(3) Suppose A and B are constants and f(x)= A sin(x) + B cos(X) (a) Calculate the first four derivatives of f(x) (b) Calculate f^(2017) (x)

Asked By mhseen On 02/25/2017 21:54


The first four derivatives, sequentially, are: f'(x)= Acos(x)-Bsin(x) f''(x)= -Asin(x)-Bcos(x) f'''(x)= -Acos(x)+Bsin(x) f''''(x)= Asin(x)+Bcos(x) Notice that, upon differentiating for the fourth time, we end up with our original function, f(x)= Asin(x)+Bcos(x). That is to say that any order derivative that is a multiple of four must be equal to the original function. 2016 is a multiple of four, so the 2016th derivative would be f(x)=Asin(x)+Bcos(x). Therefore, the 2017th derivative f^(2017)(x) would be equal to the first derivative above, that is, f^(2017)(x)=Acos(x)-Bsin(x).
Answered On 02/25/2017 22:52

The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). a) First Derivative: f'(x) = A cos(x) - B sin(x) Second Derivative: f''(x) = -A sin(x) - B cos(x) Third Derivative: f'''(x) = -A cos(x) + B sin(x) Fourth Derivative: f''''(x) = A sin (x) + B cos(x) b) After doing the derivative 4 times, you'll notice that we've come back to where we started, so the process will begin to repeat itself. 2016 is divisible by four, so doing the derivative 2016 times will just give us our original equation. Doing it one more time (2017) will give us the first derivative: f^2016(x) = A sin(x) + B cos(x) f^2017(x) = A cos(x) - B sin(x)
Answered On 02/25/2017 23:31