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AP Calculus Differential Equations
20151123
www.njctl.org
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Table of Contents
Indefinite IntegralsSlope FieldsUSubstitution USubstitution & Definite IntegralsDifferential Equations (Separable First Order)Integration by PartsPopulation Growth
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Indefinite Integrals
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Indefinite Integrals do not have bounds. They will give you an initial value and have you find C.
Example:
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If the gravity on Planet X is 10 ft/sec2 and a rock is thrown upward from the top of a 20' building. If the rock was thrown with a velocity of 5 ft/sec, when will it hit the ground?
2.562 seconds to hit the ground.
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1 Find f(7) if f(3) = 2 and
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2 Find f(5) if f(0) = 9 and
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3 If the gravity on Planet Y is 12 ft/sec2 and a rock is thrown
upward from the top of a 30' building. If the rock was thrown with a velocity of 8 ft/sec, when will it hit the ground?
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Slope Fields
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A Slope Field is of graph the slopes of an equation at specific points.
Given: , sketch the slope field.
Remember that is the slope.
3 2 1 0 1 2 3
3
2
1
1
2
3
x
y
So substituting the order pair into the equation will give you the slope at that point.
Each dash has the slope of y at that point.
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3 2 1 0 1 2 3
3
2
1
1
2
3
x
y
Now that we have our slope field, we can find a particular function.
Sketch the curve (0,1) is ony=f(x)
Graph the point given and then smoothly flow from dash to dash.
It helps if you can integrate and know what the graph should look like.
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Example: , sketch the slope field and the curve if (0,1) is on y=f(x).
2 1 0 1 2
2
1
1
2
x
y
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Q:Why are there so many dashed lines?
A: Because of the unknown constant in an indefinite integral. The slope field shows all of the family of a graph.
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A
B C
D
Does the following slope field have a horizontal asymptote? If so, where?
6 5 4 3 2 1 0 1 2 3 4 5 6
6
5
4
3
2
1
1
2
3
4
5
6
x
y
No horizontal asymptote
y = 3x = 2
y = 3
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A
B C
D
Does the following slope field have a vertical asymptote? If so, where?
6 5 4 3 2 1 0 1 2 3 4 5 6
6
5
4
3
2
1
1
2
3
4
5
6
x
y
No vertical asymptote
x = 0x = 2
y = 3
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A
B C
D
If (1,0) is on y=f(x), which these other points could be on y?
6 5 4 3 2 1 0 1 2 3 4 5 6
6
5
4
3
2
1
1
2
3
4
5
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x
y
(4,1)
(1,3)
(8,4)
(1, 1)
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A
B C
D
The family of graphs shown is for
6 5 4 3 2 1 0 1 2 3 4 5 6
6
5
4
3
2
1
1
2
3
4
5
6
x
y
circle
exponential
rational
quadratic
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8
A
B C
D
The concavity of f(x) at (4,2) is
6 5 4 3 2 1 0 1 2 3 4 5 6
6
5
4
3
2
1
1
2
3
4
5
6
x
y
positive
0
negative
undefined
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USubstitution
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Usubstitution is used to find the antiderivative of the chain rule.
Recall the chain rule:
We took the derivative of the composite of functions starting with the outer one first.
For usubstitution method we're going in reverse. We start with the inner most function and called it u.The find du/dx and make substitutions. The integral should be much easier to find.
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Example:
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Example:
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Example:
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Example:
I recalled the derivative of tan was sec2, thought this would make u and du easier to find.
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A
B
C
D
Given what should u = ?
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10
A
B
C
D
Given what should du = ?
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A
B
C
D
Given following usubstitutions is
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A
B
C
D
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A
B
C
D
Given
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A
B
C
D
Given what should u = ?
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A
B
C
D
Given what should du = ?
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A
B
C
D
Given what should 8xdx = ?
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A
B
C
D
Given following usubstitutions is
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A
B
C
D
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A
B
C
D
Given
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A
B
C
D
Given what should u = ?
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A
B
C
D
Given what should du = ?
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A
B
C
D
Given following usubstitutions is
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A
B
C
D
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USubstitution&
Definite Integrals
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For the first step in the equation below to be equal to the second the bounds have to be rewritten in terms of u.
Use u = x 4, to convert bounds.
Once bounds are converted, x is not used again.
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Example:
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A
B
C
D
Given what should u = ?
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A
B
C
D
Given what should du = ?
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26 The lower bound for
becomes what for usubstitution?
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27 The upper bound for
becomes what for usubstitution?
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A
B
C
D
Given following usubstitutions is
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Differential Equations(Separable First Order)
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Why does lead to ?
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This is why it is called separable.
Note:
Why?An unknown constant minus another unknown constant is still a constant.
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Separation of variables is used to integrate implicit differentiation.
Steps1) Separate variables
2)Integrate both sides
3)Find C as soon as possible.
