Antiderivatives (7.4, 8.2, 10.1) JMerrill, 2009. Review Info - Antiderivatives General solutions:...
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Transcript of Antiderivatives (7.4, 8.2, 10.1) JMerrill, 2009. Review Info - Antiderivatives General solutions:...
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Antiderivatives(7.4, 8.2, 10.1)
JMerrill, 2009
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Review Info - Antiderivatives
General solutions:
y f(x)dx F(x) C Integrand
Variable of Integration
Constant of Integration
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Review
Rewriting & Integrating – general solution
Original Rewrite Integrate Simplify
31
dxx 3x dx
2xC
2
2
1C
2x
x dx12x dx
32x
C32
322
x C3
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Particular Solutions
To find a particular solution, you must have an initial condition
Ex: Find the particular solution of that satisfies the condition F(1) = 0
2
1F'(x)
x
2
1F'(x) dx
x
2x dx1x 1
C C1 x
1F(1) C
1
1F(x) C
x
0 1 C
1 C
1F(x) 1
x
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Indefinite & Definite Integrals
Indefinite Integrals have the form:
Definite integrals have the form:
f (x)dxb
a
f (x)dx
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7.4 The Fundamental Theorem of Calculus This theorem represents the relationship
between antiderivatives and the definite integral
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Here’s How the Theorem Works
First find the antiderivative, then find the definite integral
2
3
1
4x dx4
3 44x4x dx x
4
24
1x 4 42 1 15
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Properties of Definite Integrals
The chart on P. 466:
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Example – Sum/Difference
Find 5
2
2
6x 3x 5 dx 5 5 5
2
2 2 2
6 x dx 3 xdx 5 dx
32 3
22
x6 x 6 2x
3
x 33 xdx 3 x
2 2
5 dx 5 x 5x
55 53 222 2
32x x 5x
2
3 3 2 232 5 2 5 2 5 5 2
2
63234 15
24352
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Less Confusing Notation?
Evaluate 2
2
0
2x 3x 2 dx 23 2
0
2x 3x2x
3 2
166 4 0 0 0
3
103
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Substitution - Review
Evaluate
Let u = 3x – 1; du = 3dx
43 3x 1 dx
4u du
455u
33
x 1 3dxx
C1
55C
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Substitution & The Definite Integral
Evaluate
Let u = 25 – x2; du = -2xdx
5
2
0
x 25 x dx
5
2
0
5 5 12
0 0
125 x 2xdx
2
1 1udu u du
2 2
32u
C3
53
2 2
0
25 x
3
32250 1
335
32
321 u
C322
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Area
Find the area bounded by the curve of f(x) = (x2 – 4), the x-axis, and the vertical lines x = 0, x = 2
2
2
0
x 4 dx23
0
x4x
3
88 0
3163
0
The answer is negative because the area is below the x-axis. Since area must be positive just take the absolute value.
163
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Finding Area
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Area – Last Example
Find the area between the x-axis and the graph of f(x) = x2 – 4 from x = 0 to x = 4.
2 4
2 2
0 2
x 4 dx x 4 dx 2 4
3 3
0 2
1 1x 4x x 4x
3 3
8 64 88 0 0 16 8
3 3 3
16
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8.2 Volume & Average Value
We have used integrals to find the area of regions. If we rotate that region around the x-axis, the resulting figure is called a solid of revolution.
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Volume of a Solid of Revolution
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Volume Example
Find the volume of the solid of revolution formed by rotating about the x-axis the region bounded by y = x + 1, y = 0, x = 1, and x = 4.
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Volume Example
4
2
1
V x 1 dx
43
1
x 1
3
3 3 1175 2
3 339
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Volume Problem
Find the volume of the solid of revolution formed by rotating about the x-axis the area bounded by f(x) = 4 – x2 and the x-axis.
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Volume Con’t
2
22
2
V 4 x dx
2
2 4
2
16 8x x dx
23 5
2
8x x16x
3 5
51264 32 64 3232 32
3 5 5 153
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Average Value
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Average Price
A stock analyst plots the price per share of a certain common stock as a function of time and finds that it can be approximated by the function
S(t)=25 - 5e-.01t
where t is the time (in years) since the stock was purchased. Find the average price of the stock over the first 6 years.
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Avg Price - Solution
We are looking for the average over the first 6 years, so a = 0 and b = 6.
6 .01
0
125 5
6 0
te dt
The average price of the stock is about $20.15
6.01
0
1 525
6 .01
tt e
.061150 500 500
620.147
e
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10.1 Differential Equations
A differential equation is one that involves an unknown function y = f(x) and a finite number of its derivatives. Solving the differential equation is used for forecasting interest rates.
A solution of an equation is a number (usually).
A solution of a differential equation is a function.
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Differential Equations
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Population Example
The population, P, of a flock of birds, is growing exponentially so that , where x is time in years.
Find P in terms of x if there were 20 birds in the flock initially.
0.05xdP20e
dx
Note: Notice the denominator has the same variable as the right side of the equation.
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Population Cont
Take the antiderivative of each side:
This is an initial value problem. At time 0, we had 20 birds.
0.05xP 20e dx0.05xdP
20edx
0.05x 0.05x20e C 400e C
0.05
0 20 400e C
380 C
0.05xP 400e 380
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One More Initial Value Problem
Find the particular solution of when y = 2, x = -1
dy2x 5
dx
dy2x 5
dx
dy2x 5dx
dx
222x
y 5x C x 5x C2
22 ( 1) 5( 1) C
6 C
2y x 5x 6
Note: Notice the denominator has the same variable as the right side of the equation.
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Separation of Variables
Not all differential equations can be solved this easily.
If interest is compounded continuously then the money grows at a rate proportional to the amount of money present and would be modeled by dA
kAdt
Note: Notice the denominator does not have the same variable as the right side of the equation.
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Separation of Variables
In general terms think of
This of dy/dx as the fraction dy over dx (which is totally incorrect, but it works!)
In this case, we have to separate the variables
dy f(x)dx g(y)
g(y)dy f(x)dx
G(y) F(x) C
(Get all the y’s on one side and all the x’s on the other)
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Example
Find the general solution of
Multiply both sides by dx to get
2dyy x
dx
2y dy x dx
2y dy x dx 2 3y x
C2 3
2 32y x 2C
3
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Lab 4 – Due Next Time on Exam Day
1. #34, P471 9. #25, P523 2. #59, P440 10. #35, P523 3. #22, P471 11. #3, P629 4. #45, P439 12. #7, P629 5. #11, P439 13. #19, P630 6. #13, P471 14. #27, P630 7. #27, P439 15. #43, P472 8. #17, P522