Derivatives of trig functions + other UNIT 2 stuff

sinx

cosx

cosx

-sinx

tanx

sec²x

cscx

-cscxcotx

secx

secxtanx

cotx

-csc²x

What are you doing if you plug a point into a derivative?

Finding the slope of the tangent line at that point

Difference quotient

limit as h approaches 0 of (f(x+h) - f(x))/h

Question gives you point on the original function. What does that mean?

The tangent line of the original function (aka, f’(x) goes through this point. The slope of the tangent line is what you get when you plug in this point to the derivative function.

What is the significance of a y-value you get after plugging an x value into the tangent line?

It means that point is on the original function.

MVT

if function is differentiable on the open interval and continuous on the closed interval then there is a value between A < K < B that is equal to the average rate of change is equal to the derivative of f’(K)

IVT

if function is continuous on the closed interval and A < B then there is a value K between A < K B that is equal to F(A) < F(K) < F(B)

FTC PART 1

the derivative of an integral gives the function inside the integral; inverse operations

FTC PART 2

the integral of f(x) from bound a to bound b where b>a is F(b) - F(a)

Squeeze theorem

if the limit of two functions, g(x) and h(x), equal the same thing and g(x) < f(x) < h(x) then the limit of f(x) exists and is also equal. g(x) and h(x) must be continuous.

EVT

if a function is continuous on the closed interval then there is a maximum and minimum

particle moves left

velocity is negative

particle moves right

velocity is positive

particle is speeding up

velocity and acceleration signs are the same

particle is slowing down

velocity and acceleration signs are different

AROC

IROC

average value

conditions for continunity

1. f(a) exists

2. the limit as x approaches a of f(x) exists

3. the two are equal

what does it mean for a limit to exist

limit value from left and right have to be equal

POI

first derivative changes from inc to dec or dec to inc

when second derivative changes signs

rel min

first der test says it changes from neg to pos

second der test says it is positive

rel max

first der test says it changes from pos to neg

second der test says its negative

absolute extrema

of the interval: if x=a is the only relative extrema on that interval

of the function: when x=a is the most extreme value

revolving over a line

dx if the line is parallel to x-axis

dy if line is parallel to y-axis

Radius for revolving around a line DX

line - radius

Radius for revolving around a line DY

radius - line

cross sections with shapes

cross sections perpendicular to x axis DX

cross sections perpendicular to y axis DY

Left reiman sum

underestimate when increasing, overestimate when decreasing

Right reiman sum

overestimate when increasing, underestimate when decreasing

midpoint reiman sum

only depends on concavity

concave up - underestimate

concave down - overestimate

trapezoid reiman sum

only depends on concavity

concave up - overestimate

concave down - underestimate

Reiman sum sigma notation

view pic

integral of sec²x

tanx

integral of csc²x

-cotx

integral of secxtanx

secx

integral of cscxcotx

-cscx

integral of 1/sqrt 1-x²

Inverse sin, inverse cos if negative

integral of 1/1+x²

Inverse tan, inverse cot if negative

Integral of 1/ |x| times sqrt of x² -1

inverse sec, inverse csc if negative

integral of e^x

e^x

integral of b^x

(b^x) / ln(b)

integral of 1/x

ln|x|

ln’(x)

1/x times derivative of x

log’(x)

1/xlna times derivative of x

a= base

derivative of b^x

b^x times ln(b)

derivative of e^x

e^x times ln(e)

derivative of (f^-1) ‘ x

1/f ‘ y