Problem 2E

How do you find the slope of the line tangent to the polar graph of r = f(θ) at a point?

Solution 2E

Step 1:

Consider the following polar equation ;

r = f(

The objective is to find the slope of the line tangent to the given polar graph.

We know that the polar point in cartesian coordinates is x = rcos( , y = r sin(

The given polar equation is r = f(

Therefore , the cartesian coordinates are given as ;

x= f( , y = f(sin(.

Hence , the parametric form in cartesian coordinates are ; x= f( , y = f(sin(.

We , know that the slope of the line tangent is ;

So, describing x and y in terms of ‘ alone.So , using the chain rule we compute

= .

Since , = .

Now , we will need the following derivatives.

Consider , x= f( , then derivative both sides with respect to ‘we get ,

= , since (uv) = , and cos(x) = -sin(x).

Consider , y = f( , then derivative both sides with respect to ‘we get ,

= , since (uv) = , and sin(x) = cos(x).

Therefore , the slope of the tangent line is;

= = .