1. AP Calculus Chapter 2, Section 1
The Derivative and the Tangent Line Problem

2. The Tangent Line Problem
Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century.
The tangent line problem
The velocity and acceleration problem
The minimum and maximum problem
The area problem
Isaac Newton (1642 – 1727) is the first to get credit for giving the first general solution to the tangent line problem.

3. What does it mean to say a line is tangent to a curve at a point?

4. Slope of the tangent line

5. Definition of Tangent Line with Slope m
If f is defined on an open interval containing c, and if the limit
lim ∆𝑥→0 Δ𝑦 Δ𝑥 = lim ∆𝑥→0 𝑓 𝑐+∆𝑥 −𝑓(𝑐) ∆𝑥 =𝑚
Exists, then the line passing through (𝑐, 𝑓 𝑐 ) with slope m is the tangent line to the graph of f at the point (𝑐, 𝑓 𝑐 ).
*Instead of using c, you can say x

6. Find the slope of the graph of 𝑓 𝑥 =2𝑥−3 at the point (2, 1)

7. Vertical Tangent Lines
The definition of a tangent line does not cover the possibility of a vertical tangent line. For vertical tangent lines, you can use the following definition: if f is continuous at x and
lim ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥 =∞ or lim ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥 =−∞
The vertical line 𝑥=𝑐 passing through (c, f(c)) is a vertical tangent line to the graph f.

9. The Derivative of a Function
The limit used to define the slope of a tangent line is also used to define the derivative of the function.
Finding the derivative refers to the slope of the tangent line
The process used to find the derivative is called differentiation.

10. Definition of a Derivative
The derivative of f at x is given by
𝑓 ′ 𝑥 = lim ∆𝑥→0 𝑓 𝑥+∆𝑥 −𝑓(𝑥) ∆𝑥
Provided the limit exists. For all x for which this limit exists, f’ is a function of x

11. Derivative Notation
𝑓 ′ 𝑥 , 𝑑𝑦 𝑑𝑥 , 𝑦 ′ , 𝑑 𝑑𝑥 𝑓 𝑥 , 𝐷 𝑥 [𝑦]

12. Find the derivative of 𝑓 𝑥 = 𝑥 2 +2𝑥

13. Find the derivative of 𝑓 𝑥 = 𝑥 3 + 𝑥 2

14. Find the slopes of the tangent lines to the graph of 𝑓 𝑥 = 𝑥 2 +1 at the points (0, 1) and (-1, 2).

15. Find 𝑓′(𝑥) for 𝑓 𝑥 = 𝑥 . Then find the slope of the graph of f at the points (1, 1) and (4, 2).

16. Using the previous derivative, discuss the behavior of f at (0, 0)

17. Find the derivative with respect to t for the function 𝑦= 2 𝑡

18. If a function is not continuous at 𝑥=𝑐, then it is not differentiable at 𝑥=𝑐
Example: 𝑓 𝑥 = 𝑥

19. A graph with a sharp turn
Graph the function 𝑓 𝑥 = 𝑥−2
Discuss the continuity of the function and its differentiation at x = 2

20. Differentiability and Continuity
If a function is differentiable at x = c, then it is continuous at x = c. So differentiability implies continuity.
It is possible for a function to be continuous at x = c and not be differentiable at x = c. So, continuity does not imply differentiability.

21. Ch. 2.1 Homework
Pg. 104 – 106: #’s 7, 11, 17, 23, 27, 33, 35, 57, 63, 81