Q. 4 - Calculus

Question

 Find a function f that has the given derivative f'  and value f(c), if possible.  

$f \text{'}\left(x\right) = 3- {\frac{4}{5}}^x , f\left(1\right) = 0$

Step-by-Step Solution

Answer

The function is $3\mathrm{x}-\frac{{\frac{4}{5}}^{\mathrm{x}}}{\log {\frac{4}{5}}^{}}-2.992$.

The value of f(2.992( is $2.992$

Step 1. Given

The given derivative of function is $f \text{'}\left(x\right) = 3- {\frac{4}{5}}^x , f\left(1\right) = 0$

Step 2. Finding the function using integration

$$\begin{gathered} f \text{'}\left(x\right) = 3- {\frac{4}{5}}^x , f\left(1\right) = 0 \\ \int f \text{'}\left(x\right) =\int 3- {\frac{4}{5}}^{x}dx \\ =3x-\frac{{\frac{4}{5}}^x}{\log {\frac{4}{5}}^{}}+c \\ \mathrm{where} \mathrm{c} \mathrm{is} \mathrm{constant} \\ \mathrm{f}\left(1\right)=3\left(1\right)-\frac{{\frac{4}{5}}^1}{\log {\frac{4}{5}}^{}}+\mathrm{c} \\ 0=3-\frac{4}{5}\times 0.0969+\mathrm{c} \\ 0=2.992+\mathrm{c} \\ \mathrm{c}=-2.992 \\ \mathrm{Hence} \mathrm{the} \mathrm{function} \mathrm{is} 3\mathrm{x}-\frac{{\frac{4}{5}}^{\mathrm{x}}}{\log {\frac{4}{5}}^{}}-2.992 \end{gathered}$$

Step 3. Finding the value of f(c).

$$\begin{gathered} f\left(x\right) =3\mathrm{x}-\frac{{\frac{4}{5}}^{\mathrm{x}}}{\log {\frac{4}{5}}^{}}-2.992 \\ Putting value of c =2.992 \\ f\left(c\right)=3\times 2.992\frac{{\frac{4}{5}}^{2.992}}{\log {\frac{4}{5}}^{2.992}}-2.992 \\ f\left(c\right)=3.45 \end{gathered}$$