Computing the slope of x3 via a limit
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lim
h® 0
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(x+h)3-x3 h
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= 3 x2 |
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A famous limit of an indeterminate form 0/0
A limit as x approaches infinity
A limit involving extra variables
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lim
x® 0
|
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ax-bx x
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= ln(a)-ln(b) |
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The limit of 1/x as x approaches 0 from the left
A limit for which L'Hopital's rule would be slow
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lim
x® 0
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(sin(x))249 (ln(1-x))251 x100 (arctan(x))400
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= -1 |
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The formula for the slope of x3
An illustration of the chain rule
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d d x
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sin(x3) = 3 x2 cos(x3) |
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A partial derivative
A 5th-order Taylor polynomial expanded about x=0
TAYLOR(ex,x,0,5) = 0.00833333 x5+0.0416666 x4+0.166666 x3+0.5 x2+x+1 |
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A 7th-order Taylor polynomial
TAYLOR(ln(cos(a x)),x,0,7) = -0.0222222 a6 x6-0.0833333 a4 x4-0.5 a2 x2 |
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An antiderivative of x2 with respect to x
An antiderivative of cosine(x) with respect to x
An antiderivative obtained by substitution
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ó õ
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x2 cos(a x3+b) dx = |
0.333333 sin(b) cos(a x3) a
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+ |
0.333333 cos(b) sin(a x3) a
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|
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A definite integral for x going from a to b
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ó õ
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b
a
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x2 dx = 0.333333 b3-0.333333 a3 |
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An integral having an infinite integration limit
An integral having an endpoint singularity
A 2-dimensional integral over a quarter disk
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ó õ
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r
0
|
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ó õ
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Ö[(r2-x2)]
0
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x y dy dx = 0.125 r4 |
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The formula for the sum of an arithmetic series
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n å
k = 0
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k = 0.5 n (n+1) |
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The formula for the sum of successive cubes
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n å
k = 0
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k3 = 0.25 n2 (n+1)2 |
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The formula for the sum of a geometric series
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n å
k = 0
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ak = |
an+1 a-1
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- |
1 a-1
|
|
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The sum of an infinite series
A sum for which iteration would be slow
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123456789 å
k = -123456788
|
|
k 370370367
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= 0.333333 |
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The product of successive even integers
File translated from TEX by TTH, version 2.30.
On 29 Jun 1999, 18:34.