Integration by Parts Logarithms and More Riemann Sums!
|
|
- Edith Hudson
- 5 years ago
- Views:
Transcription
1 Integration by Parts Logarithms and More Riemann Sums! James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 16, 2013
2 Outline 1 IbyP with logarithms 2 Graphing Riemann Sums 3 Uniform Partition Riemann Sums
3 Abstract This lecture is going to talk about the new method of integration Integration by Parts again. Then we go back and talk about Riemann sums in Matlab again so we can learn how to draw a picture that represents the Riemann sum.
4 IbyP with logarithms We can use IbyP to handle ( some polynomial of t ) ln(t) dt. Example: (2t 2 + 4) ln(t)dt. We can complicate it a bit by letting the argument inside the ln be linear. Example: (2t 2 + 4) ln(5t)dt. And we can spin out more cases, but easier to see how it plays out with some examples.
5 IbyP with logarithms Example Evaluate ln(t) dt Solution Let u(t) = ln(t) and dv = dt. Then du = 1 t dt and v = dt = t. For the antiderivative v don t add an arbitrary constant C as we will add one at the end. Applying IbyP, ln(t) dt = udv = uv vdu = ln(t) t t 1 t dt = ln(t) t dt = t ln(t) t + C
6 IbyP with logarithms Example Evaluate t ln(3t) dt Solution Let u(t) = ln(3t) and dv = tdt. Then du = 3 3t dt = 1 t dt and v = tdt = t 2 /2. Using IbyP t ln(3t) dt = udv = uv vdu = ln(3t) t 2 /2 = t2 2 ln(3t) t 2 2 t/2 dt = t2 2 ln(3t) t2 4 + C 1 t dt
7 IbyP with logarithms Example Evaluate t 3 ln(8t) dt Solution Let u(t) = ln(8t) and dv = t 3 dt. Then du = 8 8t dt = 1 t dt and v = t 3 dt = t 4 /4. Applying Integration by Parts, we have t 3 ln(8t) dt = udv = uv vdu = ln(8t) t 4 /4 t 4 /4 1 t dt = t4 4 ln(8t) t 3 /4 dt = t2 2 ln(8t) t C
8 IbyP with logarithms Example Evaluate (t 4 + 5t 2 + 8t + 9) ln(4t) dt Solution Let u(t) = ln(4t) and dv = (t 4 + 5t 2 + 8t + 9)dt. Then du = 4 4t dt = 1 t dt and v = (t 4 + 5t 2 + 8t + 9)dt = t 5 /5 + (5/3)t 3 + (8/2)t 2 + 9t. Applying Integration by Parts, we have (t 4 + 5t 2 + 8t + 9) ln(4t) dt = ln(4t)(t 5 /5 + (5/3)t 3 + (8/2)t 2 + 9t) (t 5 /5 + (5/3)t 3 + (8/2)t 2 + 9t) 1 t dt = ln(4t)(t 5 /5 + (5/3)t 3 + (8/2)t 2 + 9t) (t 4 /5 + (5/3)t 2 + (8/2)t + 9) dt = ln(4t)(t 5 /5 + (5/3)t 3 + (8/2)t 2 + 9t) (t 5 /25 + (5/9)t 3 + (8/4)t 2 + 9t) + C
9 IbyP with logarithms Homework Evaluate ln(5t) dt 26.2 Evaluate 2t ln(t 2 ) dt 26.3 Evaluate (t 4 + 5t 8) ln(7t) dt 26.4 Evaluate 5 2 (t2 + 5t + 3) ln(t) dt
10 Graphing Riemann Sums If we want to graph the Riemann sums, we need to graph those rectangles we draw by hand. To graph a rectangle, we graph 4 lines. The MatLab command plot([x1 x2], [y1 y2]) plots a line from the pair (x1, y1) to (x2, y2). So the command plot([x(i) x(i+1)],[f(s(i)) f(s(i))]); plots the horizontal line which is the top of our rectangle.
11 Graphing Riemann Sums If we want to graph the Riemann sums, we need to graph those rectangles we draw by hand. To graph a rectangle, we graph 4 lines. The MatLab command plot([x1 x2], [y1 y2]) plots a line from the pair (x1, y1) to (x2, y2). So the command plot([x(i) x(i+1)],[f(s(i)) f(s(i))]); plots the horizontal line which is the top of our rectangle. The command plot([x(i) x(i)], [0 f(s(i))]); plots a vertical line that starts on the x axis at x i and ends at the function value f (s i ).
12 Graphing Riemann Sums The command plot([x(i+1) x(i+1)], [0 f(s(i))]); plots a vertical line that starts on the x axis at x i+1 and ends at the function value f (s i ).
