91Ó°ÊÓ

The derivative of a function represents the rate at which the function's value changes at any given point. It is the slope of the tangent line to the curve at a particular point, providing insights into the behavior of the function at that location. In this exercise, the derivative given is \( \frac{dy}{dx} = x^3 \sqrt{1 + x^2} \).

This expression shows how the slope of the curve changes with respect to \( x \). Understanding the derivative is crucial as it tells us what the curve is doing at each point along its path. When you have a derivative, you can think about how a small change in \( x \) will impact the change in \( y \).

• The slope \( \frac{dy}{dx} \) can help predict the movement of the curve.
• Positive slope: curve goes up, negative slope: curve goes down.
• For more complex derivatives, integration helps us find the original curve.

Integral substitution is a technique often used to simplify the integration process. It involves changing variables to make integrals easier to evaluate. In this exercise, we aimed to find \( y \) from \( \frac{dy}{dx} = x^3 \sqrt{1 + x^2} \) using integral substitution.

Here's how it worked for our problem:

• Set \( u = 1 + x^2 \), leading to \( du = 2x \, dx \).
• Rewriting \( x \, dx = \frac{1}{2} \, du \), we can substitute these into the integral.
• Ultimately, the integral becomes \( \int (u - 1) \sqrt{u} \cdot \frac{1}{2} \, du \).
• This transformation simplifies solving the integral, making computations manageable.

Substitution is a powerful method for transforming complex integrals into forms we can handle more easily, reducing potential errors along the way.

When integrating an indefinite integral, we always add a constant of integration, \( C \). This step is crucial because there are infinitely many antiderivatives differing only by a constant. In context, the integral itself returns a family of functions, where \( C \) specifies one particular function from this family.

For our problem, after integration and back-substitution, we obtained:

• \( y = \frac{1}{5} (1 + x^2)^{5/2} - \frac{1}{3} (1 + x^2)^{3/2} + C \).

We found that the curve passes through the origin, or point (0,0), which helped determine \( C \). By substituting \( x = 0 \) and \( y = 0 \) into the equation, we solved for \( C \) as \( \frac{2}{15} \). Including \( C \) ensures we account for all potential vertical translations of the curve.

Constructing the equation of a curve is the ultimate goal when working with derivatives and integrals. In this scenario, we started with the derivative \( \frac{dy}{dx} = x^3 \sqrt{1+x^2} \) and integrated it to find an equation that describes the curve mathematically.

The final equation of the curve becomes:

• \( y = \frac{1}{5} (1 + x^2)^{5/2} - \frac{1}{3} (1 + x^2)^{3/2} + \frac{2}{15} \).

This equation provides a complete representation of the curve, illustrating its shape and how it behaves as \( x \) changes. Similarly, it reconciles our initial conditions, showing where the curve passes through the origin. In practical terms, having this equation allows for computing any point \( y \) on the curve for any given \( x \), showcasing the continuous nature of mathematical functions in visualizing a curve.