You are given the polar curve r=eθ.(a) List all of the points (r,θ) where the tangent line is horizontal. In entering your answer, list the points starting with the smallest value of r and limit yourself to 1≤r≤1000 ( note the restriction on r!) and 0≤θ<2π. If two or more points share the same value of r, list those starting with the smallest value of θ. If any blanks are unused, type an upper-case "N" in them.

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Step-by-step explanation:

REcall that given a parametric curve x(t),y(t), the tangent's line slope is given by

To find where the tangent line is horizontal, we must find where the slope is 0. That is, to find the values of t for which and at the same time.

Consider the polar curve . If we use the polar coordinates, we have that , which gives us a parametric curve with parameter . So, let us use the above to find the desired points.

We have that

Recall that the function is never 0, so, for us to have the derivative of y to be 0, we must have that . Note that if this happens, the derivative of x is different from 0. So, we must solve the following equation in the interval .

, which is equivalent to . Which gives us . This solution is out of our desired interval, then , using the fact that tangent is a periodic function with period pi, we can find the solutions in the desired interval by adding multiples of pi. Thus, the desired solutions are

Note that and ,so both solutions are inside the restriction.

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answer: d.) the total number of protons and neutrons is 15

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