Calculus Formulas

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Quotient Rule

(f'(x)g(x)-g'(x)f(x))/[g(x)]^2

d/dx[arccos u]

-1/(√1-u^2)

d/dx[arccot u]

-1/1+u^2

d/dx[arccsc u]

-1/|u|(√u^2-1)

d/dx[csc u]

-csc u cot u

d/dx[cot u]

-csc^2 u

d/dx[cos u]

-sin u

Definition of Continuity: a function is continuous at a point c when:

1. lim x→c f(x) exists. 2. f(c) is defined. 3. lim x→c f(x) = f(c)

d/dx[arcsin u]

1/(√1-u^2)

d/dx[arctan u]

1/1+u^2

d/dx[ln u]

1/u

d/dx[arcsec u]

1/|u|(√u^2-1)

sin 2θ

2sinθcosθ

Three special limits

A) lim x->0 (sin x)/x=1 B) lim x->0 (1-cos x)/x =0 C) lim x-> 0 (1+x) ^(1/x) =e

Pythagorean Identities

A) sin^2x + cos^2x= 1 B) tan^2x + 1 =sec^2x C) cot^2x + 1= csc^2x

Area of an equilateral triangle

A=(√3/4)s²

Area of a triangle

A=1/2bh

Area of a trapezoid

A=1/2h(b1+b2)

Area of a circle

A=πr²

Circumference of a circle

C=2πr

Second Derivative Test

Determines concavity- inflection points- (point where concavity changes F"(x)>0, concave upward- when f'(x) is increasing F"(x)<0, concave downward- when f"(x) is decreasing

IVT

If f is continuous on [a,b], f(a)≠ f(b) and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)= k

MVT

If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that f'(c)=(f(b)-f(a))/(b-a)

Differentiability

If f is differentiable at x = c, then the f is continuous at x=c. However, continuity does not imply differentiability: vertical tangents, cusps, sharp turns

Squeeze Theorem

If h(x) <_f(x) <_ g(x) for all x in an open interval containing c, where limx->c h(x) = L = limx->c g(x) then limx->cf(x) exists and is equal to L

Linear Approximation

L(x) = f(a) + f'(a)(x-a)

Rolle's Theorem

Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) such that f'(c)= 0. Hence, that has to be a critical point( relative min/max)

unit circle formulas

Sin-y Cos-x Tan-y/x Csc-1/y Sec-1/x Cot-x/y

Volume of a cone

V=1/3πr²h

Volume of a sphere

V=4/3πr³

Volume of a cylinder

V=πr²h

d/dx[a^u]

a^u ln a

cos θ =

adjacent/hypotenuse

cot θ =

adjacent/opposite

d/dx[sin u]

cos u

cos 2θ

cos^2 θ- sin^2 θ= 2cos^2 θ-1 =-2sin^2 θ

Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

First Derivative Test

determines critical points like relative mins/Max's when the derivative is equal to 0 or undefined

d/dx[e^u]

e^u

Chain Rule

f'(g(x))g'(x)

Definition of a Derivative

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

Product Rule

f'(x)g(x)+g'(x)f(x)

average rate of change

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

derivative of Inverse (f^1)'(x):

g'(x) = 1/f'(f^-1(x))

sec θ =

hypotenuse/adjacent

csc θ =

hypotenuse/opposite

EVT

if f is continuous on a closed interval [a,b], then f has both an absolute min and max on the interval

L'Hopital

lim x->c (f(x)/g(x)) = limx->c f ' (x)/ g ' (x)

instantaneous rate of change

lim x1->x0 (f(x1)-f(x0))/x1-x0

tanθ =

opposite/adjacent

sin θ =

opposite/hypotenuse

d/dx[sec u]

sec u tan u

d/dx[tan u]

sec^2 u

d/dx[|u|]

u/|u|

Quadratic Formula

x = -b ± √(b² - 4ac)/2a

Equation if a line

y-y1=m(x-x1)


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