Let f'''(t) = 5 t + 10 \sqrt{t}.
(a) Find the most general formula for f''(t). If an arbitrary constant must be used here, use an upper-case "C".
f''(t) =
(b) Based on your answer to (a), find the most general formula for f'(t). If another new arbitrary constant must be used here, use an upper-case "D".
f'(t) =
(c) Based on your answer to (b), find the most general formula for f(t). If another new arbitrary constant must be used here, use an upper-case "E".
f(t) =
I guess you mean
d^3y/dt^3 = 5 t + 10 t^-(1/2)
d^2y/dt^2 = (5/2)t^2 + (20/3)t^(3/2) + C
dy/dt = (5/6)t^3 + (40/15)t^(5/2) +C t+D
and so forth
To find the most general formula for f''(t), we need to integrate the given function f'''(t). Integrating will reverse the process of differentiation.
(a) Integration of f'''(t) will give us f''(t). Applying the power rule of integration to each term in the function, we integrate 5t as (5/2)t^2 and 10√t as (20/3)t^(3/2):
f''(t) = ∫(5t + 10√t) dt
= (5/2)t^2 + (20/3)t^(3/2) + C,
where C represents the constant of integration.
Now, let's move on to finding the most general formula for f'(t) based on the answer to part (a).
(b) To find f'(t), we take the derivative of f''(t) with respect to t. Taking the derivative will reverse the process of integration.
Differentiating (5/2)t^2 + (20/3)t^(3/2) + C with respect to t, we apply the power rule of differentiation:
f'(t) = d/dt[(5/2)t^2 + (20/3)t^(3/2) + C]
= 5t + (20/2)(3/2)t^(1/2)
= 5t + 10√t + D,
where D represents the arbitrary constant of integration.
Finally, let's find the most general formula for f(t) based on the answer to part (b).
(c) To find f(t), we integrate f'(t) using the reverse power rule of integration.
Integrating 5t + 10√t + D, we apply the power rule of integration:
f(t) = ∫(5t + 10√t + D) dt
= (5/2)t^2 + (20/3)t^(3/2) + Dt + E,
where E represents the new arbitrary constant of integration.
Therefore, the most general formula for f(t) is (5/2)t^2 + (20/3)t^(3/2) + Dt + E.