Calculus 2
Lia Vas
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
be the angle between the vector OP and the positive part
Calculus 2
Lia Vas
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, consider the circle x2 +y 2 = a2 .
Solving for y does
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) = f (a)(xa) y = f (a)+f (a)(xa).
The expression f (a)+f
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 to being able to solve the equation. The
process of wr
Calculus 2
Lia Vas
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 dierential. So, let us assume that the function a1 (x)
is not ze
Math 201
Lia Vas
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 sin(5x + 1) + c
7. 1/3 tan1 (3x) + c
8. 1/3 sin1 (3x) +
MATH 201
Lia Vas
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 = ce2x is a solution of dierential equation y 6y + 8y =
Math 201
Lia Vas
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
where a and b are arbitrary constants.
5.
2
6.
cos(5x + 1)
MATH 201
Lia Vas
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. Substitute y = 2ce , and
2x
y = 4ce into the equation and show
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 dierential equation.
If f is zero at a, then the horizont
Calculus 2
Lia Vas
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 integrating the length element
ds from a to b. The leng
Calculus 2
Lia Vas
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 the unknown function.
A function y is a solution of a d
Calculus 2
Lia Vas
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 subinterval, the left sum uses rectangles whose heights are obt
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 f (x) is not dened or is discontinuous at at least one
Calculus 2
Lia Vas
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
u = sin x for the rst integral
and
u = cos x for the se
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. The idea is to write a rational function
as a sum of sim
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 and dv (keep dx in dv part). Then apply
the following
MATH 201
Lia Vas
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 ) dx
1
dx
4x+1
2x
2x
6.
7.
8.
9.
(e
2x) dx
+e
x2 +1
x
MATH 201
Lia Vas
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 1)
b) Integration Evaluate the integrals:
1.
2.
3.
x9
d
MATH 201
Lia Vas
Review 2 Solutions
a) LHpitals Rule. 1. 2
o
2. 2
3. 9/2
4. e3
5. e10
6. 0
b) Integration
1. 2 ln |x + 5| ln |x 2| + c
2. x ln |x| + 2 ln |x 1| + c
3. 1/x + 2 ln |x| + 3 ln |2 + x| + c
1
4. x cos 2x + 4 sin 2x + c
2
5. x2 cos x + 2x sin 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 g on [a, b] is
b
(f (x) g(x) dx.
A=
a
Note that in this
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 x + c and
cos x dx = sin x + c.
Thus,
The derivatives
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 limit limxa f (x) of the type 0 or , consider the limit
0
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 shaded area a f (x)dx under the curve on the gure below
Calculus 2
Lia Vas
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 volume element dV which can
be obtained as the product of A(
Calculus 2
Lia Vas
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 = circumference height thickness
where r is the average radius r =
Calculus 2
Lia Vas
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 be easily calculated if the curve is given by a simple f
Calculus 2
Lia Vas
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 the rate
of change of F (x) (so that F (x) is an antid
Calculus 2
Lia Vas
Derivatives of Exponential and Logarithmic Functions.
Logarithmic Dierentiation
Derivative of exponential functions. The natural exponential function can be considered as
the easiest function in Calculus courses since
the derivative of