Find the equation of the line tangent to y=x^2+3 x-5 at x=2....

Name: Problem 70, Chapter 3: Calculus Single Variable
Uploaded: 2021-04-16T16:47:12-08:00
Duration: 1 min 56 s
Channel: Gregory Higby
Description: Find the equation of the line tangent to y=x^2+3 x-5 at x=2.

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Key Concepts

Derivative

The derivative of a function provides the instantaneous rate of change at any point on the curve. In the context of finding a tangent line, the derivative is used to calculate the slope of the tangent at a specific value of x. For any differentiable function, computing its derivative and evaluating it at a given point yields the precise slope of the line that just touches the function's graph at that point.

Point of Tangency

The point of tangency is the specific point on the function where the tangent line touches the curve. To determine this point, you evaluate the function at the given x-value. This gives both the x-coordinate and the corresponding y-coordinate, which are essential for defining the exact location where the slope is obtained and where the tangent line meets the curve.

Point-Slope Form

The point-slope form is a way to write the equation of a line when you know its slope and a point it passes through. It is expressed as y - y? = m(x - x?), where m represents the slope and (x?, y?) is the point of tangency. This form is particularly useful when deriving the equation of a tangent line because it directly incorporates the calculated slope from the derivative and the coordinates of the point of tangency.

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