91Ó°ÊÓ

In calculus, derivatives are essential for understanding how a function changes. When dealing with parametric equations like the ones provided, we define the derivatives with respect to a parameter, typically \(t\). The derivatives explain the rate at which one variable changes about another.

For the given parametric equations \(x = t + 1\) and \(y = t^2 + 3t\), the process begins by differentiating each equation with respect to \(t\). This results in \(\frac{dx}{dt} = 1\) and \(\frac{dy}{dt} = 2t + 3\). The crucial step is to find \(\frac{dy}{dx}\), which involves these derivatives:

• The chain rule connects these derivatives as: \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\)
• Substituting the values, we get \(2t + 3\)

The derivative \(\frac{dy}{dx}\) beneath the given conditions reveals insights into the rate at which \(y\) changes concerning \(x\).

The slope of a curve at a given point tells us how steep the curve is at that location. Parametric equations allow us to calculate this by substituting the specific parameter value into the derivative \(\frac{dy}{dx}\). With our example, inserting \(t = -1\) into \(2t + 3\) gives us the slope at that point.

Calculating, we find:

• For \(t = -1\), \(2(-1) + 3 = 1\)

At \(t = -1\), the calculated slope is 1, meaning the line is angled such that it rises one unit vertically for each horizontal unit. It indicates that at \(t = -1\), the curve experiences a gentle upward slope.

Concavity in mathematics describes the direction the curve bends. It informs us if the curve holds an upward or downward position at a specific point. To determine this, the second derivative \(\frac{d^2y}{dx^2}\) is used.

In our example, after finding \(\frac{dy}{dx} = 2t + 3\), we differentiate once more with respect to \(t\) and obtain the result:

• \((\frac{d}{dt}(2t + 3))/1 = 2\)

In this instance, because \(\frac{d^2y}{dx^2} = 2\) is greater than zero, the curve is concave up at \(t = -1\). This implies the shape of the curve near this point bends upwards like a U-shape.

The second derivative \(\frac{d^2y}{dx^2}\) plays a major role in understanding the curvature and concavity of the parametric equations. It provides information about how the slope \(\frac{dy}{dx}\) itself changes.

To find the second derivative, we start by recomputing the first derivative with respect to \(t\), which we then relate back to the \(x\) variable using the original parametric equations:

• The derivative of \(2t + 3\) gives us 2
• Thus, \(\frac{d^2y}{dx^2} = 2/1 = 2\)

The consistency of this second derivative as a positive number signifies a consistent upward bending in the curve. When interpreting these equations, the second derivative \(\frac{d^2y}{dx^2} = 2\) indicates a relatively stable concavity upwards. This kind of analysis allows us to visualize the curve's shape deeper.