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Chapter 4: Problem 56

Find \(f\) such that: $$f^{\prime}(x)=3 e^{4 x}, \quad f(0)=\frac{7}{4}$$

Short Answer

Expert verified
The function is \(f(x) = \frac{3}{4}e^{4x} + 1\).

Step by step solution

01

Integrate the derivative

Given the derivative of the function, integrate both sides to find the original function: \(f'(x) = 3e^{4x}\)Integrate with respect to \(x\): \[f(x) = \frac{3}{4}e^{4x} + C\]
02

Solve for the constant

Given the initial condition \(f(0) = \frac{7}{4}\), substitute \(x = 0\) into the integrated function to find the constant \(C\): \[f(0) = \frac{3}{4}e^{0} + C = \frac{7}{4}\]Since \(e^{0} = 1\):\[\frac{3}{4} + C = \frac{7}{4}\]Subtract \(\frac{3}{4}\) from both sides: \[C = \frac{7}{4} - \frac{3}{4} = 1\]
03

Write the final function

With the constant \(C\) found, write the final expression for the function \(f(x)\): \[f(x) = \frac{3}{4}e^{4x} + 1\]

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Key Concepts

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

derivative
The derivative of a function measures how the function's value changes as the input changes. In this exercise, we start with the derivative given:
  • Given:
    \(f'(x) = 3e^{4x}\)
The aim is to find the original function before differentiation. The original function can be found using a process called integration.
initial condition
An initial condition is a value that defines the particular solution of a differential equation by specifying the value of the function at a specific point. Here, the initial condition is given by the information:
  • Given:
    \(f(0) = \frac{7}{4}\)
We use this information to find the precise value of the constant when we integrate the derivative. By substituting the initial condition into the integrated function, we solve for the constant 'C'.
constant of integration
The constant of integration appears when we integrate a function. It accounts for any vertical shift in the graph of the function. In this exercise:
  • Integrate:
    \(\frac{d}{dx} \bigg[\frac{3}{4}e^{4x} + C\bigg] = 3e^{4x}\)
The constant of integration 'C' is found using the initial condition:
  • Substitute \(x=0\):
    \(f(0) = \frac{3}{4}e^{0} + C = \frac{7}{4}\)
This simplifies to find 'C':
  • \(C = 1\)
With the constant 'C' found, we write the final function as:
  • \(f(x) = \frac{3}{4}e^{4x} + 1\)

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