Question

Find f"(t) = 9/√t, f(4) = 28, f'(4) = 15. f(t) =

Answer 1

To find f(t), we need to integrate f''(t) twice. Given the known values of f'(4) and f(4), we can solve for the constants of integration and obtain the final expression for f(t).

Step-by-step explanation:

The given function is f''(t) = 9/√t. To find f(t), we need to integrate f''(t) twice. First, we integrate f''(t) with respect to t which gives us f'(t). Then, we integrate f'(t) with respect to t to find f(t).

Given that f'(4) = 15, we can find the constant of integration for the first integration. Integrating f''(t) = 9/√t gives us f'(t) = 6√t + C1, where C1 is the constant of integration.

Now, given that f(4) = 28, we can find the constant of integration for the second integration. Integrating f'(t) = 6√t + C1 gives us f(t) = 4t^(3/2) + C1t + C2, where C2 is the constant of integration.

Using the known values of f'(4) = 15 and f(4) = 28, we can solve for C1 and C2. Then, the final expression for f(t) is obtained.

Answer 2

The function f(t) is found by taking the antiderivative twice of f"(t) = 9/√t, using the initial conditions f(4) = 28 and f'(4) = 15, resulting in the function f(t) = 12t^{3/2} - 3t - 56.

Step-by-step explanation:

The student asks for the antiderivative f(t) given that its second derivative is f"(t) = \frac{9}{\sqrt{t}}, and also given the values of f(4) = 28 and f'(4) = 15. We start by integrating f"(t) to find the first derivative f'(t) and then integrating again to find f(t). The process involves finding the constants of integration by applying the given initial conditions.

To find f'(t), integrate f"(t) with respect to t:

`f'(t) = \int f`

`f'(t) = 9 \cdot 2√(t) + C`

Now, use the initial condition f'(4) = 15 to find C:

`15 = 9 \cdot 2√(4) + C`

C = 15 - 18

C = -3

Thus, f'(t) = 18\sqrt{t} - 3. We integrate f'(t) to find f(t):

`f(t) = \int (18√(t) - 3) dt`

`f(t) = 18\int t^(1/2) dt - 3\int dt`

`f(t) = 18\left((2)/(3)t^(3/2)\right) - 3t + D`

`f(t) = 12t^(3/2) - 3t + D`

Finally, apply the initial condition f(4) = 28 to find D:

`28 = 12(4)^(3/2) - 3\cdot 4 + D`

`28 = 12\cdot 8 - 12 + D`

28 = 96 - 12 + D

D = 28 - 84

D = -56

Therefore, the antiderivative
`f(t) = 12t^(3/2) - 3t - 56.`