Overview
This lecture covers solutions to advanced calculus problems from Exercise 11.1, focusing on normals and tangents to curves, parametric equations, and properties of conic sections. Step-by-step methods, formulas, and key proof techniques are highlighted for each type of question.
Finding Points of Normal Parallel to X-axis
- To find points where the normal to a curve is parallel to the x-axis, set the slope of the normal to zero.
- Substitute the obtained x-value into the original curve equation to find the corresponding y-value.
- Use the normal and tangent condition: for normal, the product of slopes is -1.
Curve, Normal, and Tangent Equations
- The general formula for a normal at point ((x_1, y_1)) is ((y - y_1) = -\frac{1}{m}(x - x_1)), where m is the tangent slope.
- Tangent slope at a point is found by differentiating the curve equation and evaluating at ((x_1, y_1)).
- For tangents parallel to the x-axis, set the derivative to zero and solve for the required points.
Parametric Equations and Their Application
- Parametric equations express (x) and (y) in terms of a parameter (e.g., (x = a \cos t + a t \sin t), (y = a \sin t - a t \cos t)).
- To find the equation of the normal or tangent in parametric form, use derivatives with respect to the parameter.
Normal Length and Perpendicular Distances
- The length of the normal at a curve point is given by (|y_1 - m x_1| \sqrt{1+m^2}), where m is the slope at the point.
- The perpendicular distance from the origin to the tangent at any point can be calculated using the line formula set to zero.
Properties and Ratios on Tangents and Normals
- For certain curves (ellipse, parabola), it can be shown that the length of the normal varies inversely with the perpendicular from the origin to the tangent.
- Prove constant ratios by expressing intercepted segments and ratios via coordinates and verifying constancy using algebraic manipulation.
Key Terms & Definitions
- Normal — A line perpendicular to the tangent at a given point on a curve.
- Tangent — A straight line that touches a curve at exactly one point without crossing it.
- Parametric Equation — Expresses coordinates of points on a curve as functions of a parameter.
- Slope (m) — The rate at which (y) changes with respect to (x), often found via differentiation.
- Intercept — The point where a curve or line crosses the x- or y-axis.
Action Items / Next Steps
- Complete homework up to question 11 in Exercise 11.1.
- Review and memorize formulas for normals, tangents, and parametric equations.
- Practice differentiating and applying these concepts to conic sections (ellipse, parabola, etc.).
- Prepare for the next class by ensuring all previous exercises are clear and solved.