Use a graphing utility to plot the curve and the tangent line....

Question

Use a graphing utility to plot the curve and the tangent line.

y = cos x / 1−cos x; x = π/3

Accepted Answer

Draw the graph of the following function on 0 to 2 pi and the tangent to the function at the given points. Y equals cosine x divided by 1 minus cosine X, where X equals pi divided by 2. I've also provided our graph here on the right. Now, to solve this graph, We can first plug in our Y equals into a graphing calculator. The black line For the black curve, I should say here is our function Y equals cosine x divided by 1 minus cosine X. So now, let's find this red tangent line. To solve this, we will find the derivative of this function first. This will help us find the slope of the tangent line. So Our derivative We will use a quotient rule to find. The quotient rule tells us if we have a function you divided by V. Its derivative is given by U multiplied by V. Minus CU multiplied by V. Divided by B squared. And in our case, we have U equals. Cosine X and V equals 1 minus cosine of X. We can first find U. Which is a derivative of cosine X. That is Negative sign of X. For V, the derivative of 1 minus cosine X. Is positive sign Xs. Let's go ahead and plug all of these. All these expressions into our function. We have negative sin of x. Multiplied by V, which is 1 minus cosine X. Minus cosine x multiplied. By sinic. All divided By 1 minus cosine of X2. We now get Negative sign of X. Plus, Sinex cosine X when distributing our first product. Minus sine X cosine x. Divided by 1 minus cosine X all squared. We know our sign cosine products cancel. This leaves us With negative sin of x divided by 1 minus cosine x 2. Now, we have X equals 5 divided by 2, so we can plug in this value. We get negative sign of pi divided by 2, all divided by 1 minus. Cosine 5 divided by 2. Squared Now, based on the unit circle. Sign of pi divided by 2 is 1. So we have -1 on the top. On the bottom 1 minus cosine pi divided by 2 is 1 minus 0 squared. We have 1 divided by 1, which is just -1. That means the slope of our tangent line is -1. To find the equation of the tangent line, we will make use of the point slope form equation. Y minus Y1 equals M multiplied by X minus X1. We have M, and we have X1, we just need to find Y1. Why one? Will be cosine, I divided by 2, divided by 1 minus cosine of pi divided by 2. Which is 0 divided by 1 minus 0, or just 0. We can finally plug this into our function. Y minus 0. Equals -1 multiplied by X minus. Pie divided by 2. We end up getting Y equals. negative X plus. Pie divided by 2. This is our equation. Of the tangent line And if you look back at our graph. This is given by the red line. OK, I hope that help you solve the problem. Thank you for watching. Goodbye.

Use a graphing utility to plot the curve and the tangent line.
y = cos x / 1−cos x; x = π/3

Key Concepts

Tangent Line

Derivative

Graphing Utility

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