7.1 Integral As Net Change Quick Review What you’ll learn about Linear Motion Revisited General...
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Transcript of 7.1 Integral As Net Change Quick Review What you’ll learn about Linear Motion Revisited General...
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7.1
Integral As Net Change
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Quick Review
2
-
2
2
Find all values of (if any) at which the function changes
sign on the given interval.
1. cos 2 on 0,1
2. 5 6 on -5,5
3. on 0,
14. on -5,5
4
x
x
x
x x
e
x
x
4
3 ,2
positive always
2 ,1 ,1 ,2
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What you’ll learn about Linear Motion Revisited General Strategy Consumption Over Time Net Change from Data Work
Essential QuestionHow can the integral be used to calculate netchange and total accumulation?
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Example Linear Motion Revisited1. v(t) = 10 – 2t is the velocity in m/sec of a particle moving along the x-
axis when 0 < t < 9. Use analytical methods to:
a. Determine when the particle is moving to the right, to the left, and stopped.
b. Find the particle’s displacement for the given time interval.
c. If s(0) = 3, what is the particle’s final position?
d. Find the total distance traveled by the particle.
right moving is particle the,0When tv
50 t stopped is particle the,0When tv
5t left moving is particle the,0When tv
95 t
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Example Linear Motion Revisited1. v(t) = 10 – 2t is the velocity in m/sec of a particle moving along the x-
axis when 0 < t < 9. Use analytical methods to:
b. Find the particle’s displacement for the given time interval.
9
0 210 dtt 9
0 210 tt
8190 00 9
It moves 9 units to the right in the first 9 seconds.
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Example Linear Motion Revisited1. v(t) = 10 – 2t is the velocity in m/sec of a particle moving along the x-
axis when 0 < t < 9. Use analytical methods to:
c. If s(0) = 3, what is the particle’s final position?
dtt 210 Ctt 210 C 20010 3
3C
ts
310 2 ttts
399109 2 s 12
Which is the original position of 3 plus displacement of 9.
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Example Linear Motion Revisited1. v(t) = 10 – 2t is the velocity in m/sec of a particle moving along the x-
axis when 0 < t < 9. Use analytical methods to:
d. Find the total distance traveled by the particle.
5
0 210 dtt
5
0 210 tt
2550 8190
25
9
5 210 tt
Total distance is 41 m.
9
5 210 dtt
2550
16 41
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Strategy for Modeling with Integrals
1. Approximate what you want to find as a Riemann sum of values of a continuous function multiplied by interval lengths. If f (x) is the function and [a, b] the interval, and you partition the interval into subintervals of length x, the approximating sums will have the form
, with kk cxcf a point in the kth subinterval.
2. Write a definite integral, here to express the limit of these sums as the norm of the partitions go to zero.
,
b
adxxf
3. Evaluate the integral numerically or with an antiderivative.
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Example Potato Consumption
2. From 1970 to 1980, the ratio of potato consumption in a particular country was C(t) = 2.2 + 1.1t million of bushels per year, with t being years since the beginning of 1970. How many bushels were consumed from the beginning of 1972 to the end of 1975? 6 ,2Step 1: Riemann sum
We partition [2, 6] into subintervals of length t and let tk be a time in the kth subinterval.The amount consumed during this subinterval is approximately C (tk ) t million bushels.
The consumption for [2, 6] is approximately C (tk ) t million bushels.
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Example Potato Consumption
2. From 1970 to 1980, the ratio of potato consumption in a particular country was C(t) = 2.2 + 1.1t million of bushels per year, with t being years since the beginning of 1970. How many bushels were consumed from the beginning of 1972 to the end of 1975? 6 ,2Step 2: Definite Integral
The amount consumed from t = 2 to t = 6 is the limit of these sums as the norms of the partitions go to zero.
6
2 dttC
6
2 1.12.2 dtt
Step 3: Evaluate
Evaluate numerically, we obtain:
6 ,2 , ,1.12.2NINT tt 692.14 mil bushels
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Work
.FdW
When a body moves a distance d along a straight line as a result of the action of a force of constant magnitude F in the direction of motion, the work done by the force is
The equation W = Fd is the constant – force formula for work.
The units of work are force x distance. In the metric system, the unit is the newton – meter or joule. In the U.S. customary system, the most common unit is the foot – pound.
Hooke’s Law for springs says that the force it takes to stretch or compress a spring x units from its natural length is a constant times x. In symbols: F = kx, where k, measured in force units per length, is a force constant.
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Example A Bit of Work3. It takes a force of 6N to stretch a spring 2 m beyond its
natural length. How much work is done in stretching the spring 5 m from its natural length?
By Hooke’s Law, F(x) = kx. Therefore k = F(x) / x.
2/6k N/m 3 xxF 3Step 1: Riemann sum
We partition [0, 5] into subintervals of length x and let xk be any point in the kth subinterval.The work done across the interval is approximately F (xk ) x.The sum for [0, 5] is approximately F (xk ) x = 3xk x.
Step 2 and 3: Evaluate Definite Integral
5
0 3 dxx
5
0
2
2
3
x N/m
2
75
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Pg. 386, 7.1 #1-29 odd