Problem 55 Find $f$ such that: $$f^{\pr... [FREE SOLUTION]

91影视

Calculus and Its Applications

Marvin L. Bittinger, David J. Ellenbogen, Scott A. Surgent

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 4: Problem 55

Find $f$ such that: $$f^{\prime}(x)=5 e^{2 x}, \quad f(0)=\frac{1}{2}$$

Short Answer

Expert verified

The function is $ f(x) = \frac{5}{2} e^{2x} - 2 $.

Step by step solution

- Integrate the given derivative

We are given the derivative of the function: $ f^{\text{'} }( x) =5e^{2x} $. To find $ f(x) $, integrate $ f^{\text{'} }( x) $ with respect to $ x $. \[ f(x) = \frac{5}{2} e^{2x} + C \], where $ C $ is the integration constant.

- Apply the initial condition

We are given the initial condition $ f(0) = \frac{1}{2} $. Substitute $ x = 0 $ and $ f(0) = \frac{1}{2} $ into the integrated function to solve for $ C $: \[ f(0) = \frac{5}{2} e^{0} + C = \frac{1}{2} \] This simplifies to: \[ \frac{5}{2} + C = \frac{1}{2} \].

- Solve for the constant C

Subtract $ \frac{5}{2} $ from both sides to find $ C $: \[ C = \frac{1}{2} - \frac{5}{2} = -2 \].

- Write the final function

Substitute the value of $ C $ back into the integrated function to get the final function: \[ f(x) = \frac{5}{2} e^{2x} - 2 \].

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Key Concepts

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

integration

Integration is a fundamental concept in calculus that involves finding a function given its derivative. In simpler terms, when you integrate, you're essentially reversing the process of differentiation. For instance, if we are given the derivative of a function, say $ f'(x) = 5e^{2x} $, integration allows us to find the original function $ f(x) $. The symbol for integration is $ \int $, and the result often includes an arbitrary constant, known as the integration constant.

initial condition

An initial condition is a piece of extra information that helps to determine the specific solution of an equation. It usually gives the value of the function at a certain point. In our case, the problem gives the initial condition $ f(0) = \frac{1}{2} $. This means that when $ x = 0 $, the function $ f(x) $ should yield $ \frac{1}{2} $. This piece of data is crucial because it allows us to solve for the integration constant, ensuring our solution fits the specific scenario described.

integration constant

When we integrate a function, the result includes an arbitrary constant known as the integration constant, often denoted by $ C $. This constant represents an infinite number of possible functions that all differ by a constant value but have the same derivative. In our exercise, after integrating $ 5e^{2x} $, we got the general solution $ f(x) = \frac{5}{2}e^{2x} + C $. To find the exact value of $ C $, we use the initial condition given, $ f(0) = \frac{1}{2} $. By substituting $ x = 0 $ and solving, we find $ C = -2 $, thus narrowing down our solution to a specific function: $ f(x) = \frac{5}{2}e^{2x} - 2 $.

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91影视

Find \(f\) such that: $$f^{\prime}(x)=5 e^{2 x}, \quad f(0)=\frac{1}{2}$$

Short Answer

Step by step solution

- Integrate the given derivative

- Apply the initial condition

- Solve for the constant C

- Write the final function

Key Concepts

integration

initial condition

integration constant

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