Calculus: Maclaurin Polynomials (x=0)
Definition
A Maclaurin polynomial is a Taylor polynomial centered at x = 0. For a function f (x),
the n-th Maclaurin polynomial is:
f ′′ (0) 2 f (3) (0) 3 f (n) (0) n
Pn (x) = f (0) + f ′ (0)x + x + x + ··· + x
2! 3! n!
Example 1: Linear Approximation
Let f (x) = ex . At x = 0:
f (0) = 1, f ′ (0) = 1
So the tangent line approximation is:
P1 (x) = 1 + x
Example 2: Quadratic Approximation
For f (x) = ex :
f ′′ (x) = ex , f ′′ (0) = 1
So:
x2
P2 (x) = 1 + x +
2
Example 3: Cubic Approximation
f (3) (x) = ex , f (3) (0) = 1
So:
x2 x3
P3 (x) = 1 + x + +
2 6
1
Definition
A Maclaurin polynomial is a Taylor polynomial centered at x = 0. For a function f (x),
the n-th Maclaurin polynomial is:
f ′′ (0) 2 f (3) (0) 3 f (n) (0) n
Pn (x) = f (0) + f ′ (0)x + x + x + ··· + x
2! 3! n!
Example 1: Linear Approximation
Let f (x) = ex . At x = 0:
f (0) = 1, f ′ (0) = 1
So the tangent line approximation is:
P1 (x) = 1 + x
Example 2: Quadratic Approximation
For f (x) = ex :
f ′′ (x) = ex , f ′′ (0) = 1
So:
x2
P2 (x) = 1 + x +
2
Example 3: Cubic Approximation
f (3) (x) = ex , f (3) (0) = 1
So:
x2 x3
P3 (x) = 1 + x + +
2 6
1
