FAD1014: MATHEMATICS II — Tutorial 11

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

BINOMIAL THEOREM

Question 1: Simplify Factorials

Simplify the following:

(a) $\frac{(n+1)!}{(n-1)!} - n!$

(b) $(n+2)!$

Question 2: Solve for n

Find the appropriate value(s) for $n$ satisfying $(n + 1)! + n! = 72(n - 1)!$.

Question 3: Pascal's Triangle Expansion

Expand $(p - 2q)^5$ using Pascal's triangle.

Question 4: Binomial Theorem Expansion

Expand $\left(x - \frac{2}{x}\right)^6$ using Binomial's Theorem.

Question 5: Find Coefficient

Find the coefficient of $x^{15}$ in the expansion of $\left(x^2 - \frac{2}{x}\right)^{10}$.

Question 6: Binomial Expansion Application

Determine $(2 + x)^5$.

(a) From expansion above, find $(2 - x)^5$.

(b) Hence, evaluate $(2.1)^5 - (1.9)^5$ correct to two decimal places.

Question 7: Series Expansion

Find the first four terms in the expansion of the following functions in ascending powers of $x$ and state the range of convergence values of $x$ for which the expansion is valid:

(a) $\frac{1}{\sqrt{1 - 2x}}$

(b) $(8 - x)^{-1/3}$

Question 8: Function Approximation

Let the following radical functions be given:

$$\sqrt{1 - \frac{x}{3}}, \quad \sqrt{2 + x}, \quad \sqrt{1 + 4x}, \quad \sqrt{4 - \frac{x}{2}}$$

Suggest the most suitable function from the list to approximate $\sqrt{2}$ and justify.

Question 9: Approximation Using Series

Expand $\sqrt{1 - x}$ as a series in ascending powers of $x$ up to the term $x^3$ and state the range of convergence values of $x$. Hence, approximate $\sqrt{0.98}$, giving to five decimal places. Compare your answer with calculator's.

Question 10: Partial Fractions and Expansion

Let the function be: $$f(x) = \frac{x + 7}{x^2 - x - 6}$$

(a) Express $f(x)$ in its partial fraction form.

(b) Determine the first four terms of expansion for $f(x)$ and state the range of convergence of values of $x$.

POWER SERIES: TAYLOR & MACLAURIN SERIES

Question 1: Taylor Series

Find the first four nonzero terms of Taylor series for the given function below expanded about the given value of $a$.

(a) $f(x) = \cos x$ ; $a = \frac{\pi}{3}$

(b) $f(x) = \frac{1}{1+x}$ ; $a = 0$

(c) $f(x) = \ln(3 + x)$ ; $a = 1$

Question 2: Taylor Series Notation

Find the first three nonzero terms of Taylor series for $f(x) = x^{3/2}$ about $a = 1$. Write in the notation of $\sum$ or $C_n$.

Hence, find the approximation for $(1.03)^{3/2}$.

Question 3: Maclaurin Series

Find the first four nonzero terms of Maclaurin series for the following functions:

(a) $f(x) = xe^{x/4}$

(b) $f(x) = x\sin 3x$

Related Concepts

Related Lectures

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