4)Sub in C if found
5)If the directions ask for y= solve for y.
and find y=
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Example: Find the general solution of the differential equation:
NOTE: Since there was not an initial value given we leave ±, had one been given C would have been found in line 3 and subbing inital value back in again line 6, would have decided + or .
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A
B
C D
and y(4)=0, separate the variables.
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A
B
C
D
and y(4)=0, the antiderivative is
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32 and y(4)=0, C=
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A
B
C D
and y(0)=3, separate the variables.
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A
B
C
D
and y(0)=3, y=
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35 and y(0)=3, C=
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Example: y(0)=3, find y.
Since an initial value is given, it can be determined whether + or is used.But which?
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Example: y(0)=3, find y.
Since an initial value is given, it can be determined whether + or is used.But which?
or Sub in initial value and see which equation is true.
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Separating the variables also works for the second derivative.
and at x=0 y'=1 and y=3. Solve the differential equation.
Start by replacing
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Integration by Parts
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Integration by Parts is used when you have the product of 2 functions that you want to integrate.
The time to use it is when usubstitution doesn't work because the one function doesn't derive to the other.
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Integration by Parts
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Example:
Make u the function that reduces when it is derived.
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Example:
Whenever there is ln(x) in an integration, that is the u.
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36 Consider the following integration by parts problem: what should u= ?
A
B
C
D
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37 Consider the following integration by parts problem: what should du= ?
A
B
C
D
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38 Consider the following integration by parts problem: what should dv= ?
A
B
C
D
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39 Consider the following integration by parts problem: what should v= ?
A
B
C
D
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40 Consider the following integration by parts problem: what should uv ∫vdu= ?
A
B
C
D
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41 Consider the following integration by parts problem:
A
B
C
D
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Sometimes the integration by parts has to be repeated because the integration is still to "unrelated" functions.
Once u=f(x) don't switch to u=g(x) for the second time of integrating by parts.
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Notice in the last example, u=xa and it was derived until the x went away. To make repeated applications of the integration by parts easier, we can use The Tabular Method of Integration.
+
+
• Start by writing the function that after multiple derivings will go to zero, derive the function to zero.• Write the other function under dv and integrate enough times so that u and dv have the same number of rows.• Multiply diagonally from u to dv. Keep the sign of the first product, change the sign of the second product. Keep alternating whether to keep or change the sign until no more diagonals left.• Add the products, don't forget your constant of integration, C.
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u dv
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In this case neither ex nor cos x will derive to zero. We will use the trig function as u and derive twice.
We've now done integration by parts twice and we've gotten back to the same integral we started with.
Now use algebra.
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Population Growth
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There are 4 types of Population Growth.
1) Linear Growth
This is direct variation so y= kt + C
As opposed to indirectly: y= k/t +C
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There are 4 types of Population Growth.
2) Sinusoidal Growth
Think of it as the population of a college town. Crests during the fall and is at a low over the summer.
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There are 4 types of Population Growth.
3) Logistic Growth
#whoheard
time
To visualize a logistic function graph think of how a rumor spreads around the school.
It starts with a couple of people then a few more and continues to spread till everyone has heard a version of it.
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There are 4 types of Population Growth.
4) Exponential Growth
The amount of growth depends on population present.
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If you recognize the model you can go to the equation and skip the integration to get there.
Example: A bacteria population grows at y'=ky, where k is constant and y is current population. P(0)=1000 and population tripled in the first 5 days.
a) write an expression for y at any time t.
Recognizing y'=ky as exponential use
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Example: A bacteria population grows at y'=ky, where k is constant and y is current population. P(0)=1000 and population tripled in the first 5 days.
b. By what factor did the population increase in the first 10 days?Using the equation from part a
Population increased by a scale factor of 9 in the first 10 days
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Example: A bacteria population grows at y'=ky, where k is constant and y is current population. P(0)=1000 and population tripled in the first 5 days.
c.How long will it take for the population to reach 6000?
It will take 8.155 days for population to reach 6000.
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If you don't recognize the growth model from the rate, seperate the variables and integrate.
Example: A sphere's volume increases at a rate proportional with the reciprocal of its radius. At t=0, r=1 and at t=15, r=2
a. Find r in terms of t
x
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Example: A sphere's volume increases at a rate proportional with the reciprocal of its radius. At t=0, r=1 and at t=15, r=2
b.Find when the Volume is 27 times its initial volume.
initial volume:
What is the radius when V is 27 times greater?
At what time does r=3?It takes 80 seconds for the volume to be 27 times the initial volume.
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42 A wolf population grows at a rate of increase that is directly proportional to 800P(t), where k is the constant of proportion. If p(2)=700, find k.
HINT
k=.549
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43 A wolf population grows at a rate of increase that is directly proportional to 800P(t), where k is the constnat of proportion. If p(2)=700, find the limit of the population as t goes to ∞.