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Math 138 Notes.pdf

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Department
Mathematics
Course
MATH 138
Professor
Rui Bin Qin
Semester
Fall

Description
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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