1. Find the instantaneous velocity for 1. Find the instantaneous velocity for the following the following time—distance function at time = 2 time—distance function at time = 2 seconds . seconds . 2 ) 2 ( 2 ) ( t t s Quiz 10-1 Quiz 10-1 4 2 ) ( 2 x x x f 2. Find the function that represents the 2. Find the function that represents the slope at any location on the following slope at any location on the following function:. function:. h x f h x f x f h ) ( ) ( lim ) ( 0 ' 2 ) 2 ( ) ( lim ) ( 2 ' t s t s t s t
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1. Find the instantaneous velocity for the following 1. Find the instantaneous velocity for the following time—distance function at time = 2 seconds. time—distance.
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1. Find the instantaneous velocity for the following 1. Find the instantaneous velocity for the following time—distance function at time = 2 seconds . time—distance function at time = 2 seconds .
2)2(2)( tts
Quiz 10-1Quiz 10-1
42)( 2 xxxf
2. Find the function that represents the slope at any 2. Find the function that represents the slope at any location on the following function:. location on the following function:.
h
xfhxfxf h
)()(lim)( 0
'
2
)2()(lim)( 2
'
t
ststs t
HOMEWORKHOMEWORK
Section 10-2 Section 10-2
(page 810)(page 810)
(evens) 2-8, 12, 14, 18,(evens) 2-8, 12, 14, 18,
22-42 even, 5022-42 even, 50
(18 problems)(18 problems)
10-2Limits and Motion
The Area problem.
What you’ll learn about• Computing Distance Traveled (Constant Velocity)• Computing Distance Traveled (Changing Velocity)• Limits at Infinity• The Connection to Areas• The Definite Integral
… and whyLike the tangent line problem, the area problem has
applications throughout science, engineering, economics and historicy.
Computing Distance Traveled
A car travels at an average rate of 56 miles per hour for 3 hours. How far does the car travel?
““The The summationsummation of ‘ of ‘a’ sub ‘ka’ sub ‘k’ for ‘’ for ‘k’ = 1 to 5k’ = 1 to 5””
We’re going to use this idea of a summation to find out the exact area under the curve.
0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)
VVEELLOOCCIITTYY
4040
3030
2020
1010
00
We will make the width of We will make the width of the rectangle the rectangle infinitesimallyinfinitesimally
small.small.
We’ll call that We’ll call that InfinitesimallyInfinitesimally small width: small width: x
The The heightheight of the of the rectangle is just therectangle is just the output value of the output value of the function. function. )(xf
The The distance traveleddistance traveled (area under the curve)(area under the curve) is the sum of all of the is the sum of all of the small rectangles.small rectangles.
We’re going to use this idea of a summation to find out the exact area under the curve.
0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)
VVEELLOOCCIITTYY
4040
3030
2020
1010
00
x
)( ixf
The The velocityvelocity (area under the curve) (area under the curve) is the sum of is the sum of all of the small rectangles.all of the small rectangles.
n
iin xxfarea
1
)(lim
Height: (Height: (function valuefunction value for for the left upper corner ofthe left upper corner ofEach of these slivers (thereEach of these slivers (there are an infinite # of them) are an infinite # of them)
n
abx
Where “b” and “a” Where “b” and “a” are the right and leftare the right and left ends of the interval.ends of the interval.
Definite Integral
1
1
Let be a function on [ , ] and let ( ) be defined as above.
The definite integral of over [ , ], denoted ( ) , is given by
( ) lim ( ) , provided the limit exists.
If t
n
ii
b
a
b a
in ia
f a b f x x
f a b f x dx
f x dx f x x
he limit exists, we say is integrable on [ , ].f a b
n
iin xxfarea
1
)(lim
n
iin xxfareadxxf
1
5
0
)(lim..)(
Limits at Infinity (Informal)
When we write "lim ( ) ," we mean that ( ) gets arbitrarily close
to as gets arbitrarily large.xf x L f x
L x
Calculating the Integral of a function (gives the area under the curve for any specified
interval)
Find: (the area under the function for the Find: (the area under the function for the interval x = [1,5]interval x = [1,5]
5
12xdx
n
iin xxfdxxf
1
5
1
)(lim)(
n
ii xxapprox
1
)1)(54321(22 30
n
abx
b = largest input valueb = largest input value
a = smalles input valuea = smalles input value
14
15
x
n = # of rectangles (bigger is more accurate) n = # of rectangles (bigger is more accurate)
Calculating the Integral of a function (gives the area under the curve for any specified
interval)
Find: (the area under the function for the Find: (the area under the function for the interval x = [1,5]interval x = [1,5]
5
12xdx
n
iin xxfdxxf
1
5
1
)(lim)(
n
ii xxapprox
1
)5.0)(55.445.335.225.115.0(22 5.27
n
abx
b = largest input valueb = largest input value
a = smalles input valuea = smalles input value
5.08
15
x
n = # of rectangles (bigger is more accurate) n = # of rectangles (bigger is more accurate)
nasarea .. Where does the infinite series “converge”?Where does the infinite series “converge”?
Calculating the Integral of a function
Find: (the area under the function for the Find: (the area under the function for the interval x = [1,5]interval x = [1,5]
5
12xdx
n
iin xxfdxxf
1
5
1
)(lim)(1010
55
5511
Area Area underunder the line (in the line (in the interval x = [1,5]the interval x = [1,5]
f(x) = 2xf(x) = 2x
Area = area large triangle – area small triangleArea = area large triangle – area small triangle