Question

(i) Using 5 points x=0, 1,2,3,4 complete the table

 x     0        1       2        3        4 
f(x) 

(ii) Find $the f’(0.5).$

(iii) Find the exact value if

$f’(x)=(-a)/100 e^(-ax/100) cos⁡〖3ax-3a〗 e^(-ax/100) sin⁡3ax$

Answer 1

$$Solution$$

$$f'(x)=\frac{−a}{100}e^{\frac{−ax}{100}}cos⁡〖3ax−3a〗e^{\frac{−ax}{100}}sin⁡3ax\\ \implies \frac{d}{dx}(e^\frac{-ax}{100}cos 3ax)$$

Integrate on both sides

$$f(x)=e^\frac{-ax}{100}cos\ 3ax$$

Now, x=0

$f(x)=e^{-0}\ cos\ 0^0=1\\ x=1\\ f(1)=e^\frac{a}{100}\ cos\ 3a$

$$\begin{matrix} x & 0 & 1 & 2 & 3 & 4 \\ f(x) & 1 & e^{-\frac{a}{100}}cos\ 3a & e^{-\frac{a}{50}}cos\ 6a & e^{-\frac{3a}{100}}cos\ 9a & e^{-\frac{a}{25}}cos\ 12a \end{matrix}$$

$$f'(0.5)=\frac{-a}{100}e^\frac{-a}{200}cos\ \frac{3a}{2}-3ae^\frac{-a}{200}sin\ \frac{3a}{2}\\ -ae^\frac{-a}{200}(\frac{cos \frac{3a}{2}}{100}+3\ sin\ \frac{3a}{2})$$