Find the points (x, y) at which the curve has: (a) a horizontal tangent: (b) a vertical tangent.

step 1 So in this question, we are given the parameter equations. X -T is equal to cause of T and Y -T

Find the points (x, y) at which the curve has: (a) a...

Key Concepts

Curve Sketching

Sketching a curve defined by parametric equations involves plotting key points, analyzing the behavior of the functions such as the occurrence of horizontal and vertical tangents, and understanding the overall structure and symmetry of the curve. This process helps in visualizing the geometric properties of the curve and is a fundamental skill in mathematical analysis and applied graphing techniques.

Horizontal and Vertical Tangents

Horizontal and vertical tangent conditions are determined by specific values of the derivative components in parametric curves. A horizontal tangent occurs when the derivative dy/dt equals zero while dx/dt is non-zero, indicating zero slope. Conversely, a vertical tangent occurs when dx/dt equals zero while dy/dt is non-zero, representing an infinite or undefined slope.

Parametric Equations

Parametric equations represent curves by expressing the coordinates as functions of a variable, often denoted by t. This method allows for a flexible description of curves that might be difficult or impossible to represent using a single functional form y = f(x) or x = g(y), and it is particularly useful in studying the geometry and motion along a path.

Derivatives in Parametric Representations

When dealing with parametric equations, the derivative dy/dx is found using the chain rule through the formula (dy/dt)/(dx/dt), provided dx/dt is not zero. This derivative provides the slope of the tangent line to the curve at any point and is critical for analyzing how the curve changes with respect to the parameter.

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