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Level 549

Calculus AB Theorems


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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