Find the slope of the tangent line to the given polar curve at the point specified by the value

Find the slope of the tangent line to the given polar curve...

Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle rather than the standard Cartesian coordinates. This system is particularly useful when dealing with curves and shapes that have symmetry around a point, and it simplifies the representation and analysis of many curves.

Conversion from Polar to Cartesian Coordinates

Converting from polar to Cartesian coordinates involves using the relationships x = r*cos(?) and y = r*sin(?). These formulas are fundamental for analyzing polar curves in a Cartesian framework, especially when calculating slopes or other derivatives related to tangent lines.

Differentiation of Polar Functions

Differentiation of polar functions involves finding the derivatives of the coordinate functions with respect to the parameter ?. This is critical in determining the slope of the tangent line by computing the derivatives of x and y with respect to ? and then applying these derivatives to find dy/dx.

Slope of the Tangent Line in Polar Coordinates

The slope of the tangent line to a polar curve can be found by taking the derivative dy/dx, which is computed as (dy/d?) divided by (dx/d?). This approach utilizes the derivatives of the polar coordinate conversion formulas and is essential for finding the instantaneous rate of change (slope) at a given point on the curve.

Transcript

00:03 So to find the slope of the tangent line at theta equals pi or 3 of the polar equation, r equals 2 minus sine theta, we will need to find the derivative of a y with respect to x at theta equals pyrr 3, which means considering parametric forms of x and y.

00:28 So let's start, because we're going to do this one in compartments, because otherwise the problem gets too large.

00:34 So let's start by stating that x equals r times cosine of theta and y equals r times sine of theta.

00:54 And in the end, we're going to want the derivative of a y with respect to x, which would be computed by finding the derivative of y with respect to theta, divided by the derivative of x with respect to theta, evaluated at pi or three in the end.

01:12 So we're going to do the numerator and the denominator separately by recognizing that x is r times cosine theta.

01:25 So in this case, x is 2 minus sine of theta times cosine of theta, which we could leave in this form or can distribute the cosine theta through, which in this case i will.

01:44 So we have x equals 2 times cosine of theta minus sine theta times.

01:52 Cosine of theta.

01:55 So that's our x, and we'll need to find the derivative of x with respect to theta to find the denominator of our d -y -d -x.

02:07 Next, let's find the y.

02:09 Y by definition, is r times sine of theta.

02:12 Our r is 2 minus sine of theta.

02:17 Again, times sine theta, or in other words, distributing, we obtain that y equals 2 times sine of theta minus sine square.

02:29 So now that we have x and y in terms of theta, we can now find the derivative of x with respect to theta and the derivative of y with respect to theta and evaluate them at pi over 3.

02:46 So next let's find the derivative of x with respect to theta.

02:53 The derivative of 2 times cosine of theta using the constant times of function rule.

02:59 We obtain negative 2 sine of theta.

03:04 Then we have minus sine theta times cosine theta, which will be minus what we obtain when we use the product rule.

03:13 So using the product rule on sine theta, cosine, theta, we get the following.

03:19 The derivative of sine theta is cosine of theta times the second function, which is cosine theta, plus the derivative of the second function, the derivative of cosine theta is negative sine theta, times the derivative of the first function, which is sine of theta.

03:43 Simplifying this just a bit, we get negative 2 times sine of theta.

03:48 Next we're going to distribute the negative sign to each term to obtain minus cosine squared theta plus sine squared theta, and that's for d x over d theta.

04:09 While we're in the derivative mode, let's next find the derivative of y with respect to theta, using a constant times of function rule.

04:19 The derivative of a 2 sine theta is 2 times cosine of theta.

04:23 And then we're going to use a chain rule, include on a power rule and a trache form on sine squared theta.

04:32 So the power comes out in front, rewrite the base, subtract one from the power times the derivative of sine theta, which is cosine of theta.

04:41 Or in other words, the derivative of a y with respect to is 2 times cosine of theta minus 2 times sine theta times cosine of theta.

04:55 Next, let's evaluate each of these derivatives at pi or 3, since we made the slope of the tangent line at pi or 3.

05:10 So next we have the derivative of x with respect to theta evaluate it at theta equals pi over 3.

05:22 We'll have negative 2 times.

05:27 The sine of pi over 3 minus the cosine squared of pi over 3 plus sine squared of pi over 3...

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