Notes

 

Percentage Rate of Change
\(\displaystyle\frac{f'(x)}{f(x)}\)

 

Questions & Solutions

\(\textbf{1)}\) \(\text{Find the percentage rate of change of } f(x)=4x^2+200 \text{ at } x=2.\)

Show Answer

\(\text{The answer is } \approx 7.41\% \)

Show Work

\(\,\,\,\,\,\,\text{Percentage rate of change is }\displaystyle \frac{f'(x)}{f(x)}\)
\(\,\,\,\,\,\,\text{Percentage rate of change is }\displaystyle \frac{8x}{4x^2+200}\)
\(\,\,\,\,\,\,\text{Plug in x=2: }\displaystyle \frac{8(2)}{4(2)^2+200}\)
\(\,\,\,\,\,\,\displaystyle \frac{16}{4 \cdot 4+200}\)
\(\,\,\,\,\,\,\displaystyle \frac{16}{216}\approx 0.074074\)
\(\,\,\,\,\,\,\text{The answer is } \approx 7.41\% \)

 

\(\textbf{2)}\) \(\text{Find the percentage rate of change of } f(x)=3x+20 \text{ at } x=3.\)

Show Answer

\(\text{The answer is } \approx 10.34\% \)

Show Work

\(\,\,\,\,\,\,\text{Percentage rate of change is }\displaystyle \frac{f'(x)}{f(x)}\)
\(\,\,\,\,\,\,\text{Percentage rate of change is }\displaystyle \frac{3}{3x+20}\)
\(\,\,\,\,\,\,\text{Plug in x=3: }\displaystyle \frac{3}{3(3)+20}\)
\(\,\,\,\,\,\,\displaystyle \frac{3}{29}\)
\(\,\,\,\,\,\,\displaystyle \frac{3}{29}\approx 0.1034\)
\(\,\,\,\,\,\,\text{The answer is } \approx 10.34\% \)

 

\(\textbf{3)}\) \(\text{Find the percentage rate of change of } f(x)=2x^3+4x+1 \text{ at } x=1.\)

Show Answer

\(\text{The answer is } \approx 142.9\% \)

Show Work

\(\,\,\,\,\,\,\text{Percentage rate of change is }\displaystyle \frac{f'(x)}{f(x)}\)
\(\,\,\,\,\,\,\text{Percentage rate of change is }\displaystyle \frac{6x^2+4}{2x^3+4x+1}\)
\(\,\,\,\,\,\,\text{Plug in x=1: }\displaystyle \frac{6(1)^2+4}{2(1)^3+4(1)+1}\)
\(\,\,\,\,\,\,\displaystyle \frac{6+4}{2+4+1}\)
\(\,\,\,\,\,\,\displaystyle \frac{10}{7}\approx 1.429\)
\(\,\,\,\,\,\,\text{The answer is } \approx 142.9\% \)

 

 

See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)

\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)

\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)

\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)

\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)

\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)

\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)

\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)

\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)

\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)

\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)

\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

 

In Summary

The percentage rate of change can be found by simply finding the derivative and dividing by the function value. It shows what percent of the total function value is changing for any given instant. It has a lot of real world applications and is usually learned in a business calculus course at the college level.