13 Graphing Riemann Sums The command plot([x(i+1) x(i+1)], [0 f(s(i))]); plots a vertical line that starts on the x axis at x i+1 and ends at the function value f (s i ). To plot rectangle, for the first pair of partition points, first we set the axis of our plot so we will be able to see it. We use the axis command in Matlab look it up using help! If the two x points are x1 and x2 and the y value is f (s1) where s1 is the first evaluation point, we expand the x axis to [x1 1, x2 + 1] and expand the y axis to [0, f (s1)]. This allows our rectangle to be seen. The command is axis([x1-1 x2+1 0 f((s1))+1]);.
14 Graphing Riemann Sums The code so far Putting this all together, we plot the first rectangle like this: >> h o l d on % s e t a x i s so we can s e e r e c t a n g l e >> a x i s ( [ P( 1 ) 1 P( 2 )+1 0 f ( E ( 1 ) ) +1]) % p l o t top, LHS, RHS and bottom o f r e c t a n g l e >> p l o t ( [ P( 1 ) P( 2 ) ], [ f ( E ( 1 ) ) f ( E ( 1 ) ) ] ) ; >> p l o t ( [ P( 1 ) P( 1 ) ], [ 0 f ( E ( 1 ) ) ] ) ; >> p l o t ( [ P( 2 ) P( 2 ) ], [ 0 f ( E ( 1 ) ) ] ) ; >> p l o t ( [ P( 1 ) P( 2 ) ], [ 0 0 ] ) ; >> h o l d o f f We have to force Matlab to plot repeatedly without erasing the previous plot. We use hold on and hold off to do this. We start with hold on and then all plots are kept until the hold off is used.
15 Graphing Riemann Sums This generates the rectange we see below: Figure: Simple Rectangle
16 Graphing Riemann Sums To show the Riemann sum approximation as rectangles, we use a for loop in MatLab To put this all together, h o l d on % s e t h o l d to on f o r i = 1 : 4 % graph r e c t a n g l e s bottom = 0 ; top = f (E( i ) ) ; p l o t ( [ P( i ) P( i +1) ], [ f ( E ( i ) ) f ( E( i ) ) ] ) ; p l o t ( [ P( i ) P( i ) ], [ bottom top ] ) ; p l o t ( [ E( i ) E ( i ) ], [ bottom top ], r ) ; p l o t ( [ P( i +1) P( i +1) ], [ bottom top ] ) ; p l o t ( [ P( i ) P( i +1) ], [ 0 0 ] ) ; end h o l d o f f % s e t h o l d o f f
17 Graphing Riemann Sums Of course, we don t know if f can be negative, so we need to adjust our thinking as some of the rectangles might need to point down. We do that by setting the bottom and top of the rectangles using an if test.
18 Graphing Riemann Sums Of course, we don t know if f can be negative, so we need to adjust our thinking as some of the rectangles might need to point down. We do that by setting the bottom and top of the rectangles using an if test. bottom = 0 ; top = f (E( i ) ) ; i f f (E( i ) ) < 0 top = 0 ; bottom = f ( E ( i ) ) ; end
19 Graphing Riemann Sums All together, we have h o l d on % s e t h o l d to on [ s i z e P,m] = s i z e (P) ; f o r i = 1 : s i z e P 1 % graph a l l t h e r e c t a n g l e s bottom = 0 ; top = f (E( i ) ) ; i f f (E( i ) ) < 0 top = 0 ; bottom = f ( E ( i ) ) ; end p l o t ( [ P( i ) P( i +1) ], [ f ( E ( i ) ) f ( E( i ) ) ] ) ; p l o t ( [ P( i ) P( i ) ], [ bottom top ] ) ; p l o t ( [ E( i ) E ( i ) ], [ bottom top ], r ) ; p l o t ( [ P( i +1) P( i +1) ], [ bottom top ] ) ; p l o t ( [ P( i ) P( i +1) ], [ 0 0 ] ) ; end h o l d o f f ; We also want to place the plot of f over these rectangles.
