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

step 1 In this question, we are given the parameter equations. XT is equal to 3 plus 2 sine of T and YT

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

Key Concepts

Curve Sketching

Sketching the curve involves plotting points from the parametric equations, particularly by considering the critical points, such as those where horizontal or vertical tangents occur. It also includes analyzing the overall range of the functions, the periodicity if trigonometric functions are involved, and the behavior of the curve as the parameter varies.

Vertical Tangents

A vertical tangent is present when the slope dy/dx is undefined, typically occurring where the denominator dx/dt equals zero while dy/dt is nonzero. By finding the parameter values for which dx/dt is zero, one can locate the points on the curve where the tangent line is vertical.

Parametric Equations

Parametric equations represent a curve by defining both x and y as functions of a third variable, usually called the parameter. This method allows one to describe more complex curves than can be represented by a single function y = f(x), and it simplifies the description of curves that may have multiple y-values for a single x-value.

Derivative of Parametric Curves

The derivative dy/dx for a parametric curve is found using the formula dy/dx = (dy/dt) / (dx/dt), which provides the slope of the tangent line at any point on the curve. This ratio of derivatives is crucial for understanding the behavior of the curve, such as identifying where the tangent is horizontal or vertical.

Horizontal Tangents

A horizontal tangent occurs when the slope of the tangent line is zero, which is achieved when dy/dt equals zero while dx/dt is nonzero. Setting the numerator of the derivative (dy/dt) to zero determines the parameter values at which the tangent to the curve is horizontal.

Please give Ace some feedback