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Level 589

Calculus BC II


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L'Hopitals rule
use to find indeterminate limits, find derivative of numerator and denominator separately then evaluate limit
indeterminate forms
All these are indeterminate (which means that if any of these are a limit of something then that limit is undefined):
6th degree Taylor Polynomial
polynomial with finite number of terms, largest exponent is 6, find all derivatives up to the 6th derivative
Taylor Series
If a function is differentiable infinitely many times in some interval around x=a, then the Taylor Series centered at a for f is:
Nth Term Test
If Limit as K approaches infinity, then the Series of K Diverges.
Alternating Series Test
If An is positive, the series ∑(-1)An converges if & only if..
converges absolutely
alternating series converges and general term converges with another test
converges conditionally
alternating series converges and general term diverges with another test
Ratio Test
limit as n→∞ of an+1/an
find interval of convergence
use ratio test, set > 1 and solve absolute value equations, check endpoints
find radius of convergence
use ratio test, set > 1 and solve absolute value equations, radius = center - endpoint
Integral Test
take integral and evaluate, if it goes to a number it is convergent, if there is ∞in it, its divergent
limit comparison test
Let A and B be series with positive terms such that p = limit A/B (to infinity) and 0 < p < infinity, then both series converge or both diverge
Geometric Series Test
general term = a₁r^n, converges if -1 < r < 1
P-Series Test
general term = 1/n^p, converges if p > 1
x(t) and y(t)
dy/dx = dy/dt / dx/dt
second derivative of parametrically defined curve
find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative, then divide by dx/dt
length of parametric curve
∫ √ (dx/dt)² + (dy/dt)² over interval from a to b
√(dx/dt)² + (dy/dt)² not an integral!
given velocity vectors dx/dt and dy/dt, find speed
given velocity vectors dx/dt and dy/dt, find total distance travelled
∫ √ (dx/dt)² + (dy/dt)² over interval from a to b
area inside polar curve
1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta
area inside one polar curve and outside another polar curve
1/2 ∫ R² - r² over interval from a to b, find a & b by setting equations equal, solve for theta.
1
sin² θ + cos² θ =
sec² θ
1 + tan² θ =
csc² θ
1 + cot² θ =
sin(-θ) =
-sin θ
cos(-θ) =
cos θ
tan(-θ) =
-tan θ
sinAcosB + sinBcosA
sin(A + B) =
sinAcosB - sinBcosA
sin(A - B) =
cosAcosB - sinAsinB
cos(A + B) =
cosAcosB + sinAsinB
cos(A - B) =
2sinθcosθ
sin 2θ =
cos 2θ =
cos² θ - sin² θ
tan θ =
sin θ / cos θ
cot θ =
cos θ / sin θ
sec θ =
1 / cos θ
csc θ =
1 / sin θ
cos² θ =
(1 / 2)(1 + cos 2θ)
sin² θ =
(1 / 2)(1 - cos 2θ)
d/dx (x^n) =
nx^(n - 1)
d/dx (fg) =
fg' + gf'
d/dx (f / g) =
(gf' - fg') / g^2
f'(g(x))g'(x)
Chain Rule
cos x
d/dx (sin x) =
-sin x
d/dx (cos x) =
sec² x
d/dx (tan x) =
-csc² x
d/dx (cot x) =
secxtanx
d/dx (sec x) =
-cscxcotx
d/dx (csc x) =
e^x
ƒ'(x) of e^x
d/dx (a^x) =
a^x * ln a
1 / x
d/dx (ln x) =
d/dx (Arcsin x) =
1 / √(1 - x²)
d/dx (Arctan x) =
1 / (1 + x²)
d/dx (Arcsec x) =
1 / (|x| * √(x² - 1))
d/dx (Arccos x) =
-1 / √(1 - x²)
d/dx (Arccot x) =
1 / (1 + x²)
d/dx (Arccsc x) =
-1 / (|x| * √(x² - 1))
0
d/dx [c] =
cf'(x)
d/dx [cf(x)] =
∫a dx =
ax + C
∫x^n dx =
(x^n+1) / (n + 1) + C,
ln |x| + C
∫1 / x dx =
∫e^x dx =
e^x + C
∫a^x dx =
a^x / ln a + C
∫ln x dx =
xln x - x + C
∫sin x dx =
-cos x + C
∫cos x dx =
sin x + C
∫tan x dx =
ln |sec x| + C
∫cot x dx =
ln |sin x| + C
∫sec x dx =
ln |sec x + tan x| + C
∫csc x dx =
-ln |csc x + cot x| + C
∫sec² x dx =
tan x + C
∫secxtanxdx =
sec x + C
∫csc² x dx =
-cot x + C
∫cscxcotx dx =
-csc x + C
∫tan² x dx =
tan x - x + C
∫1 / (a² + x²) dx =
(1 / a)(Arctan (x / a)) + C
Arcsin (x / a) + C
∫1 / (√(a² - x²)) dx =
∫1 / (x * √(x² - a²)) =
(1 / a)(Arcsec (|x| / a)) + C
lim(x→a) f(x) = L
The limit lim(x→a) exists if and only if...
A function y = f(x) is even if...
f(-x) = f(x) for every X in the function's domain.
A function y = f(x) is odd if...
f(-x) = -f(x) for every X in the function's domain.
f(x + p) = f(x) for every value of X.
A function f(x) is periodic with period p(p > 0) if...
Intermediate-Value Theorem:
A function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b).
2π / |B|
or y = Acos(Bx + C) is...
|a|
or y = Acos(Bx + C) is...
π
y = tan x is...
If f is continuous on [a,b] and f(a) and f(b) differ in sign, then...
the equation f(x) = 0 has at least one solution in the open interval (a,b).
lim(x→±∞) f(x) / g(x) = 0 if...
the degree of f(x) < the degree of g(x).
lim(x→±∞) f(x) / g(x) = ∞ if...
the degree of f(x) > the degree of g(x).
lim(x→±∞) f(x) / g(x) = [c] if...
the degree of f(x) = the degree of g(x).
lim(x→∞) f(x) = b
A line y = b is a horizontal asymptote of the graph y = f(x) if either...
lim(x→a⁺) f(x) = ±∞
A line x = a is a vertical asymptote of the graph y = f(x) if either...
((f(x₁) - f(x₀)) / (x₁ - x₀))
If (x₀,y₀) and (x₁,y₁) are points on the graph of y = f(x), then the average rate of change of y with respect to x over the interval [x₀,x₁] is...
f'(x₀).
If (x₀,y₀) is a point on the graph of y = f(x), then the instantaneous rate of change of y with respect to x at x₀ is...