Find the equation of the tangent line at the given point on each curve. x^2+y^2=100 ; (8,-6)

Name: Problem 18, Chapter 14: Finite Mathematics and Calculus with Applications
Uploaded: 2020-05-21T20:34:50-08:00
Duration: 2 min 3 s
Channel: Amy Jiang
Description: Find the equation of the tangent line at the given point on each curve. x^2+y^2=100 ; (8,-6)

step 1 Okay, so we're given our following equation and we're asked to find the equation of our tangent

Find the equation of the tangent line at the given point on...

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function defined by an equation where the dependent variable is not isolated. By differentiating each term with respect to the independent variable and applying the chain rule, one can solve for the derivative of the dependent variable, which is essential when dealing with curves defined implicitly.

Tangent Line

The tangent line to a curve at a given point is the straight line that just touches the curve at that point, with a slope equal to the derivative of the curve at that point. It serves as the best linear approximation of the curve near the point of tangency and is valuable in both analysis and application.

Point-Slope Form

The point-slope form, given by the equation y - y1 = m(x - x1), is a method for writing the equation of a line when a specific point (x1, y1) on the line and the slope (m) are known. It is frequently used in calculus to express the equation of a tangent line after finding the derivative at a given point.

Circle Equation

A circle equation, such as x^2 + y^2 = r^2, represents a set of points that are all at a fixed distance (the radius, r) from a central point (here, the origin). Understanding the properties of a circle is important when determining tangent lines, since the geometric behavior of circles influences the slope and position of tangents.

Transcript

Please give Ace some feedback