(a) Find, from first principles, the derivative of \(f(x) = (2x + 3)^{2}\). (b) Evaluate : \(\int_{1} ^{2} \frac{(x + 1)(x^{2} - 2x + 2)}{x^{2}} \mathrm {d}...

Answer Details

(a) First principles: \(f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}\) with \(f(x)=(2x+3)^2\).

\[f(x+h)-f(x)=\big[(2x+3)+2h\big]^2-(2x+3)^2\]

\[=2(2x+3)(2h)+(2h)^2=4h(2x+3)+4h^2\]

Divide by \(h\): \(4(2x+3)+4h\). Taking the limit as \(h\to0\):

\[f'(x)=4(2x+3)=8x+12\]

(b) First expand the numerator: \((x+1)(x^2-2x+2)=x^3-x^2+2\). Then

\[\frac{x^3-x^2+2}{x^2}=x-1+\frac{2}{x^2}=x-1+2x^{-2}\]

\[\int_{1}^{2}\left(x-1+2x^{-2}\right)dx=\left[\frac{x^2}{2}-x-\frac{2}{x}\right]_{1}^{2}\]

At \(x=2\): \(2-2-1=-1\). At \(x=1\): \(\tfrac12-1-2=-2.5\).

\[=-1-(-2.5)=\tfrac32\]