Find t intervals on which the curve x=3 t^2, y=t^3-t is concave up as well as concave down.

Name: Problem 98, Chapter 7: Calculus Volume 2
Uploaded: 2019-08-03T18:21:31-08:00
Duration: 1 min 31 s
Channel: Amy Jiang
Description: Find t intervals on which the curve x=3 t^2, y=t^3-t is concave up as well as concave down.

step 1 Okay, so we have x is equal to 3 t squared, y is equal to t cubed minus t. Derivative of x is th

Find t intervals on which the curve x=3 t^2, y=t^3-t is...

Key Concepts

Parametric Equations

Parametric equations express a set of related quantities as functions of an independent parameter. In these equations, both x and y are defined in terms of a third variable (often t), which allows for more flexibility in describing curves that might not be functions in the traditional sense.

First Derivative of Parametric Curves

In the context of parametric equations, the first derivative dy/dx is calculated as (dy/dt) divided by (dx/dt). This derivative represents the slope of the tangent to the curve at any given point and is essential in analyzing the behavior of the curve, such as increasing or decreasing trends.

Second Derivative of Parametric Curves

The second derivative, d²y/dx², measures the curvature or concavity of the curve. For parametric equations, it is typically computed using the formula d²y/dx² = (d/dt(dy/dx)) / (dx/dt). This derivative helps determine whether the curve is concave up or concave down at various regions along the parameter's domain.

Concavity

Concavity refers to the direction in which a curve bends. A curve is concave up if its second derivative is positive, meaning that the tangent lines all lie below the curve. Conversely, the curve is concave down if the second derivative is negative, indicating that the tangent lines lie above the curve. Analyzing concavity is important for understanding the curvature and identifying points of inflection.

Sign Analysis of the Second Derivative

After computing the second derivative for a parametric curve, sign analysis is performed to determine the intervals on which the curve is concave up or concave down. This involves solving inequalities where the second derivative is greater than or less than zero, respectively, and is a critical step in sketching the curve and understanding its behavior.

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