Calculus Formulas
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)
