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

Key Concepts

Tangent Line to a Curve

The tangent line at a given point on a curve is the line that best approximates the curve near that point. It uses the slope derived from the function's derivative and passes through the point of tangency, providing a linear approximation of the curve in a local region.

Parametric Equations

Parametric equations define curves by expressing the coordinates of the points on the curve as functions of one or more parameters. This approach allows for the representation of curves that may not be easily described by a standard function, enabling analysis of more complex motion and geometric forms.

Derivative of a Parametric Function

For curves defined by parametric equations, the slope of the tangent line is obtained by taking the derivative of the y-component with respect to the parameter and dividing it by the derivative of the x-component with respect to the parameter. This method simplifies the process of finding rates of change along the curve.

Transcript

00:01 In this question, we will be talking about chain rule.

00:04 The question gives us a parametrically defined curve and a t value, a specific t value, and asks us to find the equation of the line tangent to the curve at the point defined by this t value.

00:21 Now, in order to find this tangent line, we need to know the point at which the tangency occurs, so the tangent point.

00:31 And we also need to know the slope of the tangent line.

00:35 Once we have those pieces of information, we have enough to determine the equation of the whole line.

00:45 If you're interested to know why, the answer is simple.

00:53 To define the equation of a line, we need to know its slope and its y -intercent.

01:00 Well, actually, we don't need to, but one way to ensure that we know the whole equation is to know its slope and y -intercent.

01:08 Now, if i know the slope and the point of tangency, i obviously know the slope, and using that i can find, using the point of tangency, i have known values for y, x, and m, and i use that to find b.

01:32 That's why knowing this information is enough to know the equation of the tangent.

01:38 However, we don't even need to solve for b because we know the formula for the equation of the tangent line.

01:49 That is, given that ab is the point of tangency, the equation of the tangent is y minus b is equal to the slope times x minus a.

02:06 Okay, so let's go about finding a, b, and m.

02:18 Now, since a is the x coordinate of the point of tangency, which must occur at t equals pi over 3, a is the x value at this t value.

02:31 And so we just substitute that t value into here.

02:34 Here and for the y coordinate into here...

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