CALC 1.1.notebook
September 01, 2011
P 1
CALCULUS CHAPTER 1
We will be viewing Calculus 1) Numerically 3) Graphically
from multiple perspectives: 2) Algebraically 4) Verbally
.
.
.
** See pages 43 and 44
1.1
Secant Secant line touches curve Tangent line touches curve Tangent
CLASSIC CALCULUS PROBLEM #1
y=f(x)
Tangent
Line
Except for cases involving a vertical tangent line, the problem of finding the TANGENT LINE at a point P is equivalent to You can approximate this slope by using a line through the point of tangency and a second point on the curve. Such a line is called a secant line. If P(a,f(a)) is the point of tangency and Q(a+Δx, f(a+Δx)) is a second point on the graph of f, the slope of the secant line through these 2 points is given by
m= y2­y1
x2­x1
Q( a+Δx,f(a+Δx))
f(a+Δx)
f(a)
P(a,f(a))
a
a+Δx
MOST OFTEN THOUGH, YOU WILL SEE C REPLACING X, SO THAT msec=
f(c+Δx) ­ f(c)
c+Δx­c
=
f(c+Δx) ­ f(c)
Δx where P(c,f(c)) and Q( c+Δx, f(c+Δx)) OR YOU MIGHT SEE h REPLACING Δx, SO THAT m sec= f(c+h) ­
f(c)
h
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CALC 1.1.notebook
September 01, 2011
SECANT LiNE
SECANT LiNE
SECANT LiNE
TANGENT LiNE
As point Q approaches point P,
When such a "limiting position" exists, the slope of the tangent line
is said to be the
of the
slope of the secant line.
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CALC 1.1.notebook
September 01, 2011
EXPLORATION
THE FOLLOWING POINTS LIE ON THE GRAPH OF f(x) =x2
Q1 (1.5, f(1.5)) =
Q2 (1.1, f(1.1)) =
Q3 (1.01, f(1.01)) =
Q4 (1.001, f(1.001)) =
Q5 (1.0001, f(1.0001)) =
E
N
LI
N
TA
1
1
Each successive point gets closer to the point P(1,1). Find the slope of the secant line through:
Q1 and P
Q2 and P
Q3 and P
Use the results to estimate the slope of the tangent line to the graph of f at point P
Q4 and P
Q5 and P
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CALC 1.1.notebook
September 01, 2011
OR
P(1,1)
INE
L
TAN NE
I
C L
E
S
1
{
msec =
Q(x,f(x))
Δx = x­1
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CALC 1.1.notebook
September 01, 2011
From p47: #7 SECANT LINES
Consider the function f(x) =4x­x2 and
the point P(1,3) on the graph of f.
Note: The x value is 1, so we are using 1.
a. Graph f and the secant lines passing through
P(1,3) and Q(x,f(x)) for the x­values 2, 1.5, 0.5
4
3
2
1
1 2 3 4
b. Find the slope of each secant line
Now use the formula msec=
to find the limiting position
>
(value) of the slope as Q­ P
f(x+Δx) -f(x)
Δx
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CALC 1.1.notebook
September 01, 2011
c. Use the results of part b to estimate the slope of the tangent line of f at P(1,3). Describe how you can improve the approximation of the slope.
HW problem pg 47: #8
Note:
The x value is 4, so you will be using 4 for your x value.
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CALC 1.1.notebook
September 01, 2011
AREA UNDER A CURVE
y
y=f(x)
CLASSIC CALCULUS PROBLEM #2
y
y
X
X
X
a
b
a
b
b
a
Area under Approximation Approximation
using 6
using 3
a curve
rectanlges
rectanlges
As a simple example, consider the region bounded by the graph of the function y=f(x)m, the x­axis, and the vertical lines x=a and x=b.
You can approximate the area of the region with several rectangular regions, as shown. As you increase tje number of rectangles, the approximation tends to become betten and better because the amount of area missed by the rectangles decreases. Your goal is to
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CALC 1.1.notebook
September 01, 2011
ΝΟΤΕ:
1
1
BOUNDED REGION
1
INSCRIBED RECTANGLES
1
CIRCUMSCRIBED RECTANGLES
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CALC 1.1.notebook
September 01, 2011
PRECALCULUS
1. Find the distance traveled in 15 seconds by an object traveling at a constant velocity of 20 feet per second.
2. A bicyclist is riding on a path modeled by the function f(x) = 0.08x, where x and f(x) are measured in miles. Find the rate of change of elevation when x=2.
f(x)=0.08x
CALCULUS
3. Find the distance traveled in 15 seconds by an object moving with a velocity of v(t)=20+7cost feet per second.
4. A bicyclist is riding on a path modeled by the function f(x) = 0.04(8x­x2), where x and f(x) are measured in miles. Find the rate of change of elevation when x=2.
Q(x,f(x))
P(2,48)
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CALC 1.1.notebook
September 01, 2011
FIND THE AREA OF THE SHADED REGIONS
5.
6.
(2,3)
4
3
2
(0,0)
2
(5,0)
1
1
2
3
4
6.
5.
Use the rectangles to approximate the area of the region 5
bounded by y= , y=0, x=1, and x=5
x
7.
8.
5
4
3
2
1
5
4
3
2
1
1 2 3 4 5
7. AREA
⇒ Notice how the vertex
at the top left corner
is on the curve
1 2 3 4 5
8. AREA
9. For problems 7 and 8, describe how you could continue this process to obtain a more accurate approximation of the area.
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CALC 1.1.notebook
September 01, 2011
FIND THE AREA OF THE SHADED REGIONS
5.
6.
(2,3)
4
3
2
2
(5,0)
1
(0,0)
1
2
3
4
6.
5.
Use the rectangles to approximate the area of the region 5
bounded by y= , y=0, x=1, and x=5
x
7.
8.
5
4
3
2
1
5
4
3
2
1
1 2 3 4 5
7. AREA
⇒
Notice how the vertex
at the top left corner
is on the curve
1 2 3 4 5
8. AREA
9. For problems 7 and 8, describe how you could continue this process to obtain a more accurate approximation of the area.
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CALC 1.1.notebook
September 01, 2011
5
Consider the length of the graph of f(x) = x
from (1,5) to (5,1)
10. 5
9. 5
4
3
2
1
4
3
2
1
1 2 3 4 5
1 2 3 4 5
9. Approximate the length of the curve by finding the distance between its endpoints, as shown in figure 9.
10. Approximate the length of the curve by finding the sum of the lengths of 4 line segments, as shown in figure 10.
11. How could you obtain a more accurate approximation of the length of the curve?
NOTE: TO ENSURE ACCURACY OF AN ANSWER, DO NOT ROUND UNTIL YOU HAVE YOUR FINAL ANSWER. USE STORAGE STO IF NEEDED.
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CALC 1.1.notebook
September 01, 2011
5
Consider the length of the graph of f(x) = x
from (1,5) to (5,1)
10. 5
9. 5
4
3
2
1
4
3
2
1
1 2 3 4 5
1 2 3 4 5
9. Approximate the length of the curve by finding the distance between its endpoints, as shown in figure 9.
10. Approximate the length of the curve by finding the sum of the lengths of 4 line segments, as shown in figure 10.
11. How could you obtain a more accurate approximation of the length of the curve?
NOTE: TO ENSURE ACCURACY OF AN ANSWER, DO NOT ROUND UNTIL YOU HAVE YOUR FINAL ANSWER. USE STORAGE STO IF NEEDED.
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