20 Graphing Riemann Sums h o l d on % s e t h o l d to on [ s i z e P,m] = s i z e (P) ; f o r i = 1 : s i z e P 1 % graph a l l t h e r e c t a n g l e s bottom = 0 ; top = f (E( i ) ) ; i f f (E( i ) ) < 0 top = 0 ; bottom = f ( E ( i ) ) ; end p l o t ( [ P( i ) P( i +1) ], [ f ( E ( i ) ) f ( E( i ) ) ] ) ; p l o t ( [ P( i ) P( i ) ], [ bottom top ] ) ; p l o t ( [ E( i ) E ( i ) ], [ bottom top ], r ) ; p l o t ( [ P( i +1) P( i +1) ], [ bottom top ] ) ; p l o t ( [ P( i ) P( i +1) ], [ 0 0 ] ) ; end y = l i n s p a c e (P( 1 ),P( s i z e P ), 101) ; p l o t ( y, f ( y ) ) ; x l a b e l ( x a x i s ) ; y l a b e l ( y a x i s ) ; t i t l e ( Riemann Sum w i t h f u n c t i o n o v e r l a i d ) ; h o l d o f f ;
21 Graphing Riemann Sums We generate this figure: Figure: Riemann Sum for f (x) = x 2 for Partition {1, 1.5, 2.1, 2.8, 3.0}
22 Uniform Partition Riemann Sums To save typing, let s learn to use a Matlab function. In Matlab s file menu, choose create a new Matlab function which gives f u n c t i o n [ v a l u e 1, v a l u e 2,... ] = MyFunction ( arg1, arg2,... ) % s t u f f i n h e r e end MyFunction is the name of the function. This function must be stored in the file MyFunction.m.
23 Uniform Partition Riemann Sums To save typing, let s learn to use a Matlab function. In Matlab s file menu, choose create a new Matlab function which gives f u n c t i o n [ v a l u e 1, v a l u e 2,... ] = MyFunction ( arg1, arg2,... ) % s t u f f i n h e r e end [value1, value2,...] are returned values the function calculates that we want to save. MyFunction is the name of the function. This function must be stored in the file MyFunction.m.
24 Uniform Partition Riemann Sums To save typing, let s learn to use a Matlab function. In Matlab s file menu, choose create a new Matlab function which gives f u n c t i o n [ v a l u e 1, v a l u e 2,... ] = MyFunction ( arg1, arg2,... ) % s t u f f i n h e r e end [value1, value2,...] are returned values the function calculates that we want to save. (arg1, arg2,...) are things the function needs to do the calculations. They are called the arguments to the function. MyFunction is the name of the function. This function must be stored in the file MyFunction.m.
25 Uniform Partition Riemann Sums Our function returns the Riemann sum, RS, and use the arguments: our function f, the partition P and the Evaluation set E. Since only one value returned [RS] can be RS. f u n c t i o n RS = RiemannSum ( f, P, E ) % comments alway b e g i n w i t h a % matlab l i n e s h e r e end The name for the function RiemannSum must be used as the file name: i.e. we must use RiemannSum.m as the file name.
26 Uniform Partition Riemann Sums The Riemann sum function: 1 f u n c t i o n RS = RiemannSum ( f, P, E ) % f i n d Riemann sum dx = d i f f (P) ; RS = sum ( f ( E). dx ) ; [ s i z e P,m] = s i z e (P) ; %g e t s i z e o f P a r t i t i o n 6 c l f ; % c l e a r t h e o l d graph h o l d on % s e t h o l d to on f o r i = 1 : s i z e (P) 1 % graph r e c t a n g l e s % p l o t r e c t a n g l e code... end 11 % p l o t f u n c t i o n code... y = l i n s p a c e (P( 1 ),P( s i z e P ), 101) ; h o l d o f f ; end
27 Uniform Partition Riemann Sums Now to see graphically how the Riemann sums converge to a finite number, let s write a new function: Riemann sums using uniform partitions and midpoint evaluation sets. 1 f u n c t i o n RS = RiemannUniformSum ( f, a, b, n ) % s e t up a u n i f o r m p a r t i t i o n w i t h n+1 p o i n t s d e l t a x = ( b a ) /n ; P = [ a : d e l t a x : b ] ; % makes a row v e c t o r f o r i =1:n 6 s t a r t = a+( i 1) d e l t a x ; s t o p = a+i d e l t a x ; E ( i ) = 0. 5 ( s t a r t+s t o p ) ; end % send i n t r a n s p o s e o f P and E so we use column v e c t o r s 11 % b e c a u s e o r i g i n a l RiemannSum f u n c t i o n u s e s columns RS = RiemannSum ( f, P, E ) ; end
28 Uniform Partition Riemann Sums We can then generate a sequence of Riemann sums for different values of n. We generate a sequence of figures which converge to a fixed value. >> f x ) s i n ( 3 x ) ; >> RS = RiemannSumTwo ( f, 1,4, 10) ; >> RS= RiemannSumTwo ( f, 1,4, 20) ; >> RS = RiemannSumTwo ( f, 1,4, 30) ; >> RS= RiemannSumTwo ( f, 1,4, 40) ;
29 Uniform Partition Riemann Sums Figure: The Riemann sum with a uniform partition P 10 of [ 1, 4] for n = 10. The function is sin(3x) and the Riemann sum is
30 Uniform Partition Riemann Sums Figure: Riemann sum with a uniform partition P 20 of [ 1, 4] for n = 20. The function is sin(3x) and the Riemann sum is
31 Uniform Partition Riemann Sums Figure: Riemann sum with a uniform partition P 40 of [ 1, 4] for n = 40. The function is sin(3x) and the Riemann sum is
32 Uniform Partition Riemann Sums Figure: Riemann sum with a uniform partition P 80 of [ 1, 4] for n = 80. The function is sin(3x) and the Riemann sum is
33 Uniform Partition Riemann Sums Homework 27 For the given function f, interval [a, b] and choice of n, you ll calculate the corresponding uniform partition Riemann sum using the functions RiemannSum in file RiemannSum.m and RiemannUniformSum in file RiemannUniformSum.m. You can download these functions as files from the class web site. Save them in your personal class directory. Create a new word document in single space with matlab fragments in bold font. The document starts with your name, MTHSC 111, Section, HW number and the date.
