AB Flashcards

Intermediate Value Theorem

If f (x) is continuous on [a,b], then f (x) takes on every (y) value between f (a) and f (b).

Definition of Derivative

m = f`(x) = lim (h-0) (f(x+h) - f (x))/h

Alternate Definition of Derivative/Derivative at a point

f`(a) = lim (x-a) (f(x) - f (a))/(x-a)

tanx

sec²x

cotx

-csc²x

secx

sectanx

cscx

-csccotx

arcsin & arccos

+- 1/(1-x²)^-1/2

arctan & arccot

+- 1/(x²+1)^-1/2

arcsec & arccsc

+- 1/|x|((x²-1)^-1/2)

d/dx a^x

a^xlna

log_bx

1/xlnb

Extreme Value Theorem

If f(x) is continuous on [a,b], then f(x) has both an absolute max and absolute min on that interval.

Mean Value Theorem

If f(x) is continuous at every point of the closed interval [a,b] and is differentiable at every point (a,b) then there is one point, c in (a,b) at which:

f`(c)= f(b) - f(a)/ (b-a)

Average Value

if f is intergrtable on [a,b], it’s average value is:

(1/(b-a)) (f~(b-a)f(x)dx)

The Fundamental Theorem of Calculus, Part 1

If f(x) is continuous on [a,b], then the function

f(x)= f~f(t)dt has a derivative at every point x in [a,b], and

dy/dx(f~f(t)dt) = f(x) ←basiclly f~ is anti derv (derv) = cancel out

The Fundamental Theorem of Calculus, Part 2

(Integral Evaluation Theorem)

If f(x) is continuous at every point [a,b], and if F is any antiderivative of f(x) on [a,b], then

(f~(b-a)f(x)dx) = f(b)-f(a)\

Or (area under velocity tells you how much position changed by)

Y changes at a rate proportional to the amount present

dy/dt = ky

y=Ae^kt

The Fundamental Theorem

f(x) = f(a)+(f~(x-a)f(x)dx)

where f(a) value is known, and f(x) is unknown

mostly used with velocity and position

Displacement & Total Distance Traveled

(f~(b-a)v(t)dt & (f~(b-a)|v(t)|dt

Derivative of Inverses

If g(x) is the inverse of f(x) and f(a) = b then g`(b) = 1/f(a)

Area between 2 graphs

(f~(b-a)f(x₁)dx - (f~(b-a)f(x₂)dx

where f(x₁) = graph on the top

where f(x₂) = graph below

Trapezoid theorem

½(h)(f(x₀)+2f(x₁)+2f(x₂)…+f(x_n)) or

½(h)(f(x₀) + 2(f(x₁) + 2f(x₂)…) + f(x_n))

h= b-a/n

n= how many sections u wanna split into

b-a=range