Power Series: Taylor & Maclaurin Series

Find the Maclaurin series of $\ln(1+x)$ up to the term of $x^3$. Then, determine the Maclaurin series for the following functions:
- (a) $\ln(1-x)$
- (b) $\ln(2+x)$
Hence, show that
$$\ln(2-x-x^2) = \ln 2 - \frac{1}{2}x - \frac{2}{3}x^2 - \frac{3}{4}x^3 + \cdots$$
Find the coefficient of $x^3$ in the Maclaurin expansion of $(1-x+x^2)e^{x^2}$.
Prove the trigonometric identity $\cos 2x = 1-2\sin^2 x$ by Maclaurin series approach.
Suppose the Maclaurin series of $e^x = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \cdots$, prove that
$$e^{\ln(1+x)} = 1+x$$
Use the power series of Maclaurin to find
$$\int \frac{\cos x - 1}{x^2} , dx$$
Use Maclaurin series to estimate the following definite integrals correct to four decimal places:
- (a) $\int_0^{0.5} \frac{1}{\sqrt{1+x^2}} , dx$
- (b) $\int_0^{0.5} e^{-x^2} , dx$
- (c) $\int_0^{0.3} \cos(x^2) , dx$

Geometry: Circle

State the centre and radius of the following circles:
- (a) $(x-3)^2 + (y-4)^2 = 9$
- (b) $(y+2)^2 + (5-x)^2 = 3$
- (c) $x^2 + y^2 - 4x + 2y - 4 = 0$
- (d) $2x^2 + 2y^2 - 5x + 6y - 7 = 0$
Find the equation of the circle that passes through the points $(-1,0)$, $(1,4)$, and $(1,-4)$. Hence, state its centre and radius.
A circle has equation $x^2 + y^2 - 20x - 24y + 195 = 0$. State its centre. Thus, find the length from the centre of the circle to point $N(25,32)$.
Verify whether each pair of circles intersect, touch, or do not touch:
- (a) $C_1: x^2 + y^2 - 4x - 4y + 4 = 0$, $C_2: x^2 + y^2 - 8x - 4y + 16 = 0$
- (b) $C_1: x^2 + y^2 - 2x - 2y + 7 = 0$, $C_2: x^2 + y^2 - 14x - 2y + 41 = 0$
- (c) $C_1: x^2 + y^2 - 2x - 4y + 1 = 0$, $C_2: x^2 + y^2 - 10x - 10y + 41 = 0$
Consider a circle with equation $x^2 + y^2 - 6x + 8y + 21 = 0$ and a straight line with equation $y = kx - 1$. Find the value(s) of $k$ if the line is tangent to the circle.

Source: FAD1014 TUTORIAL 12 25-26.pdf