Input
n n
Result
n^2
Plot
Values
n | 1 | 2 | 3 | 4 | 5
n^2 | 1 | 4 | 9 | 16 | 25
Geometric figure
parabola
Root
n = 0
Property as a function
even
Derivative
d/dn(n n) = 2 n
Indefinite integral
integral n^2 dn = n^3/3+constant
Global minimum
min{n n} = 0 at n = 0
Input
n n-(n-1) (n-1)
Result
2 n-1
Plots
Values
n | 1 | 2 | 3 | 4 | 5
n^2-(n-1)^2 | 1 | 3 | 5 | 7 | 9
Geometric figure
line
Root
n = 1/2
Derivative
d/dn(n n-(n-1) (n-1)) = 2
Indefinite integral
integral (-(-1+n)^2+n^2) dn = n^2-n+constant
Input
n n n-(n-1) (n-1) (n-1)
Result
n^3-(n-1)^3
Plots
Values
n | 1 | 2 | 3 | 4 | 5
n^3-(n-1)^3 | 1 | 7 | 19 | 37 | 61
Geometric figure
parabola
Alternate forms
3 n^2-3 n+1
Complex roots
n = 1/6 (3-i sqrt(3))
Roots in the complex plane
Derivative
d/dn(n n n-(n-1) (n-1) (n-1)) = 6 n-3
Indefinite integral
integral (-(-1+n)^3+n^3) dn = n^3-(3 n^2)/2+n+constant
Global minimum
min{n n n-(n-1) (n-1) (n-1)} = 1/4 at n = 1/2
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