Derivatives and the Tangent Line

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Presentation transcript:

Derivatives and the Tangent Line

Blast from the past…

The New vs. The Old Pre-Calculus concept: Slope of a line. This is a static or constant concept. Uses a formula. Calculus concept: Slope of a curve at a given point. This is a variable concept. Uses formulas.

The Process…In Theory Theoretically, we are “drawing” a secant line through that point and a nearby point (Δy/Δx). We then "slide" the second point along the curve toward the first (remember…math in motion!); The secant line gets closer and closer to being a tangent line.

The Process…In Theory f(x+x) f(x) x x+x

The Process…In Theory x In the limit, as the distance between the two points goes to 0, the secant line becomes the tangent line. x

Side note: Does the formula for this limit look vaguely familiar? What if the x were changed to an h? Difference Quotient (version 2.0):

The Process…In Theory Its slope becomes the slope of the curve at that point, called the derivative. Expressed: x

Symbols for Derivative

Sample Problem: Find the derivative of f(x)=x2+4x+4 Picture from http://tutorial.math.lamar.edu/Classes/Alg/Parabolas.aspx

The Process

The Process

Try This One On Your Own Calculate the derivative of

Sample Problem Find the slope of the tangent to the curve at the point (6,4).

The Process

The Process