Class Notes (838,076)
Mathematics (1,919)
MATH 138 (181)
Lecture

# Math 138 Notes.pdf

23 Pages
379 Views

School
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
More Less

Related notes for MATH 138
Me

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.