The Derivative and the Tangent Line Problems

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Presentation transcript:

The Derivative and the Tangent Line Problems Section 2.1

Definition of Derivative lim ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥 lim ℎ→0 𝑓 𝑥+ℎ −𝑓(𝑥) ℎ lim 𝑥→𝑐 𝑓(𝑥)−𝑓(𝑐) 𝑥−𝑐

Notation 𝑓 𝑥 = 𝑥 2 𝑓 𝑡 = 𝑡 2

When is a function differentiable? lim ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥 Example: 𝑦= 𝑥 Example: 𝑦= 3 𝑥

Finding Derivatives #23 Find the derivative by the limit process. 𝑓 𝑥 = 𝑥+1

Finding Slopes of Tangent Lines #7 Find the slope of the tangent line to the graph of the function at the given point. 𝑔 𝑥 = 𝑥 2 −4, (1, −3)

Finding Equations of Tangent Lines #27 Find an equation of the tangent line to the graph of 𝑓 at the given point. Then use a graphing utility to graph the function and its tangent line at the point. Use the derivative feature of a graphing utility to confirm your result. 𝑓 𝑥 = 𝑥 3 , (2, 8)

Curve Sketching Given the graph of 𝑓, sketch the graph of 𝑓 ′ .

Curve Sketching Given the graph of 𝑓, sketch the graph of 𝑓 ′ .

Curve Sketching Given the graph of 𝑓, sketch the graph of 𝑓 ′ .

Curve Sketching Given the graph of 𝑓, sketch the graph of 𝑓 ′ .

Curve Sketching Given the graph of 𝑓, sketch the graph of 𝑓 ′ .