Partial fractions: notes and practice questions

Let $g(x) = \frac{7x+17}{x^2+5x+6}$, for $x \in \mathbb{R}, x \neq -2, x \neq -3$.

Express $g(x)$ in partial fractions.

Hence, show that $g(x)$ is a decreasing function.

Hence, find the exact value of $\int_{0}^{1} g(x) \,dx$. Give your answer in the form $\ln k$, where $k$ is a rational number.

The rate of change of a certain quantity $R$ with respect to a variable $x$ is given by $R'(x)=\frac{1}{x(M-x)}$, $x \in \mathbb{R}$, $x \neq 0$, $x \neq M$ where $M$ is a positive constant.

The expression for $R'(x)$ can be written in the form $\frac{A}{x} + \frac{B}{M-x}$, where $A, B \in \mathbb{R}$.

Find $A$ and $B$ in terms of $M$.

Hence, find an expression for $R(x)$.

The concentration of a certain chemical product, $C$ (in mol/L), in a reaction vessel at time $t$ (in minutes) can be modelled by the differential equation $\frac{dC}{dt} = \frac{C(L-C)}{8L}$, where $L$ is the maximum possible concentration and $C(0) = 0.2$ mol/L is the initial concentration.

By solving the differential equation, show that $C = \frac{0.2 L}{(L-0.2)e^{-\frac{t}{8}}+0.2}$.

At $t=12$ minutes, the concentration of the product has reached $0.6$ mol/L.

Find the value of $L$, giving your answer correct to four significant figures.

Find the value of $t$ when the rate of change of the concentration is at its maximum.

Let $g(x) = \frac{3x - 1}{x^2 + x - 6}$, for $x \in \mathbb{R}, x \neq 2, x \neq -3$.

(a) Express $g(x)$ in the form $\frac{A}{x-2} + \frac{B}{x+3}$, where $A, B \in \mathbb{Z}$

(b) Find an expression for $g'(x)$, the derivative of $g(x)$.

(c) Hence, find the exact value of $\int_{3}^{4} g(x) \,dx$. Give your answer in the form $\ln k$, where $k$ is a rational number.

The growth of a bacterial colony, $B$, in a petri dish can be modelled by the logistic differential equation

$\frac{\text{d}B}{\text{d}t} = k B \left(1 - \frac{B}{N}\right)$

where $t$ is the time measured in hours and $k, N$ are positive constants.

The constant $N$ represents the maximum number of bacteria the petri dish can sustain indefinitely due to limited nutrients.

In the context of this bacterial growth model, interpret the meaning of $\frac{\text{d}B}{\text{d}t}$.

Show that $\frac{\text{d}^2B}{\text{d}t^2} = k^2B\left(1-\frac{B}{N}\right)\left(1-\frac{2B}{N}\right)$.

Hence show that the bacterial colony will grow at its maximum rate when $B = \frac{N}{2}$. Justify your answer.

Hence determine the maximum value of $\frac{\text{d}B}{\text{d}t}$ in terms of $k$ and $N$.

Let $B_0$ be the initial number of bacteria.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt = \ln\left(\frac{B(N-B_0)}{B_0(N-B)}\right)$.

After 5 hours, the number of bacteria is $2B_0$. It is known that $N = 3B_0$.

Find the value of $k$ for this bacterial growth model.

Let $g(x) = \frac{5x-5}{x^2-3x-4}$, for $x \in \mathbb{R}, x \neq -1, x \neq 4$.

(a) Find the partial fraction decomposition of $g(x)$.

(b) Hence, show that $g(x)$ is a decreasing function.

(c) Hence, find the exact value of $∫_0^2 g(x) \,dx$, giving your answer in the form $\ln k$, where $k \in \mathbb{Q}$.

A team of engineers is designing a new roller coaster ride. The path of a certain section of the ride can be modelled by the function $g(x)=\frac{x^2 + 3x - 10}{x-4}$, where $x$ is the horizontal distance in metres from the starting point and $g(x)$ is the vertical height in metres. The domain of the function is $x \in \mathbb{R}, x\neq 4$.

Find the coordinates where the path of the roller coaster crosses the horizontal ground (x-axis).

Find the coordinates where the path of the roller coaster crosses the vertical axis (y-axis).

Write down the equation of the vertical asymptote of the graph of $g$.

The oblique asymptote of the graph of $g$ can be written as $y = ax + b$ where $a, b \in \mathbb{Z}$.

Find the value of $a$ and the value of $b$.

Sketch the graph of $g$ for $-20 \le x \le 20$, clearly indicating the points of intersection with each axis and any asymptotes.

Engineers want to analyse the inverse of the roller coaster's height function, $h(x) = \frac{1}{g(x)}$.

Express $h(x)$ in partial fractions.

Hence find the exact value of $\int_{0}^{1} h(x) dx$, expressing your answer as a single logarithm.

Consider the function $f(x) = \frac{3x+5}{1+2x-3x^2}$.

(a) Express $f(x)$ in partial fractions.

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

(c) State the interval of convergence for this expansion.

(a) In a controlled biological experiment, the rate of change of the population $P$ of a certain microorganism with respect to time $t$ is modeled by the differential equation $t^2 \frac{dP}{dt} = P^2 - 2tP + 2t^2$, where $t > 0$ is in hours and $P$ is in thousands of organisms. It is known that at $t = 1$ hour, the population is $P = 4$ thousand.

Use Euler's method, with a step length of 0.1, to find an approximate value of $P$ when $t = 1.4$.

(b) Use the substitution $P = vt$ to show that $t\frac{dv}{dt} = v^2 - 3v + 2$.

(c.i) By solving the differential equation from part (b), and given that $P > 2t$, show that $P = \frac{6t - 2t^2}{3 - 2t}$.

(c.ii) Find the actual value of $P$ when $t = 1.4$.

(c.iii) Using the graph of $P = \frac{6t - 2t^2}{3 - 2t}$, suggest a reason why the approximation given by Euler's method in part (a) is not a good estimate to the actual value of $P$ at $t = 1.4$.

Consider the identity $\frac{5x-5}{(2x+1)(x-2)} \equiv \frac{P}{2x+1} + \frac{Q}{x-2}$, where $P, Q \in \mathbb{Z}$.

Find the value of $P$ and the value of $Q$.

Hence, find the Maclaurin series for $\frac{5x-5}{(2x+1)(x-2)}$ in ascending powers of $x$, up to and including the term in $x^2$.

Explain why the Maclaurin series found in part (b) is not valid for $x=0.6$.

Let $f(x) = \frac{3x^2 + 5x + 1}{(1+x)^2(2-x)}$.

(a) Express $f(x)$ in the form $\frac{A}{1+x} + \frac{B}{(1+x)^2} + \frac{C}{2-x}$.

(b) Hence, find the Maclaurin series for $f(x)$ up to and including the term in $x^3$.

(c) State the interval of convergence for this series.

Consider the function $f(x) = \frac{4x-2}{x^2-x-2}$, for $x \in \mathbb{R}, x \neq 2, x \neq -1$.

(a) Express $x^2-x-2$ in the form $(x+h)^2+k$.

(b) Express $f(x)$ in partial fractions.

(c) Hence find the exact value of $\int_3^4 f(x) \,dx$.

(d) Find the area of the region enclosed by the graph of $y = f(|x|)$, the x-axis and the lines with equations $x = -4$ and $x = -3$.