Find the equation of the line tangent to the curve at the point defined by the given value of t.

Find the equation of the line tangent to the curve at the...

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Key Concepts

Parametric Equations

Parametric equations define a curve by expressing the coordinates (x, y) as functions of a parameter. This representation allows for a clear description of complex curves and is particularly useful when the relationship between x and y is not a function in the traditional sense.

Differentiation of Parametric Functions

When a curve is defined parametrically, the derivative dy/dx is obtained by differentiating both x and y with respect to the parameter t and then taking the quotient (dy/dt)/(dx/dt). This method relies on the chain rule and is essential for finding the slope of the tangent line to the curve at any point.

Tangent Line

The tangent line to a curve at a given point is the line that just touches the curve at that point. Its slope is determined by the derivative of the function (or the derivative dy/dx for parametric equations) at that point, and the tangent line is typically found using the point-slope form of a line.

Transcript

00:01 In this question, where we will be using chain rule, we are given a curve that is defined by these parametric equations and asked to find the line that is tangent to this curve at this t value.

00:22 So what do we need to know for our tangent line? first of all, we need to know the slope at the point we are looking at, and the coordinates of the point, the point of tangency.

00:36 Now, since the curve is defined by these equations, each point x, y is going to be determined by a certain value of t.

00:46 And so if we simply just substitute this t value, we're going to get the corresponding values of x and y, the coordinates.

00:56 So the coordinates are taken care of, and all we have to do now is worry about the derivative.

01:05 The line is in the xy plane, so we are interested in the derivative.

01:09 Of y with respect to x.

01:12 That's going to be the slope of our line.

01:16 However, we don't have an easy way of finding d -y -d -x from the parametric curves, directly, i mean, because we're not given y as a function of x, that we can differentiate with respect to x.

01:30 We're given each of these in terms of t.

01:35 And so we have to use derivatives with respect to t instead.

01:43 The good thing is that the chain rule tells us how to do this.

01:48 D .ydx is simply the ratio of d .y.

01:51 D .t...

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