Use the curve defined by the parametric equations x=t-cost,...

Use the curve defined by the parametric equations $x=t-\cos t, y=-1+\sin t$.

At which of the following values of $t$ is $d y / d x=0 ?$
(A) $t=\pi / 4$
(B) $t=\pi / 2$
(C) $t=3 \pi / 4$
(D) $t=\pi$
(E) $t=2 \pi$

Key Concepts

Trigonometric Equations

Trigonometric equations arise when the parametrization involves trigonometric functions. Solving these equations is essential for finding critical points such as where the derivative equals zero. This involves understanding the properties and solutions of sine, cosine, and other trigonometric functions over specific intervals.

Horizontal Tangency in Parametric Curves

A horizontal tangent corresponds to a point on the curve where the slope (dy/dx) is zero. In the context of parametric equations, this occurs when the derivative of the y-component with respect to the parameter is zero, provided that the derivative of the x-component is not zero at that point. Identifying horizontal tangents is important for characterizing the geometry of the curve.

Parametric Derivatives

The derivative dy/dx of a curve defined parametrically is obtained by taking the derivative of y with respect to the parameter and dividing it by the derivative of x with respect to the same parameter. This method uses the chain rule and allows us to find the slope of the curve at any point, which is crucial for understanding the behavior of the curve.

Parametric Equations

Parametric equations define curves where each coordinate is expressed as a function of an independent parameter. This concept allows the description of more complex curves that may not be easily represented by a single function, and is widely used in calculus for analyzing motion and geometric properties of curves.

Transcript

00:01 In this question, which is about the chain rule, we want to know at which value of t this parametric curve has a derivative of 0.

00:18 That is, the derivative of y with respect to x.

00:24 To find this out, we can simply find the derivative of y with respect to x and set that equal to zero.

00:33 Or substitute in values of t and see which one gives us 0.

00:38 0.

00:39 That's even easier because we don't have to solve the equation.

00:45 Now how do we find a dy -d -x? well chain rule tells us that dy -d -x is the ratio of the derivative of y with respect to t to the derivative of x with respect to t from this fact that d x d d x d t times d y d x d t is d y d t.

01:16 Okay so if we differentiate y with respect to t, this sine function, we get coast -t.

01:26 The derivative of the minus 1 is 0.

01:28 That's why i didn't mention it.

01:32 And then the derivative of x is 1 minus the derivative of cost -t, which is negative sine of t.

01:47 Okay.

01:48 Now that we have our derivative, we can see which value of t or values give us that this expression is equal to 0.

02:05 Okay, first of all, pi over 4.

02:13 Cosine of pi or 4, we know, is going to be the adjacent over the hypotenuse for this pi over 4 angle.

02:24 And we know that is not zero, obviously.

02:33 So this fraction is a non -zero number divided by some other number.

02:39 And now, there's no way that is going to be equal to zero, because dividing by zero is not even defined...

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