Problem 53 - Calculus and its applications

Question

Find f such that: $$ f^{\prime}(x)=3 x^{2}-5 x+1, \quad f(1)=\frac{7}{2} $$

Step-by-Step Solution

Answer

The function is \( f(x) = x^3 - \frac{5}{2}x^2 + x + 3 \).

Step 1: Integrate to Find the General Function

To find the function \( f(x) \), start by integrating \( f'(x) = 3x^2 - 5x + 1 \). The integral of \( 3x^2 \) is \( x^3 \), the integral of \( -5x \) is \( -\frac{5}{2}x^2 \), and the integral of \( 1 \) is \( x \). Thus,\[f(x) = x^3 - \frac{5}{2}x^2 + x + C\]where \( C \) is the constant of integration.

Step 2: Apply the Initial Condition

Use the initial condition \( f(1) = \frac{7}{2} \) to solve for \( C \). Substitute \( x = 1 \) into the function:\[f(1) = 1^3 - \frac{5}{2}(1)^2 + 1 + C = \frac{7}{2}\]Simplify to find:\[1 - \frac{5}{2} + 1 + C = \frac{7}{2}\]This simplifies to:\[2 - \frac{5}{2} + C = \frac{7}{2}\]\[C = \frac{7}{2} - 2 + \frac{5}{2} = 3\]So, \( C = 3 \).

Step 3: Write the Particular Function

Substitute \( C = 3 \) back into the general function:\[f(x) = x^3 - \frac{5}{2}x^2 + x + 3\]This is the particular solution of the function \( f(x) \) that satisfies the given conditions.

Key Concepts

Integration Techniques

When tackling problems in integral calculus, selecting the appropriate integration technique is crucial. For the example provided, we faced a derivative function \( f'(x) = 3x^2 - 5x + 1 \). Each term was integrated individually to revert to the original function \( f(x) \).

• The integral of \( 3x^2 \) became \( x^3 \), as power rule for integration tells us to increase the exponent by one and divide by the new exponent.
• Similarly, for the linear term \( -5x \), the integral becomes \( -\frac{5}{2}x^2 \), increasing the power from 1 to 2 and dividing by 2.
• Finally, integrating a constant, like 1, gives us a linear term, in this case, \( x \).

By integrating each term separately and summing them up, the general form \( f(x) = x^3 - \frac{5}{2}x^2 + x + C \) is achieved, where \( C \) is the constant of integration. This technique is foundational in integral calculus as it aids in systematically reversing differentiation.

Initial Conditions

Initial conditions play a pivotal role in finding specific solutions from general integrals. After finding the general form of \( f(x) = x^3 - \frac{5}{2}x^2 + x + C \), we use the given initial condition \( f(1) = \frac{7}{2} \) to pinpoint the exact function.

• Substitute \( x = 1 \) into the general form: \( 1^3 - \frac{5}{2}(1)^2 + 1 + C = \frac{7}{2} \).
• Solving this equation allows us to find what \( C \) must be to satisfy the initial condition, ensuring that our function is tailored to fit the particular problem.

Initial conditions transform a general antiderivative into a specific function, which often responds to a real-world scenario described by the problem. This condition is key to concluding which specific path the function follows amidst many possibilities.

Finding Antiderivatives

Finding antiderivatives is about working backwards from a derivative to discover the original function. In the given exercise, we started with the derivative \( f'(x) = 3x^2 - 5x + 1 \) and needed to find \( f(x) \), the original function.

• The process involves integrating each term to find the antiderivative, resulting in \( f(x) = x^3 - \frac{5}{2}x^2 + x + C \).
• The constant \( C \) represents all possible vertical shifts of the function that still satisfy the derivative equation.
• Antiderivatives are essential in calculus for solving various kinds of equations, particularly when describing how quantities change over time or space.

Finding antiderivatives not only involves computation but also understanding the theory behind inverse operations in calculus. This skill is crucial in not just academia, but in fields such as physics and engineering, where it helps translate rates of change back into actual behaviors or conditions.