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AP Calculus Chapter 2, Section 1

The Derivative and the Tangent Line Problem

2013 - 2014

The Tangent Line Problem • Calculus grew out of four major problems that

European mathematicians were working on during the seventeenth century. 1. The tangent line problem 2. The velocity and acceleration problem 3. The minimum and maximum problem 4. The area problem

• Isaac Newton (1642 – 1727) is the first to get credit for giving the first general solution to the tangent line problem.

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

Slope of the tangent line

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

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

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.

What do you know about vertical lines?

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.

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

Derivative Notation

• 𝑓′ 𝑥 , 𝑑𝑦

𝑑𝑥, 𝑦′,

𝑑

𝑑𝑥𝑓 𝑥 , 𝐷𝑥[𝑦]

Find the derivative of 𝑓 𝑥 = 𝑥2 + 2𝑥

Find the derivative of 𝑓 𝑥 = 𝑥3 + 𝑥2

Find the slopes of the tangent lines to the graph of 𝑓 𝑥 = 𝑥2 + 1 at the points (0, 1) and (-1, 2).

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

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

Find the derivative with respect to t for

the function 𝑦 =2

𝑡

Alternative Limit form of Derivatives

𝑓′ 𝑐 = lim𝑥→𝑐

𝑓 𝑥 − 𝑓(𝑐)

𝑥 − 𝑐

• This alternative form requires that the one-sided limits exist and are equal.

• Useful in investigating the relationship between differentiability and continuity.

Use the alternative form of a derivative to find the derivative at the given value of c.

𝑓 𝑥 = 𝑥3 + 2𝑥2 + 1, 𝑐 = −2

If a function is not continuous at 𝑥 = 𝑐, then it is not differentiable

at 𝑥 = 𝑐

Example: 𝑓 𝑥 = 𝑥

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

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.

Ch. 2.1 Homework

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