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Definition of a Derivative:
f'(x) = lim(h→0) (f(x + h) - f(x)) / h
e
lim(n→∞) (1 + (1 / n))^n =
e
lim(n→0) (1 + n)^(1 / n) =
Mean Value Theorem:
If f is continuous on [a,b] and differentiable on (a,b), then there is at least on number c in (a,b) such that (f(b) - f(a)) / (b - a) = f'(c).
Extreme-Value Theorem:
If f is continuous on a closed interval [a,b], then f(x) has both a maximum and minimum on [a,b].
To locate the points of inflection of y = f(x), find...
the points where f''(x) = 0 OR where f''(x) fails to exist. Then, test these points to make sure that f''(x) < 0 on one side and f''(x) > 0 on the other.
Euler's Method:
Starting with the given point (x₁,y₁),
slower
Logarithm functions grow (slower/faster) than any power function (xⁿ).
faster
Among power functions, those with higher powers grow (faster/slower) than those with lower powers.
faster than
or if lim(x→∞) g(x) / f(x) = 0
at the same rate as
f(x) grows (faster than/slower than/at the same rate as) g(x) if lim(x→∞) f(x) / g(x) = L ≠ 0.
L'Hôpital's Rule:
If lim(x→a) f(x) / g(x) is of the form 0 / 0 or ∞ / ∞, and if lim(x→a) f'(x) / g'(x) exists, then
Two functions f and g are inverses of each other if...
f(g(x)) = x for every x in the domain of g
Horizontal Line Test
inverse of a function is also a function if and only if no horizontal line intersects more than one
has
If f is strictly increasing or decreasing in an interval, then f (has/does not have) an inverse.
Trapezoidal Rule:
If a function f is continuous on the closed interval [a,b] where [a,b] has been EQUALLY partitioned into n subintervals,
Properties of the Definite Integral:
Let f(x) and g(x) be continuous on [a,b].
Fundamental Theorem of Calculus:
∫a→b f(x) dx = F(b) - F(a),
Second Fundamental Theorem of Calculus:
d/dx ∫a→x f(t) dt = f(x)
Mean Value Theorem for Integrals:
The average value of f(x) on [a,b] is
Area Between Curves:
If f and g are continuous functions such that f(x) ≥ g(x) on [a,b], then the area between the curves is
Integration by parts
If u = f(x) and v = g(x), and if f'(x) and g'(x) are continuous, then
Revolving around a horizontal line:
Volume of Solids of Revolution - Washer Method:
Revolving around a vertical line:
Volume of Solids of Revolution - Shell Method:
Logistic Growth:
y = C / (1 + Ae^(-kt)).
r² =
x² + y²
Definition of Arc Length:
If the function given by y = f(x) represents a smooth curve on the interval [a,b], then the arc length of f between a and b is given by
First derivative:
Parametric Form of the Derivative:
Arc Length in Parametric Form:
s = ∫a→b √([dx/dt]² + [dy/dt]²) dt
Speed in Parametric Form:
speed = √([f'(t)]² + [g'(t)]²)
Position Vector:
r(t) = <x(t),y(t)>
Velocity Vector:
v(t) = <dx/dt,dy/dt>
Speed Vector:
speed = |v(t)| = √([dx/dt]² + [dy/dt]²)
Acceleration Vector:
a(t) = <d²x/dt²,d²y/dt²>
dy/dx r(θ) =
(r'(θ) * sin θ + r(θ) * cos θ) /
Length of a Curve:
L = ∫a→b √[r² + (dr/dθ)²] dθ
ac + bd
<a,b> ° <c,d> =
Unit Vector:
<a / |v|,b / |v|>,
Angle Between 2 Vectors:
θ = Arccos ([u ° v] / [|u| * |v|])
Area Between a Curve and the Pole:
A = (1 / 2) * ∫a→b r² dθ
Area Between 2 Curves:
A = (1 / 2) * ∫a→b R² - r² dθ
Surface Area of Revolution:
Revolving around the x-axis (θ = 0):
b / a.