34 Uniform Partition Riemann Sums Homework 27 Continued Create a new word document for this homework. Do the document in single space. Do matlab fragments in bold font. The document starts with your name, MTHSC 111, Section Number, Date and Homework number. For each value of n, do a save as and save the figure with a filename like HW#Problem#a[ ].png where [ ] is where you put the number of the graph. Something like HW17a.png, HW#Problem#b.png etc. Insert this picture into the doc resizing as needed to make it look good. Explain in the doc what the picture shows.
35 Uniform Partition Riemann Sums Homework 27 Continued Something like this: Jim Peterson MTHSC 111, Section Number today s date and HW Number, Problem 1: Let f (t) = sin(5t) on the interval [1, 3] with P = {1, 1.5, 2.0, 2.5, 3.0} and E = {1.2, 1.8, 2.3, 2.8}. % add e x p l a n a t i o n h e r e >> f x ) s i n ( 5 x ) ; % add e x p l a n a t i o n h e r e >> RS = RiemannUniformSum ( f, 1,4,10) % add e x p l a n a t i o n h e r e >> RS = RiemannUniformSum ( f, 1,4,20) % add e x p l a n a t i o n h e r e >> RS = RiemannUniformSum ( f, 1,4,40) % add e x p l a n a t i o n h e r e >> RS = RiemannUniformSum ( f, 1,4,80)
36 Uniform Partition Riemann Sums Homework 26 Continued 26.1 Let f (t) = t 2 2t + 3 on the interval [ 2, 3] with n = 8, 16, 32 and Let f (t) = sin(2t) on the interval [ 1, 5] with n = 10, 40, 60 and Let f (t) = t 2 + 8t + 5 on the interval [ 2, 3] with n = 4, 12, 30 and 50.
Riemann Integration Theory
Riemann Integration Theory James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 3, 2017 Outline 1 Uniform Partition Riemann Sums 2 Refinements
More informationLecture 5b: Starting Matlab
Lecture 5b: Starting Matlab James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University August 7, 2013 Outline 1 Resources 2 Starting Matlab 3 Homework
More informationRiemann Integration. James K. Peterson. February 2, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Riemann Integration James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 2, 2017 Outline 1 Riemann Sums 2 Riemann Sums In MatLab 3 Graphing
More informationRiemann Integration. Outline. James K. Peterson. February 2, Riemann Sums. Riemann Sums In MatLab. Graphing Riemann Sums
Riemann Integration James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 2, 2017 Outline Riemann Sums Riemann Sums In MatLab Graphing
More informationAn Introduction to Matlab
An Introduction to Matlab James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University August 25, 2013 Outline Starting Matlab Matlab Vectors and Functions
More informationRiemann Sums. Outline. James K. Peterson. September 15, Riemann Sums. Riemann Sums In MatLab
Riemann Sums James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 15, 2013 Outline Riemann Sums Riemann Sums In MatLab Abstract This
More informationAntiderivatives! Outline. James K. Peterson. January 28, Antiderivatives. Simple Fractional Power Antiderivatives
Antiderivatives! James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 28, 2014 Outline Antiderivatives Simple Fractional Power Antiderivatives
More informationAntiderivatives! James K. Peterson. January 28, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
! James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 28, 2014 Outline 1 2 Simple Fractional Power Abstract This lecture is going to talk
More informationDefining Exponential Functions and Exponential Derivatives and Integrals
Defining Exponential Functions and Exponential Derivatives and Integrals James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 19, 2014
More informationProject One: C Bump functions
Project One: C Bump functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 2, 2018 Outline 1 2 The Project Let s recall what the
More informationSolving systems of ODEs with Matlab
Solving systems of ODEs with Matlab James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 20, 2013 Outline 1 Systems of ODEs 2 Setting Up
More informationMAT137 - Term 2, Week 2
MAT137 - Term 2, Week 2 This lecture will assume you have watched all of the videos on the definition of the integral (but will remind you about some things). Today we re talking about: More on the definition
More informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationLogarithm and Exponential Derivatives and Integrals
Logarithm and Exponential Derivatives and Integrals James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 3, 2013 Outline 1 Exponential
More informationIntegration by Parts. MAT 126, Week 2, Thursday class. Xuntao Hu
MAT 126, Week 2, Thursday class Xuntao Hu Recall that the substitution rule is a combination of the FTC and the chain rule. We can also combine the FTC and the product rule: d dx [f (x)g(x)] = f (x)g (x)
More informationMathematical Induction Again
Mathematical Induction Again James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 12, 2017 Outline Mathematical Induction Simple POMI Examples
More informationMathematical Induction Again
