Numerical methods: Applications to modelling

In a nutshell

The numerical methods used to find roots have direct applications in modelling real life scenarios. These models can be very useful in fields of science and engineering.

Example 1

The price of a motorcycle in $£$ , $x$ years after purchase is modelled using the following function:

$f(x)=5\thinspace000(0.75)^x-300\cos(x), x\gt0$ 

(i) Use the model to find the value of the motorcycle after $5$ years, to the nearest hundred.

(ii) Show that $f(x)$ has a root between $11$ and $12$ .

(iii) Find $f'(x)$ .

(iv) Taking $11.6$ as an initial approximation, apply the Newton-Raphson method once to $f(x)$ to get a second approximation for the time at which the motorcycle's value reaches $0$ . Give your answer to $3$ decimal places.

(v) Comment on the validity of this model as an approximation for the motorcycle's value.

(i) Substitute $x=5$ into $f(x)$ .

$\begin{aligned}f(5)&=5 \thinspace 000(0.75)^5-300\cos(5)\\&=1\thinspace 101.424782\end{aligned}$ 

The motorcycle is worth $\underline{£1100}$ after five years, rounded to the nearest hundred.

(ii) Substitute $x = 11$ and $x=12$ into $f(x)$ .

$\begin{aligned}f(11)&=5\thinspace000(0.75)^{11}-300\cos(11)\\&=209.847... \gt 0 \end{aligned}$ 

$\begin{aligned}f(12)&=5\thinspace000(0.75)^{12}-300\cos(12)\\&=-94.774... \lt 0 \end{aligned}$ 

$f(x)$ is continuous, and there is a change in sign in the interval $[11,12]$ . $f(x)$ must equal $0$ within this interval, therefore:

$f(x)$ has a root between $\underline{11}$ and $\underline{12}$ .

(iii) Differentiate with respect to $x$ .

$f'(x)=\underline{300\sin(x)+(5\thinspace000)(0.75)^x(\ln0.75)}$ 

(iv) Substitute $11.6$ into $f(x)$ and $f'(x)$ .

$\begin{aligned}f(11.6)&=5\thinspace000(0.75)^{11.6}-300\cos(11.6)\\&=7.210... \\f'(11.6)&={300\sin(11.6)+(5\thinspace000)(0.75)^{11.6}(\ln0.75)}\\&=520.357...\\x_{n+1}&=x_n-\dfrac{f(x)}{f'(x)}\\&=11.6-\dfrac{7.210...}{520.357...}\\&=11.6241...\\&=\underline{11.624} \ (3 \ s.f.)\end{aligned}$

(v) Use your quality of written communication to write a sentence about the real life implications of the model and its limitations.

Objects in real life cannot have a negative value. This model is not valid for motorcycles that are $12$ years older or more.