(R^2-r^2)dx

finding volume with shells

same process but use 2(pi)rh(dx) *you can use instead of washer method if you can still go through 2 curves

to find distance traveled using intergals

find out time it is going through and input equation into derivative

order of derivatives for motion

x(t)

v(t)

a(t)

v(t)

a(t)

to determine slope fields with f'(x)

take antiderivative of f'(x) and match with graph

to solve for C

set the same variables to their respective sides, take the antiderivative, solve for C with given variable values

Exponential Growth and Decay

refer to page 363

for volume of cross sections

semircircle: pi/2

equilateral triangle: root 3/4

and then do top minus bottom squared

equilateral triangle: root 3/4

and then do top minus bottom squared

e^0=1

sin(0)=0

cos(0)=1

sin(0)=0

cos(0)=1

...