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MATH 138
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Rui Bin Qin
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Mathematics

MATH 138

Rui Bin Qin

Fall

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Fundamental Theorem of Calculus
1. If ( ) ∫ ( ) , then ( ) ( ), where a is an arbitrary constant.
2. ∫ ( ) ( ) ( ), where F is antiderivatives of f, i.e. F’ = f
Note: ( ) ( ) ( ) ( ) ( ( )) ( ) ( ( )) ( )
∫ ( ) ∫ ( )
Substitution Rule
If u = g(x) is differentiable and its range I is the domain of a continuous function f, then
∫ [ ( ) ] ( ) ∫ ( ) ( ) ( )
Integration by parts
Let u = f(x), v = g(x),∫then ∫ .
Trigonometric Substitution
Substitution Domain Note
√
√
√
Integration of Rational functions
Proper rational functions:
∫ | |
∫ ( ) ( )( )
∫
∫ ( )
Improper rational functions:
∫ ( ) ∫ ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
Note: We say ( ) ( ) is improper if ( ) ( ), where ( )
( ) Volume
Definition: Let S is a solid between a and b. If the cross sectional area of S is A(x) (continuous), then
volume of S is ∑ ( ) (Riemann Sum)
∫ ( )
Sphere:
Using Pythagorean Theorem, y is given by √ .
Therefore the area of the cross-sectional area A is given by
( ) (√ ) .
Then ∫ ( )
∫ ( )
Approximation of volume:
Divide S into n pieces of equal widthby
using the planes where the
planes are perpendicular to x-axis and
through . If we choose sample points in
[ ], then we can approximate the i-th
slice by ( ) . Adding volumes of these
slices, then we will get ∑ ( ) .
The approximation becomes more accurate
as
∫ ( ) .
Washer:
inner radius - , outer radius - , height – h
Then ( ) Example:
Find the volume of the solid obtained by rotating the region bounded by y = x, about y = 2.
Solution:
,
Then ∫ ( ) ( ) ]
Wedges:
Given a cylinder with radius r and height h, when we cut the
cylinder at an angle, a wedge is produced. To find the volume of
the wedge, we sum the area of each triangle formed by the
wedge.
Example:
A wedge is cut out of a cylinder of radius 4 units by two planes.
The cutting plane intersects the base at an angle of 30° along the
diameter. Find the wedge’s volume.
Solution:
First, ( ) (√ )(√ )
√ ( )
Then, ∫ ( )
√
∫ ( ) √
Cylindrical Shell:
( )
( )( )
̅ ([average circumference]*[height]*[thickness]) Where ̅ ( )
Let S be the solid obtained by rotating ( ) about the y-axis,
where ( ) , y = 0, x = a and x=b.
Divide [a,b] into intervals [ ] of the length . Let ̅ be the
midpoint of[ ]. Then the volume of i-th cylindrical shell is
( ̅) ( ) .
Volume of the solid S is ∑ ̅ ( ̅
∫ ( ) , where is the circumference, f(x) is the height
and dx is the thickness.
Improper Integrals
Definition:
If f(x) is continuous at [ ], then∫ ( ) ∫ ( ) If the limit exists, integral
converges. Otherwise, integral diverges.
Similarly, ∫ ( ) ∫ ( ) .
For ∫ ( ) ∫ ( ) ∫ ( ) exist iff both limits exist.
If f(x) is continuous on ( ), then∫ ( ) ∫ ( ) .
Similarly, ∫ ( ) ∫ ( ) .
If f(x) has discontinuity at point x = c, a < c < b,
∫ ( ) ∫ ( ) ∫ ( ) ∫ ( ) ∫ ( ) iff both limits
exist.
Comparison Theorem:
Suppose f and g are continuous with ( ) ( ) , for ,
1) If ∫ ( ) is convergent, the∫ ( ) is also convergent.
2) If ∫ ( ) is divergent, the∫ ( ) is also divergent. Differential Equations
Definition:
A Differential Equation (D.E.) is an equation that contains an unknown function and its
derivatives.
The order of a D.E. is the order of the highest derivatives that occurs in the equation.
A function f(x) is called a solution to a D.E. if the equation is satisfied when y=f(x) and its
derivatives are substituted into the equation.
Initial Value Problem (IVP) – Find a solution of a D.E. that satisfy the condition of the form
( ) .
Direction field is a field where each point is associated with direction.
Direction Field:
Definition:
A field where each point is associated with a distance ( ) at point ( ).
The slope of a solution is ( ).
Note: Use maple to plot the direction fields. Alternatively, calculate manually.
Separable Equations:
( ) ( )
∫ ∫ ( )
( )
( ) ( )
[ ( ) ]
Mixing Problem:
Example:
A tank contains 20 kg of salt dissolved in 5000Lof water. Input 0.03kg/L at a rateof 25L/min. The
solution is well mixed, anddrains from the tank at the same rate. How much salt is left in tank after half
an hour?
Solution:
[Initial concentration] = 0.04kg/L
Let y(t) be the amount of salt remains in the tank at time t.
( ) Input: 0.3 x 25 = 0.75 kg/min
( )
Output: kg/min
Rate of change:
∫ ∫
Let u=150-y, du = -dy, y = 150+u
∫
| |
( )
Population Model problem:
Law of Natural growth:
Rate of growth of the population is proportional to the population.
Variable – time, t Population – P(t)
( ) ( )
Where ( ) , P(0) is the initial population.
Logistic Model:
When the maximum capacity is M, then
( )
Other Models:
( )
e.g. ( )( )
First Order Linear Differential Equations: ( ) ( ) ( ) ( )
Integrating factor: I(x) which satisfies I(x)[y’+P(x)y]=I(x)y.
Usually, we take ( ) ∫ ( ) [∫ ( ) ].
Sequences:
Definition:
A list of numbers written in a definite order i.e.
We can write a sequence as or where ( )
