AP CalculusChapter 2, Section 1THE DERIVATIVE AND THE TANGENT LINE PROBLEM

2013 – 2014

UPDATED 2015 - 2016

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 problem2. The velocity and acceleration problem3. The minimum and maximum problem4. 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

Exists, then the line passing through with slope m is the tangent line to the graph of f at the point .

*Instead of using c, you can say x

Translation: The function must be continuous on the interval.

The derivative can be found by this limit:

The derivative of a function is the slope of the function.◦ Will be an equation or a value at a specific point

Derivative Notation

𝑓 ′ (𝑥 ) , 𝑑𝑦𝑑𝑥, 𝑦 ′ ,

𝑑𝑑𝑥

[ 𝑓 (𝑥 ) ] ,𝐷𝑥 [𝑦 ]

Find of the

Find the slope of the function at the point (2, 1)

Find the slope of the tangent line to the graph of the function at the point (3, -4)

Find the derivative of using the limit definition of a derivative

Find the slopes of the tangent lines to the graph of at the points (0, 1) and (-1, 2).

Find for . 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

If a function is not continuous at , then it is not differentiable at

Example:

A graph with a sharp turn

Graph the function

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.

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

or

The vertical line passing through (c, f(c)) is a vertical tangent line to the graph f.

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