Tangent Line Equations, Slope, & Derivatives In Polar Form | Calculus 2 | Summary and Q&A

TL;DR

The video explains how to find the slope and equation of the tangent line for a given polar equation, using the point-slope formula.

Key Insights

• 🐻‍❄️ Polar equations can be represented using the formula r = a + b*cos(theta), where r depends on theta.
• ❣️ The x and y coordinates for a given point on a polar curve are determined by substituting the value of theta into the equations x = rcos(theta) and y = rsin(theta).
• 🫡 The derivative of the polar equation with respect to theta is calculated to find the slope of the tangent line.
• 😥 The point-slope formula is used to find the equation of the tangent line, using the slope and a point on the polar curve.

Transcript

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Questions & Answers

Q: How do you find the x and y coordinates for a given point on a polar curve?

To find the x and y coordinates, substitute the given value of theta into the equations x = rcos(theta) and y = rsin(theta), where r is the value of the polar equation at that theta.

Q: What is the process for finding the slope of the tangent line to a polar curve?

To find the slope, calculate the derivative of r with respect to theta and evaluate it at the given theta. Then use the equations for x and y to find dx/dtheta and dy/dtheta. Finally, use these values to calculate dy/dx.

Q: How is the equation of the tangent line derived using the point-slope formula?

The point-slope formula, y - y1 = m(x - x1), is used with the point (x1, y1) on the polar curve and the slope (m) of the tangent line. Substitute the values and simplify to obtain the equation of the tangent line.

Q: How is the polar curve and tangent line graphed?

The polar curve is graphed by substituting different values of theta into the equation and plotting the resulting (x, y) points. The tangent line is graphed using the equation obtained, with the slope and point plotted on the graph.

Summary & Key Takeaways

• The video demonstrates the process of finding the x and y coordinates for a given point on a polar curve, as well as the derivative of the polar curve with respect to theta.

• Using the derived values, the slope of the tangent line is calculated and used to determine the equation of the tangent line using the point-slope formula.

• The equation of the tangent line is then determined and graphed along with the polar curve.