Find an equation of the tangent line to the curve at the given point. y=(2 x+1)/(x+2),(1,1)

Name: Problem 8, Chapter 2: Calculus
Uploaded: 2021-03-10T02:22:33-08:00
Duration: 2 min 42 s
Channel: Heather Eichman
Description: Find an equation of the tangent line to the curve at the given point. y=(2 x+1)/(x+2),(1,1)

step 1 Today we are going to be finding the equation of a line that is tangent to the curve 2x plus 1 o

Find an equation of the tangent line to the curve at the...

Key Concepts

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it at that vicinity and has the same slope as the curve at the point of contact. In calculus, finding the tangent line involves computing the derivative of the function at the given point to determine this slope, which is then incorporated into the line's equation using an appropriate point-slope formulation.

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its variable. It is fundamentally defined as the limit of the difference quotient as the increment approaches zero. This concept is central in calculus as it allows us to determine slopes of tangent lines to curves, optimize functions, and solve problems involving motion and growth.

Quotient Rule

The quotient rule is a method used in calculus for differentiating functions that are expressed as a ratio of two differentiable functions. It states that the derivative of a quotient is found by taking the derivative of the numerator multiplied by the denominator, subtracting the numerator multiplied by the derivative of the denominator, and then dividing the result by the square of the denominator. This rule is particularly useful when dealing with rational functions.

Point-Slope Form

The point-slope form is a way of expressing the equation of a line when a point on the line and its slope are known. It is written as y - y? = m(x - x?), where (x?, y?) is a point on the line and m is the slope. This form is especially important in calculus when forming the equation of the tangent line to a curve at a specific point.

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