Mathematical Induction Again James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 2, 207 Outline Mathematical Induction 2 Simple POMI Examples
More informationHölder s and Minkowski s Inequality
Hölder s and Minkowski s Inequality James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 1, 218 Outline Conjugate Exponents Hölder s
More informationMath 112 Section 10 Lecture notes, 1/7/04
Math 11 Section 10 Lecture notes, 1/7/04 Section 7. Integration by parts To integrate the product of two functions, integration by parts is used when simpler methods such as substitution or simplifying
More informationCalculus Dan Barbasch. Oct. 2, Dan Barbasch () Calculus 1120 Oct. 2, / 7
Calculus 1120 Dan Barbasch Oct. 2, 2012 Dan Barbasch () Calculus 1120 Oct. 2, 2012 1 / 7 Numerical Integration Many integrals cannot be computed using FTC because while the definite integral exists because
More informationNewton s Cooling Model in Matlab and the Cooling Project!
Newton s Cooling Model in Matlab and the Cooling Project! James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 10, 2014 Outline Your Newton
More informationOBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph.
4.1 The Area under a Graph OBJECTIVES Use the area under a graph to find total cost. Use rectangles to approximate the area under a graph. 4.1 The Area Under a Graph Riemann Sums (continued): In the following
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.5 (additional techniques of integration), 7.6 (applications of integration), * Read these sections and study solved examples in your textbook! Homework: - review lecture
More informationMathematical Induction
Mathematical Induction James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 12, 2017 Outline Introduction to the Class Mathematical Induction
More informationProject Two. James K. Peterson. March 26, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Project Two James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 26, 2019 Outline 1 Cooling Models 2 Estimating the Cooling Rate k 3 Typical
More informationProject Two. Outline. James K. Peterson. March 27, Cooling Models. Estimating the Cooling Rate k. Typical Cooling Project Matlab Session
Project Two James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 27, 2018 Outline Cooling Models Estimating the Cooling Rate k Typical Cooling
More information1 Introduction; Integration by Parts
1 Introduction; Integration by Parts September 11-1 Traditionally Calculus I covers Differential Calculus and Calculus II covers Integral Calculus. You have already seen the Riemann integral and certain
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.3 (the definite integral +area), 7.4 (FTC), 7.5 (additional techniques of integration) * Read these sections and study solved examples in your textbook! Homework: - review
More informationLinear Systems of ODE: Nullclines, Eigenvector lines and trajectories
Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 203 Outline
More informationDay 2 Notes: Riemann Sums In calculus, the result of f ( x)
AP Calculus Unit 6 Basic Integration & Applications Day 2 Notes: Riemann Sums In calculus, the result of f ( x) dx is a function that represents the anti-derivative of the function f(x). This is also sometimes
More informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 9, 2018 Outline 1 Extremal Values 2
More informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 8, 2017 Outline Extremal Values The
More informationLinear Systems of ODE: Nullclines, Eigenvector lines and trajectories
Linear Systems of ODE: Nullclines, Eigenvector lines and trajectories James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 2013 Outline
More informationSubstitutions and by Parts, Area Between Curves. Goals: The Method of Substitution Areas Integration by Parts
Week #7: Substitutions and by Parts, Area Between Curves Goals: The Method of Substitution Areas Integration by Parts 1 Week 7 The Indefinite Integral The Fundamental Theorem of Calculus, b a f(x) dx =
More informationIntegration by Parts
Calculus 2 Lia Vas 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
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predator - Prey Model Trajectories and the nonlinear conservation law James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 28, 2013 Outline
More information1 Lesson 13: Methods of Integration
Lesson 3: Methods of Integration Chapter 6 Material: pages 273-294 in the textbook: Lesson 3 reviews integration by parts and presents integration via partial fraction decomposition as the third of the
More informationTaylor Polynomials. James K. Peterson. Department of Biological Sciences and Department of Mathematical Sciences Clemson University
James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 24, 2013 Outline 1 First Order Approximation s Second Order Approximations 2 Approximation
More informationComplex Numbers. Outline. James K. Peterson. September 19, Complex Numbers. Complex Number Calculations. Complex Functions
Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline Complex Numbers Complex Number Calculations Complex
More informationComplex Numbers. James K. Peterson. September 19, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline 1 Complex Numbers 2 Complex Number Calculations
More informationHölder s and Minkowski s Inequality
Hölder s and Minkowski s Inequality James K. Peterson Deartment of Biological Sciences and Deartment of Mathematical Sciences Clemson University Setember 10, 2018 Outline 1 Conjugate Exonents 2 Hölder
More informationPredator - Prey Model Trajectories are periodic
Predator - Prey Model Trajectories are periodic James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 4, 2013 Outline 1 Showing The PP
More informationCable Convergence. James K. Peterson. May 7, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Cable Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 2018 Outline 1 Fourier Series Convergence Redux 2 Fourier Series
More informationMAT137 - Term 2, Week 4
MAT137 - Term 2, Week 4 Reminders: Your Problem Set 6 is due tomorrow at 3pm. Test 3 is next Friday, February 3, at 4pm. See the course website for details. Today we will: Talk more about substitution.
More informationRegression and Covariance
Regression and Covariance James K. Peterson Department of Biological ciences and Department of Mathematical ciences Clemson University April 16, 2014 Outline A Review of Regression Regression and Covariance
More informationFinal Problem Set. 2. Use the information in #1 to show a solution to the differential equation ), where k and L are constants and e c L be
Final Problem Set Name A. Show the steps for each of the following problems. 1. Show 1 1 1 y y L y y(1 ) L.. Use the information in #1 to show a solution to the differential equation dy y ky(1 ), where
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.4 (FTC), 7.5 (additional techniques of integration), 7.6 (applications of integration) * Read these sections and study solved examples in your textbook! Homework: - review
More informationMA 1125 Lecture 15 - The Standard Normal Distribution. Friday, October 6, Objectives: Introduce the standard normal distribution and table.
MA 1125 Lecture 15 - The Standard Normal Distribution Friday, October 6, 2017. Objectives: Introduce the standard normal distribution and table. 1. The Standard Normal Distribution We ve been looking at
More informationEvaluating Integrals (Section 5.3) and the Fundamental Theorem of Calculus (Section 1 / 15 (5.4
Evaluating Integrals (Section 5.3) and the Fundamental Theorem of Calculus (Section (5.4) Evaluating Integrals (Section 5.3) and the Fundamental Theorem of Calculus (Section 1 / 15 (5.4 Intro to 5.3 Today
More informationDifferentiating Series of Functions
Differentiating Series of Functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 30, 017 Outline 1 Differentiating Series Differentiating
More informationVariation of Parameters
Variation of Parameters James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 13, 218 Outline Variation of Parameters Example One We eventually
More informationMath 122 Fall Unit Test 1 Review Problems Set A
Math Fall 8 Unit Test Review Problems Set A We have chosen these problems because we think that they are representative of many of the mathematical concepts that we have studied. There is no guarantee
More informationPredator - Prey Model Trajectories are periodic
Predator - Prey Model Trajectories are periodic James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 4, 2013 Outline Showing The PP Trajectories
More informationLecture 5: Integrals and Applications
Lecture 5: Integrals and Applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2012 Lejla Batina Version: spring 2012 Wiskunde 1 1 / 21 Outline The
More informationSolving Linear Systems of ODEs with Matlab
Solving Linear Systems of ODEs with Matlab James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 27, 2013 Outline Linear Systems Numerically
More informationLecture 4: Integrals and applications
Lecture 4: Integrals and applications Lejla Batina Institute for Computing and Information Sciences Digital Security Version: autumn 2013 Lejla Batina Version: autumn 2013 Calculus en Kansrekenen 1 / 18
More informationA Simple Protein Synthesis Model
A Simple Protein Synthesis Model James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 3, 213 Outline A Simple Protein Synthesis Model
More informationSin, Cos and All That
Sin, Cos and All That James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 9, 2017 Outline 1 Sin, Cos and all that! 2 A New Power Rule 3
More informationMAT137 - Term 2, Week 5
MAT137 - Term 2, Week 5 Test 3 is tomorrow, February 3, at 4pm. See the course website for details. Today we will: Talk more about integration by parts. Talk about integrating certain combinations of trig
More informationAssignment 16 Solution. Please do not copy and paste my answer. You will get similar questions but with different numbers!
