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Chapter 6: Problem 43

Find all functions \(f(t)\) that satisfy the given condition. $$f^{\prime}(x)=x, f(0)=3$$

Short Answer

Expert verified
\(f(t) = \frac{t^2}{2} + 3\)

Step by step solution

01

Understand the Problem

The problem asks to find a function \(f(t)\) given its derivative and an initial condition. Given \(f^{\backprime}(x) = x\) and \(f(0) = 3\), we need to integrate the derivative to obtain the original function.
02

Integrate the Derivative

To find \(f(t)\), integrate the given derivative. \(\frac{d}{dt} f(t) = t\). Therefore: \[f(t) = \int t \, dt = \frac{t^2}{2} + C\]
03

Apply the Initial Condition

Use the initial condition \(f(0) = 3\) to find the constant \(C\). Substitute \(t = 0\): \(\frac{0^2}{2} + C = 3\) so \(C = 3\)
04

Write the Final Solution

Combine the result from the integration and the constant to get the final function: \[f(t) = \frac{t^2}{2} + 3\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

initial condition
Sometimes, when solving problems involving differential equations, you will be given an initial condition. This is a piece of information that tells you the value of the function at a specific point. It is crucial for determining the constant of integration. In this exercise, the initial condition is given as \(f(0) = 3\). This means that when \(t = 0\), the function \(f(t)\) is equal to 3. To use this initial condition, we first integrate the given derivative and then substitute the point into the general solution to solve for the constant.
definite integral
The process of integration helps us move from a derivative back to the original function. A definite integral finds the exact value of the area under a curve between two points. However, in this exercise, we are more concerned with an indefinite integral, which gives us a family of functions. When integrating the derivative \(f'(t) = t\), we get: \[f(t) = \int t \, dt = \frac{t^2}{2} + C\] Here, \(C\) is the constant of integration. Notice that without additional information, there are infinitely many possible functions, all differing by a constant.
antiderivative
An antiderivative is a function whose derivative is the given function. For the derivative \(f'(t) = t\), the antiderivative is any function \(f(t)\) that satisfies this relationship. Integrating \(t\) with respect to \(t\) gives us \(\frac{t^2}{2} + C\). To express the particular solution that fits the problem, we use the initial condition \(f(0) = 3\). By substituting \(0\) for \(t\), we can solve for \(C\):
  • \(\frac{0^2}{2} + C = 3\)
  • \(C = 3\)
Putting it all together, the specific function that satisfies both the derivative and the initial condition is: \[f(t) = \frac{t^2}{2} + 3\]

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