FAD1014: MATHEMATICS II — Tutorial 12

Centre for Foundation Studies in Science, Universiti Malaya
Session 2025/2026

POWER SERIES: TAYLOR & MACLAURIN SERIES

Question 1: Maclaurin Series for Logarithm

Find the Maclaurin series of $\ln(1 + x)$ up to the term of $x^4$. Then, determine the Maclaurin series for the following function:

(a) $\ln(1 - x)$

(b) $\ln(2 + x)$

Hence, show that: $$\ln(2 - x - x^2) = \ln 2 - \frac{1}{2}x - \frac{5}{8}x^2 - \frac{7}{24}x^3 + \cdots$$

Question 2: Coefficient in Maclaurin Expansion

Find the coefficient of $x^4$ in the Maclaurin expansion of $(1 - x + x^2)e^{x^2}$.

Question 3: Prove Trigonometric Identity

Prove the trigonometry identity of $\cos 2x = 1 - 2\sin^2 x$ by Maclaurin series approach.

Question 4: Maclaurin Series Proof

Suppose the Maclaurin series of $e^x = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \cdots$, prove that: $$e^{x\sin(x)} = 1 + x^2$$

Question 5: Power Series Integration

Use the power series of Maclaurin to find: $$\int \frac{\cos x - 1}{x^2} dx$$

Question 6: Definite Integral Approximation

Use Maclaurin series, estimate the following definite integral correct to four decimal places:

(a) $\int_0^{0.2} e^{-x^2} dx$

(b) $\int_0^{0.5} e^{x^2} dx$

(c) $\int_0^{0.5} \cos(x^2) dx$

GEOMETRY: CIRCLE

Question 1: Centre and Radius

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) $2x^2 + 2y^2 - 5x + 6y - 7 = 0$

Question 2: Circle Through Three Points

Find the equation of circle that passes through the points $(-1, 0)$, $(1, 4)$ and $(1, -4)$. Hence, state its centre and radius.

Question 3: Distance from Centre to Point

A circle has equation of $x^2 + y^2 - 20x - 24y + 195 = 0$ and state its centre. Thus, find the length from centre of circle to a point $N(25, 32)$.

Question 4: Intersection of Circles

Verify whether each pair of circle intersects, touches or does 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) $C_1: x^2 + y^2 - 2x - 4y + 1 = 0$, $C_2: x^2 + y^2 - 10x - 10y + 41 = 0$

Question 5: Tangent Line to Circle

Consider a circle with equation of $x^2 + y^2 - 6x + 8y + 21 = 0$ and a straight line with equation of $y = kx - 1$. Find the value(s) of $k$, if the line is tangent to the circle.

Related Concepts

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