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∫k f(u) du=
k ∫f(u) du
∫[f(u) ± g(u)] du=
∫f(u) du ± ∫g(u) du
∫kdu=
ku+C
∫u^n du =
(u^(n+1))/(n+1) +C , n≠ -1
∫cosu du=
sinu + C
∫sinu du=
-cosu + C
∫sec^2 u du=
tanu + C
∫secu tanu du=
secu + C
∫csc^2 u du=
-cotu + C
∫cscu cotu du=
-cscu + C
∫e^u du=
e^u + C
∫a^u du=
(a^u/ ln a) + C
∫tanu du=
-ln |cosu| + C
∫cotu du=
ln |sinu| + C
∫secu du=
ln |secu + tanu| + C
∫cscu du=
-ln |cscu + cotu| + C
∫1/u du=
ln |u| + C
∫lnu du=
ulnu-u + C
arcsin(u/a) + C
∫du/ √(a^2 - u^2 ) =
(1/a)arctan(u/a) + C
∫du/ (a^2 + u^2) =
(1/a)arcsec(|u|/a) + C
∫ du/u√u²-a²
(f(b)-f(a))/(b-a)
Slope of secant line of f(x) on [a,b].
Mean Value Theorem (MVT)
If f is continuous on [a,b] and differentiable on (a,b), then there exists a number c in (a,b) such that
intermediate value theorem (IVT)
suppose f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b). Then there exists a number "e" such that f(e)=N.
Definition of Continuity
At a point:
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 Derivative test
If the first derivative changes from negative to positive at c then f has a relative minimum at the point (c,f(c)) If it goes from positive to negative it is a relative maximum
Second derivative test
If the second derivative is greater than zero then the function has a relative minimum
concavity
a graph is concave up is the tangent lines are below the graph; a graph is concave down if the tangent lines are above the graph
Test for Concavity
Let f be a function whose second derivative exists on an open interval I.
Inflection Point
a point in the domain of the function at which a tangent line exists and the concavity of the function changes.
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= ∫(from a to b) A(x)dx
Volume by cross sections perpendicular to the x-axis
velocity
The rate at which distance is changing with time.
Speed
D / T
acceleration
a=v¹= dx/dt
Displacement
Fluid being pushed back out of the way by an object and talking its place.
Total Distance (from x=a to x=b)
∫(from a to b) |v(t)dt| or |∫(from a to c) v(t)dt| + |∫(from c to b) v(t)dt|
Integration by parts
If u = f(x) and v = g(x), and if f'(x) and g'(x) are continuous, then
Length of Arc for functions
s= ∫(from a to b) √1+[f'(x)]^2 dx
dP/dt= kP(L-P)
Logistic Growth Formula
P= L/[1+Ce^(-Lkt)]
Solution of Logistics Differential Equation
LaGrange Formula for Error Bound
|R(n)|≤ | [M/(n+1)!]* (b-c)^(n+1) |
Parametrics
dy/dx= (dy/dt)/ (dx/dt)
Position Vector
(x(t), y(t))
Velocity Vector
(x'(t), y'(t))
Acceleration Vector
(x"(t), y"(t))
|v(t)|= √[(dx/dt)^2 + (dy/dt)^2]
Speed (Magnitude of Velocity Vector)
Distance traveled from t=a to t=b (arc length)
s= ∫(from a to b) √[(dx/dt)^2 + (dy/dt)^2] dt
x= rcosθ, y= rsinθ
Relationship of Polar and Rectangular
Slope of a Polar Curve
dy/dx= (rcosθ+ r'sinθ)/ (-rsinθ + r'cosθ) or
Area of a Polar Curve
A= (1/2) ∫ (from alpha to beta) r^2 dθ
Arc Length of a Polar Curve
s= ∫(from alpha to beta) √[r^2 + (dr/dθ)^2] dθ
nth term test for divergence
lim (as n approaches infinity) of a{n} ≠ 0, then the series diverges
geometric series
sum of the terms of an arithmetic sequence
P-Series
Sum where 1/(n^p) converges if and only if p > 1
Alternating Series
∑(from n=1 to infinity) (-1)^n-1 a{n}. Converges if 0<a{n+1}<a{n} and lim (as n approaches infinity) a{n}= 0
Integral Test
take integral and evaluate, if it goes to a number it is convergent, if there is ∞in it, its divergent
Ratio Test
limit as n→∞ of an+1/an
Direct Comparison
Series converges if 0<a{n}<b{n} and ∑(from n=1 to infinity) b{n} converges; Series diverges if 0<b{n}<a{n} and
limit comparison
Series converges if lim (as n approaches infinity) a{n}/ b{n}= L>0 and
Power series for e^x
1+ x + x^2/2! + x^3/3! + x^4/4! +...=∑(from n=1 to infinity) x^n/n!
Power Series for sin x
x - x^3/3! + x^5/5! - x^7/7! + x^9/9! -...=