The Derivative and the Tangent Line Problem..
• Why did it arise?
• A secant line to a curve versus a tangent line to a
curve
• Definition of the Derivative of a function (What
differential calculus is about)
You’ve run across at least one tangent line
problem in Geometry.
P
The Tangent Line to a circle is perpendicular to the radius at
point P
Recall that a tangent line touches a curve at just one point.
Determining the Tangent Line
for a general curve became an
interesting esoteric problem for
mathematicians.
The problem had importance in
the 1600’s with the advent of
artillery and optical lenses…..
The tangent line of the curve equals the direction of the
curve …….so how a projectile strikes a target and where it
goes after it hits the target depends on the tangent line of
the projectile and the surface.
Direction
projectile goes
Target
Shot fired
Tangent line to
curve at point of
impact
The tangent line of the curve equals the direction of the
curve …….so how a projectile strikes a target and where it
goes after it hits the target depends on the tangent line of
the projectile and the surface.
Direct impact
Target
Shot fired
Tangent line to
curve at point of
impact.
Perpendicular.
Telescopes and microscopes
added a second major
application for the tangent line
problem.
To determine the course a light ray after it strikes the surface
of a lens, we must know the angle the light ray makes with the
lens.
If you know the tangent line created by the light ray striking
the lens, then you can discover that angle.
Finding the slope of line that touches a curve twice is a
SECANT LINE problem.
f(c)
c
This equation is called a DIFFERENCE QUOTIENT. It tells
you the AVERAGE RATE OF CHANGE between two points
on a curve
Let : f (x) = x 2
x = hours
f (x) = miles travelled
What is the average rate of
change (called speed when we
talk of distance and time)
from hour 3 to hour 7?
How can we use the slope of the secant to find
the slope of a tangent line at a point?
f(c)
c
BY LETTING Δx GET VERY,VERY,VERY SMALL!!
Demonstrate that the closer the two points are the
closer the secant’s slope is to the tangent line’s
slope
• 
How a secant line becomes a
Let the two points get
tangent
very close together, so
the difference between
the two points approaches
zero
•  The limit definition
of the derivative is below
Definition of a TANGENT LINE with slope m
If f is defined on an open interval containing c and
the limit….
Then, we call the line passing through (c,f(c)) with the slope
m the tangent line to the graph of f at the point (c,f(c)).
The slope of the tangent line is the slope of the curve at x = c.
How tangent lines to the curve look…..
Using the limit definition of the derivative to
find the slope at a specific x-value
Find the slope of the curve at x=2.
Almost
done.
Using the limit definition of the derivative to
find the slope at a specific x-value
The slope of the curve at x = 2 is 1.
Using the limit definition of the derivative to
find the slope at a general x-value
for f (x) = x 2 − 3x + 2
* Don’t replace x with a value, let x stand for any x. Now, plug this information into the quotient and take
the limit.
Using the limit definition of the derivative to
find the slope at a general x-value
f '(x) = 2x − 3
so,the slope at x = 2 f '(2) = 2(2) − 3 = 1
the slope at x = 5
f '(5) = 2(5) − 3 = 7
The general solution gives us a new function, the derived
function
Problems for you …..
From page 103
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