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

step 1 In this question, we are talking about the chain rule. Specifically, we are given this parametri

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

Key Concepts

Parametric Differentiation

In the parametric form of a curve, the derivative dy/dx is obtained by dividing the derivative of y with respect to the parameter (dy/dt) by the derivative of x with respect to the parameter (dx/dt). This use of the chain rule is essential for finding the slope of the tangent to the curve at a specific point.

Parametric Equations

Parametric equations define curves using a parameter, typically denoted by t, to express both the x and y coordinates. This form allows for a more flexible representation of curves that may be difficult or impossible to express as a single function y = f(x). Understanding how to work with parametric forms is key when analyzing the behavior of a curve, such as determining its tangent line.

Tangent Line to a Curve

A tangent line to a curve at a given point is a straight line that touches the curve without crossing it at that point. The slope of the tangent line for a parametric curve is found by computing dy/dx using parametric differentiation. Once the slope and the point of tangency are known, the point-slope form of a line can be used to construct the equation of the tangent line.

Differentiation of Trigonometric Functions

When working with curves expressed in terms of trigonometric functions, it is important to utilize the differentiation rules for these functions. Knowledge of the derivatives of functions such as tan(t), sec(t), and others is necessary for accurately computing dy/dt and dx/dt, which in turn are used to determine the slope and equation of the tangent line.

Transcript

00:01 In this question, we are talking about the chain rule.

00:05 Specifically, we are given this parametrically defined curve with the parameter t, and asked to find the line tangent to the curve at this value of t.

00:23 So, how do we do this? well, what is the formula for the tangent line, first of all? well, say our point of tangency is a, b, and in that case, the formula is, say also that the point happens at t equals t not.

00:56 Then the formula is y minus b is equal to the derivative of y with respect to x at t equals t0 times x minus a.

01:09 Notice that t equals t not, because of the parametric curve, defines what a and b are.

01:19 This t value determines our x and y coordinates.

01:24 And once we know the coordinates and the derivative at that t value, we can use this formula to determine the tangent line.

01:35 Okay, and let's just, let's do just that.

01:41 First, what is the value of a? it has a secant function, which is the reciprocal of cosine.

01:51 So first i'm going to find cosine of minus pi over four.

02:03 That is one over the root, one over root two, or root two over two.

02:12 That's also an acceptable form.

02:18 That gives us that cost squared at that value is just one half.

02:27 And since secan squared is the reciprocal of that, our value of x is 2 minus 1.

02:53 Next, the value of y.

02:57 So again, we can use the unit circle diagram to see that at the angle minus pi over 4, the opposite side length is one and the adjacent side length is one.

03:11 Therefore, 10 is also 1.

03:15 Oh, sorry, these are not necessarily 1.

03:18 They are root 2 over 2 on the unit circle.

03:25 That is if the radius is 1.

03:27 But either way, no matter how large the radius is, these two side lengths will be equal to each other.

03:34 And so opposite over adjacent will be equal to 1.

03:40 Therefore, b is 1.

03:44 And finally, we're going to find d, y, d, x, the meat of this problem, using chain rule.

03:56 Now, we don't know why in terms of x right away.

04:00 We might be able to find it by somehow combining these functions, but that's going to take a lot of work.

04:05 And so, we can instead use our knowledge of y in terms of t and x in terms of t and the chain rule to determine this...

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