1.11 Partial fractions questions for IB Maths AA

Express $g(x)$ in partial fractions.

Hence find ${\displaystyle {\int }_2^{5}g(x)\mathrm{d}x}$, giving your answer in the form $\ln k$, where $k\in {\mathbb{Z}}^{+}$.

Show that ${\displaystyle \frac{2000}{P(2000-P)}}\equiv {\displaystyle \frac{1}{P}}+{\displaystyle \frac{1}{2000-P}}$.

By solving the differential equation, show that $P={\displaystyle \frac{2000}{1+7e^{-0.4t}}}$.

Find the time taken for the number of fish to reach 1500.

Write down the long-term number of fish predicted by this model.

Find the greatest rate at which the number of fish increases.

When the number of fish reaches 1200, fishing begins. Fish are then removed at a constant rate of 150 fish per year, and the number of fish is modelled by

Find the values of $P$ for which ${\displaystyle \frac{\mathrm{d}P}{\mathrm{d}t}}=0$.

Hence describe what happens to the number of fish in the long term. Justify your answer.

Let $z=y^{1-n}$. Show that the Bernoulli equation can be written as

Consider the differential equation ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}-{\displaystyle \frac{y}{x}}=-y^2$, for $x>0$, where $y={\displaystyle \frac{1}{2}}$ when $x=1$.

Use part (a) to show that $z={\displaystyle \frac{1}{y}}$ satisfies ${\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}+{\displaystyle \frac{z}{x}}=1$.

Hence show that $y={\displaystyle \frac{2x}{x^2+3}}$.

Find the exact coordinates of the maximum point on the graph of $y={\displaystyle \frac{2x}{x^2+3}}$.

The logistic equation ${\displaystyle \frac{\mathrm{d}N}{\mathrm{d}t}}=rN(1-{\displaystyle \frac{N}{K}})$, where $r$ and $K$ are positive constants, models a population $N$ at time $t$. It is given that $N=N_0$ when $t=0$, where $0<N_0<K$.

Show that the logistic equation is a Bernoulli equation, stating the values of $n$, $P$ and $Q$.

Use the substitution $z={\displaystyle \frac{1}{N}}$ to show that $N={\displaystyle \frac{K{N}_0}{N_0+(K-N_0)e^{-rt}}}$.

Write down ${\displaystyle \underset{t\to \mathrm{\infty }}{\lim }N}$.

Consider the differential equation ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}+y=e^x\sqrt{y}$, where $y=1$ when $x=0$.

Show that the substitution $z=\sqrt{y}$ gives ${\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x}}+{\displaystyle \frac{1}{2}}z={\displaystyle \frac{1}{2}}{e}^x$.

Hence find $y$ in terms of $x$.

Verify that $y=0$ also satisfies the differential equation in part (d), and suggest why the method of part (a) does not produce this solution.

Express ${\displaystyle \frac{1}{(2r-1)(2r+1)}}$ in partial fractions.

Find the value of $S_2$ and the value of $S_3$.

Given that $S_1=\frac{1}{3}$, suggest an expression for $S_n$ in terms of $n$.

Prove by mathematical induction that your expression for $S_n$ is correct for all $n\in {\mathbb{Z}}^{+}$.

Find the least value of $n$ for which $S_n>{\displaystyle \frac{499}{1000}}$.

Express $f(x)$ in partial fractions.

Hence find the binomial expansion of $f(x)$ in ascending powers of $x$, up to the term in $x^3$.

Find the values of $x$ for which the expansion of $f(x)$ is valid.

Use your answer to part (b) to find an approximation for $f(0.1)$, and find the percentage error in this approximation.

Show that the coefficient of $x^n$ in the expansion of $f(x)$ is $2^n+{(-\frac{1}{2})}^n$.

Find the least value of $n$ for which the coefficient of $x^n$ is greater than ${10}^6$.

Write down the values of $F_3$, $F_4$, $F_5$ and $F_6$.

Let $f(x)={\displaystyle \frac{1}{1-x-x^2}}$, for $\mathrm{\mid }x\mathrm{\mid }<{\displaystyle \frac{\sqrt{5}-1}{2}}$.

Find $f(0)$ and $f^{\mathrm{\prime }}(0)$.

Show that $f^{\mathrm{\prime }\mathrm{\prime }}(0)=4$, and hence write down the first three terms of the Maclaurin series for $f(x)$.

Let $f(x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots$.

By considering the identity $(1-x-x^2)(a_0+a_{1}x+a_2{x}^2+\cdots )=1$, show that $a_0=1$, $a_1=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n\ge 2$.

Hence write down $a_n$ in terms of a Fibonacci number.

Let $\varphi ={\displaystyle \frac{1+\sqrt{5}}{2}}$ and $\psi ={\displaystyle \frac{1-\sqrt{5}}{2}}$.

Show that $1-x-x^2=(1-\varphi x)(1-\psi x)$.

Show that $f(x)={\displaystyle \frac{1}{\sqrt{5}}}({\displaystyle \frac{\varphi }{1-\varphi x}}-{\displaystyle \frac{\psi }{1-\psi x}})$.

By expanding each fraction in part (f) as a geometric series, show that $F_n={\displaystyle \frac{{\varphi }^n-{\psi }^n}{\sqrt{5}}}$ for $n\ge 1$.

State the condition on $x$ for both geometric series in part (g) to converge, and hence explain the domain given for $f$ in part (b).

Explain why $F_n$ is the integer nearest to ${\displaystyle \frac{{\varphi }^n}{\sqrt{5}}}$ for all $n\ge 1$.

Use your GDC and the result of part (i) to find $F_{30}$.

Find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{F_{n+1}}{F_n}}$.