#Maths # Infinite Series and Approximations ## A Rundown on Approximations If there is mention of a function moving by a small step $h$, we can take the total change as equal to the numerical value of the instantaneous derivative at that point. $\frac {dy}{dx} \approx \frac{\delta y}{\delta x}$ Where $\delta y$ and $\delta x$ are small changes in the y and x variables respectively. >[!EXAMPLE]- Small Changes Example Question >Given a function $x^{\frac{3}{2}}$ with derivative $dy/dx = \frac{3}{2}\sqrt{x}$, find the approximate change in $x$ when y increases from $8$ to $8+h$ by the small amount $h$. >To do this, we can use the formula listed above. $\delta y$ is thus equal to h. >We can find the numerical value of $x$ when $y = 8$ through substitution. >$ 8 = x^{\frac{3}{2}}, x = 4$ >Therefore: >$\frac{dy}{dx} = \frac{3}{2}\sqrt{4} = 6$ >$3 = \frac{h}{\delta x}$ >$\delta x = h/3$ >We now have an expression for the sensitivity of $x$ when compared to $y$. ## Infinite Series A couple of you more mathematically minded may ### The Maclaurin Series Take the arbitrary value within the function as 0! cool taylor series approximations: $\sin x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!}\dots$ ### The Taylor Series ### The Binomial Series ### Extending the Factorials to the Reals # Continuing with Limits