Problem 10 - Calculus Early Transcendentals

Question

For a given function \(f\), explain the steps used to solve the initial value problem \(F^{\prime}(t)=f(t), F(0)=10\)

Step-by-Step Solution

Answer

Question: Solve the initial value problem \(F^{\prime}(t)=f(t), F(0)=10\) for the given function \(f(t)\). Answer: To solve the initial value problem, follow these steps: 1. Integrate the function f(t) to find F(t): \(F(t) = \int f(t)dt + C\) 2. Apply the initial condition F(0)=10 to find the value of C: \(C = 10\) 3. Write the final solution for the function F(t): \(F(t) = \int f(t)dt + 10\) The complete solution for the initial value problem is \(F(t) = \int f(t)dt + 10\).

Step 1: Integrate the Function f(t) to find F(t)

Integrate the function \(f(t)\) with respect to \(t\): \(F(t) = \int f(t)dt + C\) where \(C\) is the integration constant.

Step 2: Apply the Initial Condition F(0)=10

To find the value of the integration constant \(C\), we need to apply the initial condition \(F(0)=10\): \(10 = \int f(0)d(0) + C\) Solve for \(C\): \(C = 10\)

Step 3: Write the Final Solution

Write the final solution for the function \(F(t)\), using the found value for integration constant \(C\): \(F(t) = \int f(t)dt + 10\) Now, we have the complete solution for the initial value problem for the given function \(f(t)\).