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This questions asks you to investigate the recursive nature of the derivatives of 1−x1 and how this impacts the Maclaurin series expansion.
Consider the function f(x)=1−x1, x=1.
Find the value of f(0.5).
To earn a crown, get your answer ready before you reveal the options!
Show that (1−x)f′(x)−f(x)=0.
By differentiating the equation in (b), show that (1−x)f′′(x)−2f′(x)=0.
Let f(n) be the nth derivative of f.
Using mathematical induction, prove that (1−x)f(n)(x)−nf(n−1)(x)=0 for all n∈N.
Hence solve the differential equation dxdy=1−x3y.
Using (d), show that f(n)(0)=n! for all n∈N.
Hence find the Maclaurin series expansion for f(x).
Use the Maclaurin series expansion to find the value of f(21), and comment on the result.
This questions asks you to investigate the recursive nature of the derivatives of 1−x1 and how this impacts the Maclaurin series expansion.
Consider the function f(x)=1−x1, x=1.
Find the value of f(0.5).
To earn a crown, get your answer ready before you reveal the options!
Show that (1−x)f′(x)−f(x)=0.
By differentiating the equation in (b), show that (1−x)f′′(x)−2f′(x)=0.
Let f(n) be the nth derivative of f.
Using mathematical induction, prove that (1−x)f(n)(x)−nf(n−1)(x)=0 for all n∈N.
Hence solve the differential equation dxdy=1−x3y.
Using (d), show that f(n)(0)=n! for all n∈N.
Hence find the Maclaurin series expansion for f(x).
Use the Maclaurin series expansion to find the value of f(21), and comment on the result.