Equations of tangent lines by definition ( 2 ) a. Use...

Equations of tangent lines by definition ( 2 )
a. Use definition (2) (p. 129 ) to find the slope of the line tangent to the graph of $f$ at $P$.
b. Determine an equation of the tangent line at $P$.
$$f(x)=\sqrt{x-1} ; P(2,1)$$

Key Concepts

Point-Slope Form

The point-slope form is a method of writing the equation of a line when a point on the line and the slope are known. It is especially useful in deriving the equation of the tangent line once the slope from the derivative is calculated.

Derivative as the Limit of the Difference Quotient

The derivative of a function at a point is defined as the limit of the difference quotient as the interval approaches zero. This fundamental concept in calculus quantifies the instantaneous rate of change of a function and provides the slope of the tangent line at that point.

Tangent Line

The tangent line to a curve at a particular point is the straight line that touches the curve only at that point and has the same slope as the curve does there. This concept is essential for understanding local linear approximations and the behavior of functions at specific points.

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