2.1 Derivative:
Slope of a Tangent Line
Objective: You will be able to:
• find the slope of the tangent line to a curve
• use the limit definition to find the derivative
• sketch the graph of a derivative
Sep 20­1:25 PM
You will be able to be able to find the slope of a tangent line to a curve
Warm Up...
Algebra and Geometry Review:
(x2, y2)
Formula for slope of a line:
(x1, y1)
Define:
secant:
tangent:
Sep 20­1:27 PM
1
You will be able to be able to find the slope of a tangent line to a curve
Calculus: How do you find the slope of a graph that is not a straight line?
Calculus In Motion
Define Derivative & NDER.gsp
Sep 20­1:28 PM
You will be able to be able to find the slope of a tangent line to a curve
Sep 20­1:28 PM
2
You will be able to be able to find the slope of a tangent line to a curve
DERIVATIVE: Tangent Line with Slope m
f(x + Δx) ­ f(x)
Δx
The line passing through (x, f(x)) with slope m is the tangent line to the graph f at the the point (x, f(x))
equivalent form
Sep 20­1:28 PM
Stand and Deliver
Calculus the Musical: Differientabul
2.1
Limit Definition to Find a Derivative
Sep 2­7:57 AM
3
secant line
tangent line as h
Δx
0
0
Notation: (all mean the same operation)
[
[
f '(x), dy, y', d f(x)
dx
dx *Derivative = Slope*
Aug 31­5:07 PM
You will be able to be able to find the slope of a tangent line to a curve
Find the slope of the tangent line for f(x) = 3x + 1
Sep 20­1:29 PM
4
You will be able to be able to find the slope of a tangent line to a curve
a. Find the derivative of f(x) = x2 + 2
b. Use it to find the slope at (­1, 3) and (0, 2)
Sep 20­1:29 PM
You will be able to be able to find the slope of a tangent line to a curve
a. Write an equation of the tangent line to the graph at the point.
b. Graph the function and it's tangent line at the point.
c. Use the derivative feature of your calculator (I'll show you this later)
f(x) = x2 + 2x + 1, (­3, 4)
Sep 20­1:29 PM
5
Stand and Deliver
2.1
3 Scenarios when function is not differentiable
1. not continuous
2. corner or cusp
3. vertical tangent line
Sep 2­7:57 AM
1. not continuous
2. corner or cusp
scroll down
3. vertical tangent line
Sep 2­8:01 AM
6
Differentiability and Continuity
Differentiability Implies Continuity If f is differentiable, then it is continuous.
However, continuity does NOT imply differentiability!
corner, cusp and vertical tangent line are all continuous, but not differentiable.
Sep 20­1:29 PM
I've taught you enough to do through #27 on hw, and 81­90
Aug 31­8:33 AM
7
Find the derivative by the limit process.
f(x) = √x - 1
Sep 20­1:29 PM
Find an equation of the line that is tangent to the graph of f and parallel to the given line.
f(x) = x3 + 2
line: 3x ­y ­ 4=0
Aug 31­8:20 AM
8
The graph of f is given. Sketch the graph of f '.
Sep 7­8:34 AM
The graph of f is given. Sketch the graph of f '.
Sep 7­8:34 AM
9
The graph of f is given. Sketch the graph of f '.
Sep 7­8:34 AM
Let f be the function given by f(x) = |x|.
Which of the following statements about f are true?
I.
f is continuous at x = 0.
II. f is differentiable at x = 0.
III. f has an absolute minimum at x = 0.
(A) I only
(B) II only
(C) III only
(D) I and III only (E) II and III only
Jun 28­1:27 PM
10
AP Question
(A)
Aug 5­3:07 PM
AP Question
(B)
Aug 5­3:11 PM
11
AP Question
(A)
Aug 5­3:13 PM
AP Question
(E)
Aug 5­3:25 PM
12
Write an equation of a tangent line to the graph at the given point.
g(x) = 5 - x2, (2, 1)
Aug 31­8:34 AM
Aug 31­8:16 AM
13
Attachments
Define Derivative & NDER.gsp