Calculus Rules, Equations, and General concepts

d/dx (arcsin)

1/sqrt(1-x^2) * x'

d/dx (arccos)

-(1/sqrt(1-x^2) * x'

d/dx (arctan)

1/1+(x)^2 * x'

d/dx (arccot)

-(1/1+(x)^2) * x'

d/dx (arcsec)

1/|x|sqrt(x^2-1) * x'

d/dx(arccsc)

-(1/|x|sqrt(x^2-1) * x'

Intermediate Value Theorum

Suppose f(x) is a continuous function on the interval [a,b] with f(a) ≠ f(b). If N is a number between f(a) and f(b), then there is a point: C, in (a,b), such that f(C)=N

True or False: Suppose f(x) is a continuous function on the interval [4,12] with f(4)= 5 and f(12)= 8. Then there exists a number: C on the interval [4,12] where f(c)= 6

True

Rolle's Theorem: Name the three conditions, and what happens when all three conditions are met.

1. f(x) is continuous on [a,b]

2. f(x) is differentiable on (a,b)

3. f(a)=f(b)

Then there exists at least one C on [a,b] such that f'(C)=0

Jump Discontinuity

The left and right hand limits both exist but they do not equal each other.

Infinite Discontinuity

The limit of the function approaches positive or negative infinity as the input approaches a specific point.

This is also a Vertical Asymptote.

Removable Discontinuity

The function is undefined at a particular point, but the limit of the function at the same point exists and is finite.

Jump Discontinuity

(How it looks on a graph)

Infinite Discontinuity

(How it looks on a graph)

Removable Discontinuity

(How it looks on a graph)

sinh(x) =

(e^x-e^-x)/2

cosh(x)=

(e^x+e^-x)/2

tanh(x)=

(e^x-e^-x)/(e^x+e^-x)

or

sinh(x)/cosh(x)

coth(x)=

(e^x+e^-x)/(e^x-e^-x)

or

cosh(x)/sinh(x)

sech(x)=

2/(e^x + e^-x)

or

1/cosh(x)

csch(x)

2/(e^x-e^-x)

or

1/sinh(x)

d/dx(sinhx)

coshx

d/dx(coshx)

sinhx

d/dx(tanhx)

sech^2(x)

d/dx(cothx)

-(cschx)^2

d/dx(sechx)

-sech(x)tanh(x)

d/dx(cschx)

-csch(x)coth(x)

Definition of Continuity: List the three requirements

1. f(a) is defined

2. lim f(x) as x-->a exists

3. lim f(x) as x-->a =f(a)

What does Newton's method approximate for? And what is its equation

Roots/ x-intercepts

Definition of the derivative

lim (f(a+h)-f(a))/h as h-->0

How to find a Vertical Asymptote

Set denominator equal to zero and solve for x

How to find a Horizontal Asymptote

• degree of numerator smaller than degree of denominator

• H.A.= y=0

• degree of numerator same as degree of denominator

• H.A. = y= leading coefficient of numerator/ leading coefficient of denominator

-degree of numerator greater than degree of denominator

• no horizontal asymptote

Draw full unit circle

d/dx (a^x)

a^x *ln(a) * d/dx(x)

d/dx(e^x)

e^x * d/dx(x)

d/dx (e^x^7)

7e^7x *x^6