AP Calculus AB/BC Master Review
Quotient Rule
(low)(dhigh) - (high)(dlow) / low^2
Acceleration Vector
(x''(t), y''(t))
Velocity Vector
(x'(t), y'(t))
Position vector
(x(t), y(t))
First/Second Derivative Test
+ - + / increasing decreasing increasing / concave up concave down concave up Shift over one as you add a derivative
d/dx Arccot
-1 / 1+u^2 du/dx
d/dx Arccos u
-1 / √(1-u^2) du/dx
d/dx Csc u
-Csc u Cot u du/dx
d/dx Cot u
-Csc^2 u du/dx
e^x (series)
1 + x/1! + x^2/2! + x^3/3! + ..... x^n/n!
cos x (series)
1 - x^2/2! + x^4/4! - x^6/6! + ... ((-1)^n)(x^2n) / (2n)!
Average Value of f(x) on [a,b]
1 / (b-a) ∫(from a to b) f(x)dx
d/dx Arctan u
1 / 1+u^2 du/dx
d/dx logau
1 / u ln a du/dx
d/dx Arcsin u
1 / √(1-u^2) du/dx
Cos^2 x
1+cos2x / 2
Sin^2 x
1-cos2x / 2
Cos 2x
1. cos^2 x - sin^2 x 2. 1 - 2sin^2 x 3. 2cos^x - 1
Definition of Continuity
1. f(c) exists. 2. lim x→c f(x) exists. 3. lim x→c f(x) = f(c)
Area inside a polar curve
1/2 ∫(from a to b) r²dθ
∫du / u√(u^2-a^2)
1/a Arcsec(|u|/a) + C
∫du / a^2 + u^2
1/a Arctan(u/a) + C
Inverse Derivative
1/f'(f^-1(x))
d/dx ln u
1/u du/dx
Sin 2x
2 sin x cos x
Definition of a Point of Inflection
A point at which the graph changes concavity
Indefinite Integrals
ADD +C
Definition of a Sequence
An ordered list of numbers
∫du / √(a^2 - u^2 )
Arcsin(u/a) + C
Intermediate Value Theorem
If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c between a and b such that f(c)=k
ALL TERMS PAST THIS POINT ARE FOR BC ONLY
If you are from Calc AB just deselect these terms
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
d/dx Sec u
Sec u Tan u du/dx
d/dx Tan u
Sec^2 u du/dx
Definition of a Series
The sum of a sequence (∑)
Volume around a horizontal axis (washer method)
V = π ∫(from a to b) ([R(x)]^2 - [r(x)]^2) dx
Volume around a horizontal axis (disc method)
V = π ∫(from a to b) [r(x)]^2 dx
Volume by cross sections taken perpendicular to the x-axis
V = ∫(from a to b) A(x) dx where A is the area of the shape
Rules for doing integrals by hand from a graph
When moving right, add when above x-axis and subtract when below When moving left, add when below x-axis and subtract when above
Acceleration
a(t) = v'(t) = s''(t) (and vice versa)
∫a^u du
a^u / ln a
Chain Rule Version of Fundamental Theorem of Calculus
d/dx ∫ (a to g(x)) f(t)dt = f(g(x)) g'(x)
Second Fundamental Theorem of Calculus
d/dx ∫ (from a to x) f(t)dt= f(x)
Slope of a polar curve
dy/dx= (rcosθ + r'sinθ) / (-rsinθ + r'cosθ)
d/dx e^u
e^u du/dx
Average rate of change of f(x) on [a,b]
f(b)-f(a) / b-a
Rolle's Theorem
if f(x) is continuous on [a,b] and differentiable on (a,b), and if f(a)=f(b), then there is at least one point (x=c) on (a,b) [DON'T INCLUDE END POINTS] where f'(c)=0
Mean Value Theorem
if f(x) is continuous on [a,b] and differentiable on (a,b), there is at least one point (x=c) where f'(c)= F(b)-F(a)/b-a
Limit definition of the Derivative
limit (as h approaches 0)= F(x+h)-F(x)/h
Alternate form of the limit derivative
limit (as x aproaches a) = f(x)-f(a) / x-a
Arc Length for functions
s = ∫(from a to b) √(1+[f'(x)]^2) dx
Product Rule
uv' + vu'
sin x (series)
x - x^3/3! + x^5/5! - x^7/7! + ... ((-1)^n)(x^2n+1) / (2n+1)!
In polar curves
x = rcosθ y = rsinθ
Speed
|v(t)|
Speed (or magnitude of velocity vector)
√((dx/dt)^2 + (dy/dt)^2)
First Fundamental Theorem of Calculus
∫ (from a to b) f(x)= F(b)-F(a)
Displacement from x=a to x=b
∫(from a to b) v(t) dt
Total distance
∫(from a to b) |v(t) dt|
Distance traveled from x=a to x=b or length of arc
∫(from a to b) √((dx/dt)^2 + (dy/dt)^2)
Integration by parts
∫u dv = uv - ∫v du
