Download 2.1: The Derivative and Tangent Line Problem

AP CALCULUS
Section Number:
LECTURE NOTES
Topics: Definition of Derivative
MR. RECORD
Day: 10
2.1 A
1. What is a tangent line?
Examples of Tangent Lines drawn to a curve f(x) at a point P.
Note: Although many times, we might say that a tangent line drawn to a curve may only “touch” the curve
one time, that is not entirely true.
A secant line, by definition is a line that can intersect a curve at least twice.
2. The Infamous Tangent Line Problem
Begin with a graph of y=f(x)
y
Step 1: Let’s begin with an arbitrary point and call it  c, f (c)  .
Then, let’s choose another point on the curve that has a horizontal distance of
x away from our initial x value, c.
We would call that new point  c  x, f (c  x)  .
What is the slope of that secant line joining those two points?
x
=======================================================
Step 2: Now, just for kicks, let’s let our second x-value from above move
closer to our original x-value of c.
Do we still see a secant line?
Is there any difference in the way we calculate its slope?
y
x
=======================================================
Step 3: Since we are having such a blast moving our second x-value closer
to c, let’s move it so that it almost touches c.
y
Do we still see a secant line?
x
How could we perceive the slope now using a very important concept from
Chapter 1 ?
Definition of a Tangent Line with Slope m
If f is defined on an open interval containing c, and if the limit
y
f (c  x)  f (c)
lim
 lim
m
x  0 x
x  0
x
Exists, then the line passing through the point  c, f (c)  with slope m is the tangent line to the graph of f
at the point  c, f (c)  .
Example 1: The Slope of the Graph of a Linear Function.
Find the slope of the graph of f ( x)  2 x  3 .
Example 2: The Slope of the Graph of a Nonlinear Function.
Find the slope of the tangent lines to the graph of f ( x)  x 2  1 .
3. Vertical Tangent Lines
The definition of a tangent line to a curve does not cover the possibility of a vertical tangent line.
For vertical tangent lines, it’s possible to use this definition:
Definition of a Vertical Tangent Line
If f is continuous at c, and
f (c  x)  f (c)
f (c  x)  f (c)
lim
  or lim
,
x  0
x  0
x
x
The vertical line x = c passing through  c, f (c)  is a
vertical tangent line to the graph of f.
Definition of the Derivative of a Function
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 the limit exists, f  is a function of x.
The process of finding a derivative is called differentiation. Written as an imperative verb, it would be
differentiate.
A Side Note on “Notation”
In addition to f ( x) , which is read “f prime of x,” other notations are used to denote the derivative of
y  f ( x) . The most common are:
Whenever we read the second one above, we say “the derivative of y with respect to x.”
Example 3: Finding Derivatives by the Limit Process.
Find the derivative of f ( x)  x3  2 x .
Example 4: Finding the Derivative of a Function.
Find the derivative with respect to t for the function y 
2
.
t
4. Alternate Form of Derivative
The following alternative limit form of the derivative is useful in investigating the relationship between
differentiability and continuity. The derivative of f at c is
f (c)  lim
xc
f ( x )  f (c )
xc
Note that in order for the limit above to exist,
f ( x )  f (c )
f ( x )  f (c )
lim
and lim
must exist and be equal.
xc
xc
xc
xc
We call these one-sided limits derivatives from the left and
from the right.
Example 5:
Using the Alternate Form of the Derivative to Find the Slope at a Point.
Find f ( x) for f ( x)  x . Then find the slope of the graph of f at the points (1, 1) and (4, 2)
Is it possible to find the slope at (0, 0)?
AP CALCULUS
Section Number:
LECTURE NOTES
Topics: Definition of Derivative
MR. RECORD
Day: 11
2.1 B
Example 6: Graphs With Sharp Turns.
Sketch the graph and find the derivative of f ( x)  x  2
when x  2
y
3
2
1
x
-1
1
2
3
4
-1
Example 7: Graphs With Vertical Tangent Lines
Sketch the graph and find the derivative of f ( x)  x
1
3
when x  0 .
4
y
3
2
1
x
-3
-2
-1
1
-1
-2
-3
-4
2
3
4