Problem 54 - Calculus and its applications

Question

Find f such that: $$ f^{\prime}(x)=6 x^{2}-4 x+2, \quad f(1)=9 $$

Step-by-Step Solution

Answer

The function \( f(x) = 2x^3 - 2x^2 + 2x + 7 \) satisfies the given conditions.

Step 1: Integrate the Derivative

To find the function \( f(x) \), start by integrating the given derivative \( f'(x) = 6x^2 - 4x + 2 \). The integral of \( f'(x) \) will be \( f(x) + C \) where \( C \) is the constant of integration.

Step 2: Compute the Integral

Integrate each term separately:- \( \int 6x^2 \, dx = 2x^3 \) (since \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \))- \( \int -4x \, dx = -2x^2 \)- \( \int 2 \, dx = 2x \)So, \( f(x) = 2x^3 - 2x^2 + 2x + C \).

Step 3: Apply Initial Condition

Use the initial condition \( f(1) = 9 \) to find \( C \):Substitute \( x = 1 \) into \( f(x) \):\[2(1)^3 - 2(1)^2 + 2(1) + C = 9\]Solve for \( C \):\[2 - 2 + 2 + C = 9\]\[2 + C = 9\]\[C = 7\]

Step 4: Write the Final Expression of \( f(x) \)

Substitute \( C = 7 \) back into the integrated function to find \( f(x) \):\[f(x) = 2x^3 - 2x^2 + 2x + 7\]

Key Concepts

Derivatives

Understanding derivatives is crucial in calculus. They represent the rate of change of a function with respect to a variable, usually denoted as \( f^{\prime}(x) \). For example, in our problem, we have \( f^{\prime}(x) = 6x^2 - 4x + 2 \). By knowing the derivative, you can gain insight into how the function behaves at any given point.

Here’s what each part represents:

• The term \( 6x^2 \) shows that the rate of change is influenced heavily by \( x^2 \), meaning it increases rapidly as \( x \) grows.
• The term \( -4x \) indicates a negative linear influence on the function's rate of change.
• The constant \( +2 \) affects the rate of change uniformly, adding a consistent rate across all \( x \) values.

By taking derivatives, one can determine features like where a function's rate of increase or decrease is maximal, or where it might have inflection points.

Initial Conditions

Initial conditions are vital for uniquely determining a function from its derivative. In our example, we are given \( f(1) = 9 \). This tells us that when \( x = 1 \), the function's value is exactly 9. This piece of information allows us to solve for the constant of integration when integrating the derivative.

Without initial conditions, the solution to the integral would contain an arbitrary constant \( C \). Initial conditions make the mathematical problem more constrained and ensure the derived function meets specific criteria set by the conditions.

• Application: In physics, initial conditions can dictate a system's starting status in a dynamic model.
• Mathematical necessity: In calculus, it helps finalize the indefinite integral to a specific function.

It acts almost like a "starting point" that ensures the mathematical function we construct behaves as expected from the initial constraints given.

Constant of Integration

The constant of integration \( C \) embodies the indefinite nature of the integration process. When you integrate a function, you essentially "reverse" the differentiation process, which can lead to infinitely many solutions. Each solution differs by a constant value, since derivatives of constants are zero.

In our situation, we integrated the derivative to get \( f(x) = 2x^3 - 2x^2 + 2x + C \). To identify the constant \( C \), we applied the initial condition \( f(1) = 9 \). By substituting \( x = 1 \) into \( f(x) \), we solved for \( C \), discovering that \( C = 7 \).

• Purpose: The constant reflects any missing "baseline" information lost through differentiation.
• Flexibility: Allows for the adjustment of the function to meet initial or boundary conditions.

Ultimately, knowing how to determine the constant of integration lets you derive a proper function that precisely matches the criteria outlined by its initial conditions.