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Chapter 11: Problem 2

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

Short Answer

Expert verified
Answer: The expression for the slope of the tangent line to the polar graph of a function \(r = f(\theta)\) is given by: \(m_\text{tangent} = \frac{f'(\theta)\sin(\theta) + f(\theta)\cos(\theta)}{f'(\theta)\cos(\theta) - f(\theta)\sin(\theta)}\).

Step by step solution

01

Convert polar equation to parametric form

To find the Cartesian coordinates of the polar graph, we use the following equations: \(x(\theta) = r(\theta)\cos(\theta)\) and \(y(\theta) = r(\theta)\sin(\theta)\).
02

Find the derivatives of the parametric equations

To find the slope of the tangent line, we need to find the derivatives of the parametric equations with respect to \(\theta\). We have: \(\frac{dx}{d\theta} = \frac{d(r\cos(\theta))}{d\theta}\) and \(\frac{dy}{d\theta} = \frac{d(r\sin(\theta))}{d\theta}\). Using the product rule, we get: \(\frac{dx}{d\theta} = f'(\theta)\cos(\theta) - f(\theta)\sin(\theta)\) and, \(\frac{dy}{d\theta} = f'(\theta)\sin(\theta) + f(\theta)\cos(\theta)\).
03

Find the slope of the tangent line

Now, we need to find the slope of the tangent line in the Cartesian coordinate system. The slope is the ratio of the rate of change of \(y\) to the rate of change of \(x\) with respect to \(\theta\). So, we have: \(m_\text{tangent} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}\) Substituting the derivatives of the parametric equations, we have: \(m_\text{tangent} = \frac{f'(\theta)\sin(\theta) + f(\theta)\cos(\theta)}{f'(\theta)\cos(\theta) - f(\theta)\sin(\theta)}\) This expression gives us the slope of the tangent line to the polar graph of \(r = f(\theta)\) at any point.

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