1. The Derivative and the Tangent Line Problems
Section 2.1

2. 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

3. Notation
𝑓 𝑥 = 𝑥 2
𝑓 𝑡 = 𝑡 2

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

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

6. 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)

7. 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)

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

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

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

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

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