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Intermediate Value Theorem

If f(a) > 0 and f(b) < 0 then there is a real zero between a and b

Definition of a Derivative (x=c)

f'(c)=lim x->c [f(x) - f(c)] / (x-c)

Definition of a Derivative (h->0)

f'(x)=lim h->0 [f(x+h) - f(x)] / h

Geometrical Meaning of Derivative

-slope of a line tangent to the graph at that point

Forward Difference Quotient

i.e. [f(x+h) - f(x)] / h

Backward Difference Quotient

i.e. [f(x) - f(x-h)]/ h

Symmetric Difference Quotient

i.e. [f(x+h) - f(x-h)] / 2h

sin'x = cosx

Derivatives of the 6 Trigonometric Functions

Derivatives of 6 Inverse Trigonometric Functions

deriv of inverse sin: 1 / square root of 1-x^2

Differentiability Implies Continuity

If a function is differentiable at x=c, it's continuous at x=c

Linearization of a Function

y = f(c) + f'(c)(x-c)

Riemann Sums Set-Up

Upper: sample points are the highest y-values within each pair of points

Mean Value Theorem

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b)

Fundamental Theorem of Calculus

∫ f(x) dx on interval a to b = F(b) - F(a)

Properties of Definite Integrals

+/- Integrands: integral of f(x)dx from a to b is + if f(x) is + for all val's x in [a, b] and - if f(x) is - for all val's x in [a, b]

Application of a Definite Integral

To find product of P = yx where y = f(x) and goes from a to b

Definition of the Natural Log

lnx = integral of 1/t dt from 1 to x

2nd Form of the Fundamental Theorem of Calculus

If g(x) = integral of f(t)dt from a to x where a is a constant, then g'(x) = (deriv of x)f(x)

Derivative of lnx

ln'x = 1/x times derivative of x

Integral of 1/u du

equals lnlul + C

Change of Base Property

log base b of x = lnx/lnb

Derivative/Integral of e^x

Deriv of e^x = e^x times x'

L'Hospital's Rule

If the limit of a function is indeterminate (i.e. x/0, 0^0, etc.) then you can take the derivatives of the top and bottom functions separately in order to get the actual limit

sinxdx-- -cosx + C

Integrals of the 6 Trigonometric Functions

The notation lim_(x→a)f(x)=c means...

As x goes to a, f(x) gets closer and closer to y=c.

lim(x→∞)f(x)=c

f(x) has a horizontal asymptote at y=c if...

0

lim(x→±∞)( sinx/x)

f(x) has a vertical asymptote at x=c if...

x-c is a factor of the denominator that does not cancel out of the numerator

f(x) has a hole at x=c if...

x-c is a factor of the denominator and numerator with equal multiplicity (and cancels out)

Show that

exists at x=a

(f(b)-f(a))/(b-a)

Slope of secant line of f(x) on [a,b].

lim(h→0)(f(a+h)-f(a))/h

Derivative of f(x) at x=a

Prove:

To Prove a function f(x) is differentiable at point x=a

tanѲ=

sinѲ=0

tanѲ=

sinѲ=1/2

tanѲ=

sinѲ=1/√2

tanѲ=

sinѲ=√3/2

tanѲ=

sinѲ=1

tanѲ=

sinѲ=-1

sin(2x)=

2sinxcosx

cos(2x)=

(cos^2x-sin^2x )

cos^2x

(1+cos2x)/2=

sin^2x

(1-cos2x)/2=

1

sin^2x+ cos^2x

1+tan^2x

sec^2x

cot^2x+1

csc^2x

1/cos x

secx

1/sin x

cscx