Question

Parametric equations for a curve are given. Find \(\frac{d^{2} y}{d x^{2}},\) then determine the intervals on which the graph of the curve is concave up/down. \(x=t, \quad y=t^{2}\)

Short answer

The graph is concave up everywhere since \( \frac{d^2y}{dx^2} = 2 \).

Step-by-step solution

Compute the first derivative

The first step is to find the first derivative \( \frac{dy}{dx} \). Since both \( x \) and \( y \) are given as functions of \( t \), use the chain rule: \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \). Here, \( \frac{dy}{dt} = 2t \) and \( \frac{dx}{dt} = 1 \). Therefore, \( \frac{dy}{dx} = \frac{2t}{1} = 2t \).

Compute the second derivative

To find \( \frac{d^2y}{dx^2} \), take the derivative of \( \frac{dy}{dx} \) with respect to \( t \) and divide by \( \frac{dx}{dt} \). We have \( \frac{d}{dt}(2t) = 2 \). Therefore, \( \frac{d^2y}{dx^2} = \frac{2}{1} = 2 \).

Determine the concavity of the curve

The sign of \( \frac{d^2y}{dx^2} \) determines the concavity. Since \( \frac{d^2y}{dx^2} = 2 \) is positive for every \( t \), the graph is concave up on its entire domain.

Key concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus used to differentiate composite functions. When dealing with parametric equations, where both x and y are functions of a third variable, often represented as t, the Chain Rule assists in finding derivatives with respect to x. For instance, given functions x(t) and y(t), the derivative \( \frac{dy}{dx} \) is found using the formula \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \).
In our exercise, we have:

• \( x = t \)
• \( y = t^2 \)

To apply the Chain Rule here, we compute \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \):

• \( \frac{dy}{dt} = 2t \) because the derivative of \( t^2 \) with respect to t is 2t.
• \( \frac{dx}{dt} = 1 \) because the derivative of t with respect to itself is 1.

Thus, \( \frac{dy}{dx} = \frac{2t}{1} = 2t \). This calculation shows how the Chain Rule transforms derivatives of parametric functions into derivatives with respect to x.

First Derivative

The first derivative is crucial in understanding the rate of change of the function. For parametric equations, like in the exercise with \( x(t) = t \) and \( y(t) = t^2 \), the first derivative \( \frac{dy}{dx} \) essentially measures how y changes with respect to x as the parameter t changes.
After applying the Chain Rule, we found that \( \frac{dy}{dx} = 2t \). This derivative indicates that the slope of the tangent to the curve at any point is proportional to 2t. If t increases, the slope of the curve steepens; if t decreases, the slope of the curve flattens. Understanding this behavior helps predict how the curve will move and inflect at different parameter values.

Second Derivative

The second derivative gives insights into the curvature of the graph, making it possible to determine concavity. The formula to find the second derivative \( \frac{d^2y}{dx^2} \) from a parametric form involves differentiating \( \frac{dy}{dx} \) again with respect to t, and then dividing by \( \frac{dx}{dt} \).
In this exercise:

• Start with the first derivative, \( \frac{dy}{dx} = 2t \).
• Differentiating 2t with respect to t yields 2, giving \( \frac{d}{dt}(2t) = 2 \).
• Finally, dividing by \( \frac{dx}{dt} = 1 \) gives \( \frac{d^2y}{dx^2} = \frac{2}{1} = 2 \).

Thus, the second derivative \( \frac{d^2y}{dx^2} \) tells us whether the curve opens up or down and helps determine intervals of concavity.

Concavity

Concavity indicates whether a curve is bending upwards or downwards. It's determined by the sign of the second derivative \( \frac{d^2y}{dx^2} \). For this exercise, since \( \frac{d^2y}{dx^2} = 2 \), which is consistently positive, the curve is always concave up on its entire domain. This means the curve bends upwards, resembling a U-shape pattern.
Understanding concavity is essential for sketching graphs accurately, optimizing functions, and finding points of inflection, which are points where the curvature changes direction.