Harmonic sequence:
{ }
Fibonacci sequence:
Limits of sequence:
Definition:
The ε-definition of limits.
| |
Definition:
Example: Find the limit of the sequence { } .
Solution:
From observation, .
( ) ( )
| |
If we choose , then whenever n>N, we have | | .
Bounded sequence:
Definition:
A sequence is bounded if , where m is the lower bound and M is the upper bound.
Theorem:
Every convergent sequence is bounded.
Some Limits to remember:
1.
2.
3. √
4.
5. ( )
Theorem:
1. Let f be any function that is continuous at point p, if is a sequence in the domain of f,
( ) ( ). [ ( ) ( )]
2. If ( ) as and a sequence as , then ( ) [ ( )
( ) ]
Proposition: If f is a function s.t. as and is the sequence given by ( ) then
. [ ( ) ( ) ]
Monotonic sequence:
Increasing sequence , i.e. . If the inequalities are strictly increasing (<
instead of ), we say is strictly increasing.
Decreasing sequence , i.e. . If the inequalities are strictly decreasing
(> instead of ), we say is strictly decreasing.
A sequence is monotonic when it is either increasing or decreasing.
Monotonic sequence theorem:
If a sequence is bounded and monotonic, then has a limit, and therefore convergent. For
increasing sequence, the supremum is the limit of the sequence. For decreasing sequence, the infemum is
the limit of the sequence.
Completeness Axiom for ℝ :
If S is a non-empty set of real numbers that has an upper bound M, then S has a least upper
bound
Similarly, if m is a lower bound of a non-empty set of real numbers, then it has a greatest lower
bound N.
Sup(s) – supremum S is the least upper bound, Inf(s) – infemum S is the greatest lower bound.
Series:
Definition:
Given a series∑ , let denotes the n-th partial sum of the series i.e.
∑ . If the sequence converges, and exists as a real number,
then ∑ is convergent. If the sequence diverges, then the series is divergent.
Geometric series:
∑ , where r is the common ratio and .
The series converges when |r|<1, diverges when | .
Harmonic series:
∑
P-series: The p-series is convergent if p>1 and divergent i.
Theorem:
If the series is convergent, then .
Contrapositive: (Test of divergent)
If ( ) or , then the series is divergent.
Theorem:
∑ ∑ ∑( ) ∑ ∑
Integral Test:
Suppose f is (i) positive, (ii) continuous and (iii) decreasing fun), andon[ ( ), then the
series is convergent if and only if the improper i∫tegral is convergent, i.e.:
1. ∫ ( ) ∑ .
2. ∫ ( ) ∑ .
Note: While using integral test, it is not necessary to start the series or integral at n=1. It is also not
necessary that f is always decreasing. As long as f is ultimately decreasing, then the test can be used.
Error/Estimates of the remainder:
Let ∑ ∑ ,
Suppose ( ) , where f is continuous, positive and decreasing fand ∑ is
convergent.
Then ∫ ( ) ∫ ( ) .
∑ ∫ ( ) ∑ ∫ ( )
More Accurate Estimates: Since , and we know that ∫ ( ) ∫ ( ) .
Then we have [ ∫ ( ) ] [ ∫ ( ) ].
Choose the mid point of [ ∫ ( ) ]and [ ∫ ( ) ] as estimate. (Smaller error)
The Comparison Test:
Suppose that ∑ ∑ are series with positive terms,
1. If ∑ is convergent and , then∑ is also convergent.
2. If ∑ is divergent and , then∑ is also divergent.
Note: Usually we use two types of series to prove convergence.
1. p-series, , convergent whenp>1, divergent when .
2. Geometric series, ∑ , convergent when |r|<1, divergent when| .
Limit Comparison Test:
Suppose ∑ ∑ are series with positive terms. If , where c is a finite number, c>0, then
either both series converge or both series diverge.
Error/Estimating the sum of series
∑ ∑
Suppose ∑
So , then we use to estimate the error bound.
Alternating Series
Definition
A series whose terms are alternatingly positive and negative.
Alternating Series Test
If the alternating series ( ) satisfies (i) ,
(ii) , then the series is convergent. Error/Estimating the sum of series
Alternating Series Estimation Theorem:
If ∑( ) satisfies (i) , (ii) , then| | | | .
Absolute convergence
Definition
A series ∑ is absolutely convergent if the series ∑| | is convergent.
A series ∑ is conditionally convergent if it is convergent but not absolutely convergent.
Theorem
If ∑ is absolutely convergent, then ∑ is convergent. (If ∑| | is convergent, then ∑ is convergent.)
Ratio & Root Test
Ratio Test
| |
If L<1, then ∑ is absolutely convergent.
If L>1, then ∑ is divergent.
If L=1, the ratio test is inconclusive.
Root Test
√ | |
If L<1, then ∑ is absolutely convergent.
If L>1, then ∑ is divergent.
If L=1, the root test is inconclusive.
Note:
1. If we see a factorial(!) sign or the series is recursive i.is defined by , then ratio test is
useful.
2. If is a term raised to the power of n, then consider root test.
3. If the ratio test is inconclusive i.e. L=1, so is the root test. Power Series
Definition
Series in the form of ( ) ( ) ( ) , where are coefficients
and a is an arbitrary constant. It is called the power series centereor a power series about .
Domain of this function f(x) is the set of x for which the power series converges. Power series always
converges at .
Theorem
Given

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