Assignment 6 Solution Please do not copy and paste my answer. You will get similar questions but with different numbers! Suppose f is a continuous, positive, decreasing function on [, ) and let a n = f
More informationMatrices and Vectors
Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix
More informationAssignment 1b: due Tues Nov 3rd at 11:59pm
n Today s Lecture: n n Vectorized computation Introduction to graphics n Announcements:. n Assignment 1b: due Tues Nov 3rd at 11:59pm 1 Monte Carlo Approximation of π Throw N darts L L/2 Sq. area = N =
More informationSin, Cos and All That
Sin, Cos and All That James K Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 9, 2014 Outline Sin, Cos and all that! A New Power Rule Derivatives
More informationGetting Started With The Predator - Prey Model: Nullclines
Getting Started With The Predator - Prey Model: Nullclines James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 28, 2013 Outline The Predator
More informationINTEGRATION: THE FUNDAMENTAL THEOREM OF CALCULUS MR. VELAZQUEZ AP CALCULUS
INTEGRATION: THE FUNDAMENTAL THEOREM OF CALCULUS MR. VELAZQUEZ AP CALCULUS RECALL: ANTIDERIVATIVES When we last spoke of integration, we examined a physics problem where we saw that the area under the
More informationAP Calculus AB 2nd Semester Homework List
AP Calculus AB 2nd Semester Homework List Date Assigned: 1/4 DUE Date: 1/6 Title: Typsetting Basic L A TEX and Sigma Notation Write the homework out on paper. Then type the homework on L A TEX. Use this
More informationMA 3280 Lecture 05 - Generalized Echelon Form and Free Variables. Friday, January 31, 2014.
MA 3280 Lecture 05 - Generalized Echelon Form and Free Variables Friday, January 31, 2014. Objectives: Generalize echelon form, and introduce free variables. Material from Section 3.5 starting on page
More informationFourier Series Code. James K. Peterson. April 9, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Fourier Series Code James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 9, 2018 Outline 1 We will need to approximate Fourier series expansions
More informationUpper and Lower Bounds
James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University August 30, 2017 Outline 1 2 s 3 Basic Results 4 Homework Let S be a set of real numbers. We
More informationDerivatives and the Product Rule
Derivatives and the Product Rule James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 28, 2014 Outline 1 Differentiability 2 Simple Derivatives
More informationCalculus AB Topics Limits Continuity, Asymptotes
Calculus AB Topics Limits Continuity, Asymptotes Consider f x 2x 1 x 3 1 x 3 x 3 Is there a vertical asymptote at x = 3? Do not give a Precalculus answer on a Calculus exam. Consider f x 2x 1 x 3 1 x 3
More information8.7 MacLaurin Polynomials
8.7 maclaurin polynomials 67 8.7 MacLaurin Polynomials In this chapter you have learned to find antiderivatives of a wide variety of elementary functions, but many more such functions fail to have an antiderivative
More informationThe Method of Laplace Transforms.
The Method of Laplace Transforms. James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 25, 217 Outline 1 The Laplace Transform 2 Inverting
More information6.1 Antiderivatives and Slope Fields Calculus
6. Antiderivatives and Slope Fields Calculus 6. ANTIDERIVATIVES AND SLOPE FIELDS Indefinite Integrals In the previous chapter we dealt with definite integrals. Definite integrals had limits of integration.
More informationChapter 6 Differential Equations and Mathematical Modeling. 6.1 Antiderivatives and Slope Fields
Chapter 6 Differential Equations and Mathematical Modeling 6. Antiderivatives and Slope Fields Def: An equation of the form: = y ln x which contains a derivative is called a Differential Equation. In this
More informationMath 142, Final Exam, Fall 2006, Solutions
Math 4, Final Exam, Fall 6, Solutions There are problems. Each problem is worth points. SHOW your wor. Mae your wor be coherent and clear. Write in complete sentences whenever this is possible. CIRCLE
More information8.5 Taylor Polynomials and Taylor Series
8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES 50 8.5 Taylor Polynomials and Taylor Series Motivating Questions In this section, we strive to understand the ideas generated by the following important questions:
More informationThe Method of Undetermined Coefficients.