If a line = <a,b>, m =
Vector Velocity:
Velocity = (speed) * (direction)
(f(b)-f(a))/(b-a)
Slope of secant line of f(x) on [a,b].
Rolle's Theorem
If f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists a point c∈(a,b), where f '(c) = 0.
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)
Intermediate Value Theorem
If f(a) > 0 and f(b) < 0 then there is a real zero between a and b
(1+cos(2x))/2
Trig Property: cos^2(x)
(1-cos(2x))/2
Trig Property: sin^2(x)
cos^2(x)-sin^2(x); 1-2sin^2(x); 2cos^2(x)-1
Trig Properties (3): cos(2x)
d/dx[c]
0
d/dx[uv]
uv'+vu'
d/dx[f(g(x))]
f'(g(x))*g'(x)
d/dx[x^2]
nx^(n-1)
d/dx[u/v]
(vu'-uv')/v^2
d/dx[sin(u)]
cos(u)du/dx
d/dx[cos(u)]
-sin(u)du/dx
d/dx[tan(u)]
sec^2(u)du/dx
d/dx[cot(u)]
-csc^2(u)du/dx
d/dx[sec(u)]
sec(u)tan(u)du/dx
d/dx[csc(u)]
-csc(u)cot(u)du/dx
d/dx[ln(u)]
(1/u)du/dx
d/dx[loga(u)]
(1/(uln(a))du/dx
d/dx[e^u]
(e^u)du/dx
d/dx[a^u)
(a^u)(ln(a))du/dx
d/dx[arcsin(u)]
(1/sqrt(1-u^2))du/dx
d/dx[arccos(u)]
-(1/sqrt(1-u^2))du/dx
d/dx[arctan(u)]
(1/(1+u^2))du/dx
d/dx[arccot(u)]
-(1/sqrt(1+u^2))du/dx
f^-1(a)
1/(f^-1(f^-1(a)))
∫cos(u)du
sin(u)+C
∫sin(u)du
-cos(u)+C
∫sec^2(u)du
tan(u)+C
∫csc^2(u)du
-cot(u)+C
∫sec(u)tan(u)du
sec(u)+C
∫csc(u)cot(u)du
-csc(u)+C
∫(1/u)du
ln|u|+C
∫tan(u)du
-ln|cos(u)|+C
∫cot(u)du
ln|sin(u)|+C
∫sec(u)du
ln|sec(u)+tan(u)|+C
∫csc(u)du
-ln|csc(u)+cot(u)|+C
∫(e^u)du
(e^u)+C
∫(a^u)du
(a^u)/(ln(a))+C
∫du/sqrt(a^2-u^2)
arcsin(u/a)+C
∫du/(u^2+a^2)
(1/a)arctan(u/a)+C
∫du/(usqrt(u^2-a^2))
(1/a)arcsec(|u|/a)+C
Definition of a Critical Number
Let f be defined at c. If f'(c)=0, or if f' is undefined at c, then c is a critical number of f.
First Fundamental Theorem of Calculus
int from a to b of f'(x)dx = f(b)-f(a)
Second Fundamental Theorem of Calculus
deriv of integral from a to x of f(t)dt = f(x)
Average value of f(x) on [a,b]
(1/(b-a))*(integral from a to b of f(x)dx)
Volume around a horizontal axis by discs
V=pi(integral from a to b of [r(x)]^2dx
Volume around a horizontal axis by washers
V=pi(integral from a to b of [R(x)]^2-[r(x)]^2
V=(integral from a to b of A(x)dx)
Volume by cross sections taken perpendicular to the x-axis
Volume around a vertical axis by shells
V=2pi(integral fro a to b of x(r(x)))
Differential equation for logistic growth
dP/dt=kP(L-P) where L=lim as t->∞ P(t)
Arclength (functions)
s=(integral from a to b of sqrt(1+[f'(x)]^2)dx)