Section 2.1: The Derivative and the Tangent Line Problem ...

AP Calculus

Section 2.1: The Derivative and the Tangent Line
Problem

Goals for this Section:

Find the slope of the tangent line to a curve at a point.

Day 1

Use the limit definition to find the derivative of a
function.

Day 1

Understand the relationship between differentiability
and continuity.

Day 2

Chapter 2 1

AP Calculus

Section 2.1: The Derivative and the Tangent Line Problem

Remember from Chapter 1...

Tangent Line Problem

In the tangent line problem, you are given a
function fand a point P on its graph and are
asked to find an equation of the tangent line
to the graph at point P (figure 1.1 - page 45).
The problem of finding the tangent line at a
point P is equivalent to finding the slope of
the tangent line at P.

You can approximate this slope by using a line
through the point of tangency and a second
point on the curve. This line is called a secant
line.

As the slope of the secant line approaches the
slope of the tangent line, a "limiting position"
exists.

Therefore, the slope of the tangent line is said
to be the limit of the slope of the secant line.

Chapter 2 2

AP Calculus

Section 2.1: The Derivative and the Tangent Line Problem
Definition of Tangent Line with Slope m

If f is defined on an open interval containing c and if the
limit

lim ∆y = lim f (c + ∆x) - f (c) = m

∆x ➝ 0 ∆x ∆x ➝ 0 ∆x

exists, then the line passing through (c , f (c)) with slope m is
the tangent line to the graph of f (c) at the point (c , f (c))

Example: Find the slope of the graph of
f(x) = 2x - 3 at the point (2 , 1).

Chapter 2 3

AP Calculus

Section 2.1: The Derivative and the Tangent Line Problem
Definition of the Derivative of a Function

The derivative of f at x is given by

f '(x) = lim f (x + ∆x) - f (x)

∆x ➝ 0 ∆x

provided the limit exists. For all x for which this limit exists f '
is a function of x.

According to Wikipedia...

In calculus, a branch of mathematics, the derivative is a measurement of how a function changes when the values of its
inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
For example, the derivative of the position of a car at some point in time is the velocity, or speed, at which that car is
traveling (conversely the integral of the velocity is the car's position or distance traveled).

Differentiation is a method to compute the rate at which a quantity, y, changes with respect to the change in another
quantity, x, upon which it is dependent. This rate of change is called the derivative of y with respect to x.

Other notations used to denote the derivative of y = f (x):

f '(x) ; y' ; d [ f (x)] ; Dx[y] ; dy
dx dx

dy = lim ∆y = lim f (x + ∆x) - f (x) = f '(x)

dx ∆x ➝ 0 ∆x ∆x ➝ 0 ∆x

Chapter 2 4

AP Calculus

Section 2.1: The Derivative and the Tangent Line Problem

Example: Find the derivative of f(x) = x3 + 2x

Chapter 2 5

AP Calculus

Chapter 2 6

AP Calculus

Section 2.1: The Derivative and the Tangent Line
Problem

Goals for this Section:

Find the slope of the tangent line to a curve at a point.

Day 1

Use the limit definition to find the derivative of a
function.

Day 1

Understand the relationship between differentiability
and continuity.

Day 2

Chapter 2 7

AP Calculus

Section 2.1: The Derivative and the Tangent Line Problem
Alternative Form of a Derivative:

f '(c) = lim f (x) - f (c)
x-c
x ➝c

**Use this form when you are trying to find the derivative at a point...

Example: Find the slope of the tangent line
graph of f(x) = x2 - 3 at the point (2 , 1).

Chapter 2 8

AP Calculus

Section 2.1: The Derivative and the Tangent Line Problem
One-Sided Derivatives:

A function y = f (x) is differentiable on a closed interval

[a , b] if it has a derivative at every interior point of the
interval, and if the limits

lim f (x) - f (c) and lim f (x) - f (c)
x-c x ➝c+
x ➝c- x-c

exist and are equal.

Chapter 2 9

AP Calculus

Section 2.1: The Derivative and the Tangent Line Problem

Example: If f (x) = | x2 - 9 | find f '(3)

Chapter 2 10

AP Calculus

Section 2.1: The Derivative and the Tangent Line Problem
Theorem 2.1: Differentiability Implies Continuity

If f is differentiable at x = c, then f is continuous at x = c.

differentiable at point c ➔ continuous at c
continuous at c ➔ differentiable at c

Example: If f (x) = (2x)/(x-1) find f '(1)

Chapter 2 11

AP Calculus

USING YOUR GRAPHING CALC...

If you were to graph the f(x) in your calculator...
2nd TRACE
6: dy/dx
type in the "x" you are looking for

From the home screen...
MATH
8:nDerv
screen should appear as follows
nDerv( type the equation,type the variable,type the "x")
for example: nDerv(x2,x,4) = 8

Chapter 2 12