AP Calculus BC Monday, 14 September 2015 OBJECTIVE TSW (1) define the slope of a curve at a point,...
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AP Calculus BC Monday, 14 September 2015 • OBJECTIVE TSW (1) define the slope of a curve at a point, and (2) define the derivative. • Tests are graded. • TODAY’S ASSIGNMENT – Sec. 3.1: pp. 144-145 (17-31 odd) • Due on Wednesday/Thursday, 16/17 September, 2015.
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Transcript of AP Calculus BC Monday, 14 September 2015 OBJECTIVE TSW (1) define the slope of a curve at a point,...
AP Calculus BCMonday, 14 September 2015
• OBJECTIVE TSW (1) define the slope of a curve at a point, and (2) define the derivative.
• Tests are graded.
• TODAY’S ASSIGNMENT– Sec. 3.1: pp. 144-145 (17-31 odd)
• Due on Wednesday/Thursday, 16/17 September, 2015.
Sec. 3.1: Introducing the Derivative
Sec. 3.1: Introducing the Derivative
The Tangent Line Problem
Sec. 3.1: Introducing the Derivative
The Tangent Line Problem
“What is a tangent?”
Sec. 3.1: Introducing the Derivative
The Tangent Line Problem
Sec. 3.1: Introducing the Derivative
The Tangent Line Problem
Sec. 3.1: Introducing the Derivative
The Tangent Line Problem
This ratio is also called the difference quotient.
AP Calculus BCTuesday, 15 September 2015
• OBJECTIVE TSW (1) define the slope of a curve at a point, and (2) define the derivative.
• YESTERDAY’S ASSIGNMENT– Sec. 3.1: pp. 144-145 (17-31 odd)
• Due Wednesday/Thursday, 16/17 September 2015.
• TODAY’S ASSIGNMENT– Sec. 3.2: pp. 150-153 (5-13 odd, 21-24 all,
29)• Due Friday, 18 September 2015.
Sec. 3.1: Introducing the Derivative
Ex: Find the slope of the graph of f (x) = 3x – 5 @ (4, 7).
0limx
f c x f cm
x
0
4 4limx
f x f
x
0
3 5 3 5l
4im
4x x
x
0
12 3 5 7limx
x
x
0
3limx
x
x
3
Sec. 3.1: Introducing the Derivative
Ex: f(x) = x2 + 3Find the slope of the tangent line at each of the following points: (1, 4), (2, 7), and (–1, 4).
Since there are three points at which to find the slope, let’s find the general case first. (Instead of c, we’ll use x.
Then, we’ll substitute each of the three x-values to find each slope.
Sec. 3.1: Introducing the Derivative
Ex: f(x) = x2 + 3Find the slope of the tangent line at each of the following points: (1, 4), (2, 7), and (–1, 4).
0
limx
f x x f xm
x
2 2
0
3 3limx
x
x
x x
22 2
0
2 3 3limx
x x x x x
x
2
0
2limx
x x x
x
0
2limx
x x x
x
0
lim 2x
x x
2x
Sec. 3.1: Introducing the Derivative
Ex: f(x) = x2 + 3Find the slope of the tangent line at each of the following points: (1, 4), (2, 7), and (–1, 4).
2m x At (1, 4), m = 2(1) = 2
At (2, 7), m = 2(2) = 4
At (–1, 4), m = 2(–1) = –2
Sec. 3.1: Introducing the Derivative
If f (x) is continuous and
0
lim ,x
f c x f c
x
then the vertical line x = cIs the vertical tangent lineto the graph of f.
Definition: Vertical Tangent
Sec. 3.1: Introducing the Derivative
The Derivative of a Function
Sec. 3.1: Introducing the Derivative
Sec. 3.1: Introducing the Derivative
The Derivative of a Function
"Derivative" means "slope."
Sec. 3.1: Introducing the Derivative
The Derivative of a Function
Sec. 3.1: Introducing the Derivative
Ex: Find the derivative off(x) = x3 + 2x.
0
limx
f x x f xf x
x
3 3
0
2 2limx
x x x x x x
x
2 33 2 3
0
3 3 2 2 2limx
x x x x x x x x x x
x
2 32
0
3 3 2limx
x x x x x x
x
22
0lim 3 3 2x
x x x x
23 2f x x
Sec. 3.1: Introducing the Derivative Ex: Find f'(x) for Then find
the slope of the graph of f at the points (1, 1) and (4, 2). Discuss the behavior of f at (0, 0).
.f x x
0
limx
f x x f xf x
x
0limx
x x x
x
x x x
x x x
0
limx
x x x
x x x x
0limx
x
x x x x
1
x x
1
2f x
x
AP Calculus BCThursday, 17 September 2015
• OBJECTIVE: TSW use basic differentiation rules to find derivatives, slopes of tangent lines, equations of tangent lines, and rates of change.
• ASSIGNMENT DUE BY END OF PERIOD– Sec. 3.1: pp. 144-145 (17-31 odd) wire basket (I will address some “issues” first)
• TUESDAY’S ASSIGNMENT– Sec. 3.2: pp. 150-153 (5-13 odd, 21-24 all, 29)
• Due tomorrow, Friday, 18 September 2015.
• TODAY’S ASSIGNMENT– REGISTER YOUR ACCOUNT!!!– Sec. 3.3: pp. 160-161 (7-17 odd, 19-24 all omit 21, 35, 36)
• Due Monday, 21 September 2015.
• QUIZ: Sec. 3.1 – 3.3 is tomorrow, Friday, 18 September 2015.– No calculator!
Sec. 3.1: Introducing the Derivative Ex: Find f'(x) for Then find
the slope of the graph of f at the points (1, 1) and (4, 2). Discuss the behavior of f at (0, 0).
.f x x
1
2f x
x
11
2 1f 1
2
14
2 4f 1
4
At (0, 0), f '(0) = ∞, f has a vertical tangent at x = 0.
Sec. 3.1: Introducing the Derivative
Ex: Find the derivative with respect to t for the function y = 2/t.
0limt
f t t f ty
t
0
2 2
limt
t t tyt
0
2 2
limt
t t ttt tt
t tt
0
2 2 2
limt
t t tt t t
t
0
2 1limt
t
t t t t
0
2limt t t t
2
2
t
