Question

Consider the function 𝑦 = tan (𝑥)

a) Show that the first two non-zero terms in the Maclaurin series of 𝑦 are $𝑥 + 1/3 𝑥 ^3$ …

b) Use the first two terms of the Maclaurin series of 𝑦 to estimate $tan ( 1/3 )$

Answer 1

$(i) Let \ f(x)=\tan x, then \ f(0)=0 \\ f^{\prime}(x)=\sec ^{2} x=1+\tan ^{2} x \\ \Rightarrow f^{\prime}(0)=1 \\ f^{\prime \prime}(x)=2 \sec x\left(1+\tan ^{2} x\right)\\ \Rightarrow f^{\prime \prime}(0)=2 \\ The \ Maclaurin \ series \ expansion \ is \\ \tan x=f(0)+x f^{\prime}(0)+\frac{x^{2}}{2 !} f^{\prime \prime}(x)+\frac{x^{3}}{3 !} f^{\prime \prime}(x)+\ldots \\ =x+\frac{x^{3}}{3}+o\left(x^{3}\right)$

$(ii) Put \ x=\frac{1}{3}\\ \therefore tan(\frac{1}{3})=\frac{1}{3}+\frac{1}{3}(\frac{1}{3})^3+...\\ =\frac{1}{3}+\frac{1}{81}\\ =\frac{28}{81}$