We are given the second derivative of f(t)
color(red)(f''(t) = t^(-3/2))
That is, color(red)((d^2y)/(dt^2) = t^(-3/2))
We must find color(blue)(f(t) using the given details.
Solution Process being used:
We will integrate color(red)(f''(t) = t^(-3/2)) to obtain the first derivative of f(t) and get color(blue)(f'(t)
Then we will integrate color(blue)(f'(t) to obtain color(blue)(f(t)
color(green)(Step.1)
To find the first derivative, we integrate color(red)(f''(t)
Hence, color(red)(f'(t) = int " " f''(t)*dt
f'(t) = int " "t ^ (-3/2)*dt
rArr t ^ (-3/2+1)/(-3/2+1)+C
rArr t^(-1/2)/(-1/2)+C
rArr -t^(-1/2)/(1/2)+C
rArr -2t^(-1/2)+C
rArr -2/sqrt(t)+C
color(blue)(f'(t) = int " "t ^ (-3/2)*dt = -2/sqrt(t)+C)
That is, color(red)((dy)/(dt) = -2/sqrt(t)+C)
color(green)(Step.2)
To obtain the f(t), we integrate color(red)(f'(t)
Hence, color(red)(f(t) = int " " f'(t)*dt
We will now work on
color(red)(f(t) = int " " f'(t)*dt =int " " -2/sqrt(t)+C *dt
We will pull the constant term out to get
-2int " " 1/sqrt(t)+C*dt
rArr -2 int t ^ (-1/2)+C* dt
rArr -2.[t^(-1/2+1]/(-1/2+1]]+(C/1)*t^1
rArr -2.[t^(1/2]/(1/2]]+Ct
rArr -2.(2/1)*[t^(1/2)]+Ct
rArr -4.sqrt(t)+Ct
We must also add a constant term to our answer.
Hence,
color(blue)(f(t) = -4sqrt(t)+Ct+D)
is our required function
