Calculus 2
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Polar Coordinates
If P = (x, y) is a point in the xy-plane and O denotes the origin, let
r denote the distance from the origin O to the point P = (x, y). Thus, x2 + y 2 = r2 ;
y
b
Calculus 2
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Parametric Curves
In the past, we mostly worked with curves in the form y = f (x). However, this format does not
encompass all the curves one encounters in applications. For example
Calculus 2
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Taylor Polynomials
Recall that the line which approximates a function f (x) at a point (a, f (a) has the slope f (a).
By point-slope equation, the equation of this line is
y f (a) =
Calculus 2
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Modeling with Dierential Equations
In most cases, it is equally important to be able to come up with a dierential equation that
accurately describes the problem you need to solve as
Calculus 2
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Linear Dierential Equation
A rst order dierential equation is linear if it can be written in the form a1 (x)y +a0 (x)y = b(x).
Note that if a1 (x) = 0, the equation is not dierentia
Math 201
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Review for Final Exam Solutions
1) Denite and Indenite Integrals.
1. 1/21 (3x + 5)7 + c
2.
b ax2 +1
e
2a
+ c.
3. ln |x| + 1/x + c
4.
1
a
5.
1
23x+1
3 ln 2
ln |ax + b| + c.
+c
6. 1/5 s
MATH 201
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Review for Exam 3
a) General Solution. Find the general solutions of the following.
1. y = 2x 1 y 2
2. y = y 2 xe2x
3. y = x(y + 1)
4. xy + 2y = x3
5. xy + y = x cos x
6. Show that y
Math 201
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Review for Final Exam
1) Integrals. Evaluate the following integrals.
1.
(3x + 5)6 dx
2.
bxeax
3.
1
(x
4.
2 +1
dx where a and b are arbitrary constants.
1
)
x2
1
dx
ax+b
3x+1
dx
wher
MATH 201
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Review for Exam 3 Solutions
a) General Solution. 1. y = sin(x2 + c)
3
2. y =
2
1
1
1
xe2x + 4 e2x +c
2
3. y = cex
2 /2
1
2x
4. y = x /5 + c/x
5. y = sin x + (cos x)/x + c/x
6. Substit
Calculus 2
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Autonomous Dierential Equations and Population
Dynamic
If a dierential equation is of the form
y = f (y),
it is called autonomous. Note that an autonomous equation is a separable di
Calculus 2
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Arc Length. Surface Area.
Arc Length. Suppose that y = f (x) is a continuous function with a continuous derivative on
[a, b]. The arc length L of f (x) for a x b can be obtained by
Calculus 2
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Dierential Equations of First Order. Separable Dierential
Equations. Eulers Method
A dierential equation is an equation in unknown function that contains one or more derivatives of
Calculus 2
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Approximate Integration. Trapezoidal and Simpsons sums.
Recall that the Left and the Right Sums approximate the area under a curve as the sum of certain
rectangles. On each subinter
Calculus 2
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Improper Integrals
The integral
b
a
f (x) dx is improper if it is of one of the following three types:
1. At least one of the bounds is positive or negative innity.
2. The function
Calculus 2
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Trigonometric Integrals
Let us consider the integrals of the form
f (sin x) cos xdx
or
f (cos x) sin xdx
where f (x) is a function with antiderivative F (x). Using the substitution
Calculus 2
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Partial Fractions
A rational function is a quotient of two polynomial functions. The method of partial fractions is
a general method for evaluating integrals of rational function. T
Calculus 2
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Integration by Parts
Using integration by parts one transforms an integral of a product of two functions into a simpler
integral. Divide the initial function into two parts called u
MATH 201
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Review for Exam 1
a) Denite and Indenite Integrals. Evaluate the following integrals.
4
1. ( x x2 ) dx
2
2. (2 3 x + x ) dx
4
3.
4
3
1( x
4.
(3x + 5)6 dx
5.
5 x
dx
3
x2 +9
1
1
( x x2
MATH 201
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Review for Exam 2
a) LHpitals Rule. Evaluate the limits:
o
1. limx0
e4x 1
sin 2x
tan1 2x
x
1cos 3x
limx0 x2
2. limx0
3.
4. limx0 (1 + 3x)1/x
5
5. limx (1 x )2x
6. limx ln(x + 2) ln(x
Calculus 2
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Areas between Curves
If f (x) and g(x) are two continuous functions
dened on the interval [a, b] such that f (x) g(x)
for all x in [a, b], then the area between the graphs
of f and
Calculus 2
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Derivatives and Integrals of Trigonometric and Inverse
Trigonometric Functions
Trigonometric Functions.
If y = sin x, then y = cos x and
if y = cos x, then y = sin x.
sin x dx = cos
Calculus 2
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LHpitals Rule
o
LHpitals rule is used to convert limits in an indeterminate form to a determinate form. One
o
can apply it in several situations.
Basic Case 0 or . To evaluate a lim
Calculus 2
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The average value. Some physics applications.
The average value of a function. Let f be a continuous function. The average value of f on
b
[a, b] is the y-value fave such that the s
Calculus 2
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Volume - Washer method
Computing general volume. If S is a solid between x = a and x = b with cross sectional
area A(x), then the volume V of S can be found by integrating the volum
Calculus 2
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Volumes by Cylindrical Shells. Disc Method
Recall that the volume of a cylindrical shell with the inner radius r1 , outer radius r2 and the
height h is
Volume = 2r h dr = circumfere
Calculus 2
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Denite Integral. The Left and Right Sums
The denite integral arises from the question of nding the area between a given curve and x-axis
on an interval.
The area under a curve can b
Calculus 2
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The Fundamental Theorem of Calculus.
The Total Change Theorem and the Area Under a Curve.
Recall the following fact from Calculus 1 course. If a continuous function f (x) represents
Calculus 2
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Derivatives of Exponential and Logarithmic Functions.
Logarithmic Dierentiation
Derivative of exponential functions. The natural exponential function can be considered as
the easies