formula quizzes updated

d/dx (sin x)

cos x

average value

total distance traveled/displacement

dy/dx =

ky (differential equation)

y(t) = (general solution)

Ce^kt

fundamental theorem of calculus (various forms)

inverse function formula

d/dx (cos x)

-sin x

d/dx (tan x)

sec²x

d/dx (csc)

( -csc x ) ( cot x )

d/dx (sec x)

(sec x ) (tan x)

d/dx (cot x)

-csc²x

d/dx (e^x)

e^x

d/dx (ln x) 

1/x

d/dx (a^x)

d/dx (loga x)

d/dx (f/g) — quotient rule

d/dx (fg)

chain rule: if h(x) = f(g(x)), then

tan x =

sin x / cos x

quotient ID

cot x =

cos x / sin x

quotient ID

sec x =

1 / cos x

reciprocal ID

csc x =

1 / sin x

reciprocal ID

sin²x + cos²x = 1. what is the other pythagorean ID?

sec²x - tan²x = 1. what is the other pythagorean ID?

2sinxcosx = sin2x. what is the other double angle ID?

cos²x - sin²x = cos2x. what is the other double angle ID?

sin (-x) = -sin x (odd), cos(-x) = cos x (even), tan (-x) = -tanx (odd)

even-odd id’s

definition |x|

sin (A + B) =

sin A cos B + cos A sin B

cos (A + B) =

cos A cos B - sin A sin B

sin (A - B) =

sin A cos B - cos A sin B

cos (A-B)

cos A cos B + sin A sin B

distance between two points

midpoint formula

laws of logarithms ln(ab)=

ln (ab) = ln a + ln b

laws of logarithms ln (a/b)

ln (a/b) = ln a - ln b

laws of logarithms (a^n)

ln (a^n) = n ln a

ln (1/a) =

-ln a

ln (0) =

undefined

ln (1)=

0

ln (e)=

1

THE UNIT CIRCLE :)

cos is the x value and sin is the y value

one sided limits

definition of a limit

lim x→a of c =

c

average rate of change…

squeeze theorem

limit rules for e^x

limit rules for ln x

rules for lim 1/x

lim(x→0) of sinx/x =

1

lim(x→0) of 1-cosx/x =

0

limit rules for arctan x

definition of vertical asymptote

definition of horizontal asymptote

definition of continuity

intermediate value theorem (IVT):

definition of the derivative

tangent line equation

normal line

the line perpendicular to the tangent line at the point of tangency

three reasons a function f will not be differentiable at a point x=a

important derivatives

the derivative of the inverse of sin x

the derivative of the inverse of cos tx

the derivative of the inverse of tan x

the derivative of the inverse of cot x

the derivative of the inverse of sec x

the derivative of the inverse of csc x

s(t), v(t), |v(t)|, a(t) application

s(t) = position

v(t) = s’(t) = velocity

|v(t)| = speed

a(t) = v’(t) = s’’(t) = acceleration

1. particle at rest when v(t) = 0

2. speed is increasing if v and a have the same sign

3. speed is decreasing if v and a have opposite signs

linear approximation of f for values of x near x=a (a is the x-value at the point of tangency)

y=f(a) + f’(a)(x-a) —> tangent line at x=

hospital rule

note: only used when you have 0/0 or inf/inf

mean value theorem

“the slope of the tangent equals the slope of the secant at some point between a and b'“

if 1. f is continuous on [a,b]

2. differentiable on (a, b)

then there exists a number C between a and b such that f’© = f(b)-f(a)/b-a

minimum or maximum values refer to…

the y value

extreme value theorem (EVT)

1. if f is continuous on a closed interval [a,b]

2. then f has an absolute max and absolute min on the interval [a,b]

definition of critical points

a critical point of f is a number c (x-value) such that either

1. f’ c = 0 (stationary points) OR

2. f’ c does not exist (singular points)

candidates test for absolute extrema on a closed interval

1. find the y-values of critical points in the interval (a,b)

2. find the y-values of endpoints, a and b

3. the largest y-value is the maximum and the smallest y-value is the minimum

test for increasing and decreasing (DUH)

f’ > 0 means f is increasing

f’ < 0 means f is decreasing

relative extrema must occur at…

relative or local mins/maxs must occur at critical points

1st derivative test for relative extrema

1. if f’ changes from positive to negative at c, then f© is a relative max

2. if f’ changes from negative to positive at c, then f© is a relative min

3. if f’ does not change sings at c, then f© is neither.

definition of concavity

f is concave up if f’ is increasing

f is concave down if f’ is decreasing

test for concavity

f’’ > 0 means f is concave up

f’’ < 0 means f is concave down

definition of a point of inflection (POI)

a point on the graph of f where the concavity of f changes

test for inflection point

a function f has an inflection point at (c, f©) iff

f’’ changes from + to - or vice versa at x =c

2nd derivative test for relative extrema

given f’© = 0, then

1. if f’’© <0 (concave down) then f© a max

2. if f’’©> 0 (concave up) then its a min

3. if f’’© = 0 then the test is incoclusive

the integral of x^ndu

the integral of 1/u du

there is an absolute value because you can’t use negative logs

the integral of e^u du

the integral of a^u du

the integral of f(x)dx a and a

the integral of f(x)dx b and a

the integral of sin u du

the integral of cos u du

integral of sec²u du

integral of csc u cot u du

integral of sec u tan u du