I have 5 quartic functions that were found with quartic regression. Each function models a country's relationship with rotavirus vaccination rates (x) against years (y, years are integers and not real values, ex. year 1 -> x=1, etc.)
| f(x) = ax4 + bx3 + cx2 + dx + e | Domain and Range | |
|---|---|---|
| Morocco | y = -0.220x⁴ + 5.44x³ - 47.3x² + 171x - 123 | {x | 1≤x≤10} {y | 5.92 ≤y≤97} |
| Sudan | y = -0.199x⁴ + 5.27x³ - 50.1x² + 205x - 220 | {x | 2≤x≤10} {y | 28.6≤y≤100} |
| Rwanda | y = -0.318x⁴ + 9.12x³ - 95.1x² + 426x - 592 | {x | 3≤x≤10} {y | 50.6≤y≤98} |
| Ghana | y = -0.290x⁴ + 8.20x³ - 84.6x² + 377x - 518 | {x | 3≤x≤10} {y | 49.5≤y≤ 92} |
| Botswana | y = -0.034x⁴ + 1.28x³ - 16.8x² + 90.2x - 78.3 | {x | 3≤x≤10} {y | 72.9≤y≤83.7} |
I'm now trying to find instantaneous rates of change for each country. I want to find instantaneous rates for each integer x value within a country's domain. so for Rwanda, it'd be instantaneous rates of change for x=3,4,5,6,7,8,9,10. I'm super new to calculus so I'm not sure how I'd go about finding the derivative of my quartic functions to calculate instantaneous rates of change - if someone could help me through this and possibly show a sample calculation for one instantaneous rate of change for one of my countries, that'd be really appreciated!!
PS: my goal afterwards is to graph each country’s instantaneous rates of change values (y) over time (x) and perform linear regression each. and then, whichever country (instantaneous rate of change linear graph) has the highest slope value = increased the most in vaccination rates over x many years. please let me know if you think this makes sense if you can!
The rate of change over time t of a function is given by its rise over its run, so by $\frac {f(a+t)-f(a)}{t}$. Try to graph a function to see it graphically.
It's clear that by reducing $t$ more and more you get a rate of change over a smaller and smaller amount of time, so ideally as $t$ approaches $0$ (ax $x \to 0$) you'll get the instantaneous rate of change.
$f'(a)$ is none other than $lim_{t \to 0} \frac {f(a+t)-f(a)}{t}$.
Polynomials like yours are quite easy to differentiate; without going into a proof you should know that whenever $f(x)= b x^n$ then $f'(x)=b \;nx^{n-1}$; if $f(x)=c$ for some constant $c$ then $f'(x)=0$ and that $(f+g)'(a)=f'(a)+g'(a)$.
So in the first case we'll have $f'(x)=-0.220 *4 x^3+5.44*3 x^2+47.3*2x+171+0$.