The Method of Undetermined Coefficients. James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 24, 2017 Outline 1 Annihilators 2 Finding The
More informationMore Protein Synthesis and a Model for Protein Transcription Error Rates
More Protein Synthesis and a Model for Protein James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 3, 2013 Outline 1 Signal Patterns Example
More informationMECH : a Primer for Matlab s ode suite of functions
Objectives MECH 4-563: a Primer for Matlab s ode suite of functions. Review the fundamentals of initial value problems and why numerical integration methods are needed.. Introduce the ode suite of numerical
More informationVII. Techniques of Integration
VII. Techniques of Integration Integration, unlike differentiation, is more of an art-form than a collection of algorithms. Many problems in applied mathematics involve the integration of functions given
More informationMATH 408N PRACTICE FINAL
2/03/20 Bormashenko MATH 408N PRACTICE FINAL Show your work for all the problems. Good luck! () Let f(x) = ex e x. (a) [5 pts] State the domain and range of f(x). Name: TA session: Since e x is defined
More informationIntegrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61
Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up
More informationChapter 5 Integrals. 5.1 Areas and Distances
Chapter 5 Integrals 5.1 Areas and Distances We start with a problem how can we calculate the area under a given function ie, the area between the function and the x-axis? If the curve happens to be something
More information7.3 Hyperbolic Functions Hyperbolic functions are similar to trigonometric functions, and have the following
Math 2-08 Rahman Week3 7.3 Hyperbolic Functions Hyperbolic functions are similar to trigonometric functions, and have the following definitions: sinh x = 2 (ex e x ) cosh x = 2 (ex + e x ) tanh x = sinh
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationConsequences of Continuity
Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline 1 Domains of Continuous Functions 2 The
More informationEx. Find the derivative. Do not leave negative exponents or complex fractions in your answers.
CALCULUS AB THE SECOND FUNDAMENTAL THEOREM OF CALCULUS AND REVIEW E. Find the derivative. Do not leave negative eponents or comple fractions in your answers. 4 (a) y 4 e 5 f sin (b) sec (c) g 5 (d) y 4
More informationSection 5.6. Integration By Parts. MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10
Section 5.6 Integration By Parts MATH 126 (Section 5.6) Integration By Parts The University of Kansas 1 / 10 Integration By Parts Manipulating the Product Rule d dx (f (x) g(x)) = f (x) g (x) + f (x) g(x)
More informationMore On Exponential Functions, Inverse Functions and Derivative Consequences
More On Exponential Functions, Inverse Functions and Derivative Consequences James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 10, 2019
More informationLearning Scientific Notebook
Learning Scientific Notebook Reading and Writing in Scientific Notebook Viewing On-Screen: Zoom Factor, Invisibles Math and Text Producing the Math Symbols from Toolbars and from the Keyboard Using the
More informationPractice Problems: Integration by Parts
Practice Problems: Integration by Parts Answers. (a) Neither term will get simpler through differentiation, so let s try some choice for u and dv, and see how it works out (we can always go back and try
More information1.4 Techniques of Integration
.4 Techniques of Integration Recall the following strategy for evaluating definite integrals, which arose from the Fundamental Theorem of Calculus (see Section.3). To calculate b a f(x) dx. Find a function
More information1 Antiderivatives graphically and numerically
Math B - Calculus by Hughes-Hallett, et al. Chapter 6 - Constructing antiderivatives Prepared by Jason Gaddis Antiderivatives graphically and numerically Definition.. The antiderivative of a function f
More informationLab 2 Worksheet. Problems. Problem 1: Geometry and Linear Equations
Lab 2 Worksheet Problems Problem : Geometry and Linear Equations Linear algebra is, first and foremost, the study of systems of linear equations. You are going to encounter linear systems frequently in
More informationFourier Sin and Cos Series and Least Squares Convergence
Fourier and east Squares Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 28 Outline et s look at the original Fourier sin
More information5 Integrals reviewed Basic facts U-substitution... 4
Contents 5 Integrals reviewed 5. Basic facts............................... 5.5 U-substitution............................. 4 6 Integral Applications 0 6. Area between two curves.......................
More informationMATHEMATICS FOR ENGINEERING
MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL FURTHER INTEGRATION This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning
More informationSometimes the domains X and Z will be the same, so this might be written:
II. MULTIVARIATE CALCULUS The first lecture covered functions where a single input goes in, and a single output comes out. Most economic applications aren t so simple. In most cases, a number of variables
More informationMAT 1320 Study Sheet for the final exam. Format. Topics
MAT 1320 Study Sheet for the final exam August 2015 Format The exam consists of 10 Multiple Choice questions worth 1 point each, and 5 Long Answer questions worth 30 points in total. Please make sure that
More information