Math Calculus I
|
|
- Pierce Harper
- 5 years ago
- Views:
Transcription
1 Math Calculus I Christian Roettger 382 Carver Hall Mathematics Department Iowa State University November 13, 2011
2 4.1 Introduction to Area Sigma Notation 4.2 The Definite Integral 4.3 First Fundamental Theorem of Calculus 4.4 Second Fundamental Theorem of Calculus 4.5 Mean Value Theorem for integrals, symmetry
3 Intro to Area I We all know how to calculate the area of a rectangle, a parallelogram (height x base), a triangle (height x base /2), a polygon (sum of triangles). To do that, we use common sense facts about area. The area of a rectangle is height x base. Congruent regions have equal areas. Lines have area zero. The area of two regions which overlap only in a line is the sum of the area of these regions. If one region is contained in another, then the area of the second is larger than that of the first.
4 Intro to Area II Area by Inscribed Polygons The following area has been evaluated by the ancient Greek mathematician Archimedes 2000 years ago. Consider the region R bounded by the parabola y = x 2, the x-axis and the vertical line x = 2.
5 Intro to Area III We calculate the area of the red rectangles. This will be smaller than A(R), A(r 1 ) + + A(r 4 ) = 0.4( ) = 1.92 A(R). and if we divide [0, 2] into n equal parts of length h, we will get in a similar way A(r 1 ) + + A(r n 1 ) = h(0h 2 + 1h 2 + 4h (n 1) 2 h 2 ) We might guess the limit A(R) lim A(r 1) + + A(r n 1 ) n
6 Intro to Area IV but how do we compute it?? What happens if we take a polygon which contains R?
7 Intro to Area V The yellow rectangles together contain R, and their area is A(s 1 ) A(s 4 ) = 0.4( ) = 2.52 A(R). If we did the same with n rectangles, dividing [0, 2] into n equal parts, we would get A(r 1 ) A(r n ) = h(1h 2 + 4h 2 + 9h n 2 h 2 ) A(R) Visually, the difference between red and yellow rectangles is just that last yellow rectangle! Its area 8/n tends to zero as n.
8 Sigma Notation I A convenient shorthand for handling long sums involving.... Definition For given numbers a 1, a 2, a 3,..., a n, we write n a i := a a n. i=1 The variable i is called index variable, 1 and n are called limits of or bounds for i. These examples show that i can start at any integer, that a i can be the value of a function at i, and that the Sigma notation can make your life if not easier, then at least shorter.
9 Sigma Notation II 6 i = , i=3 3 sin(πk/2) = sin(0) + sin(π/2) + sin(π) + sin(3π/2), k=0 100 r 2 = r=1 Any programmers out there?
10 Sigma Notation III Many programming languages would calculate the last sum somewhat like this. S:=0; for r from 1 to 100 do S:=S+r^2; enddo; Example (constant sequence) n i=1 c = c c }{{} n times = n c.
11 Sigma Notation IV Like D x on functions, Σ can be viewed as a linear operator on finite sequences. Theorem (A - Linearity of Σ) Given finite sequences a 1,..., a n and b 1,..., b n and a constant c, n ca i = c i=1 n (a i + b i ) = i=1 n (a i b i ) = i=1 n i=1 a i n a i + i=1 n a i i=1 n i=1 n i=1 b i b i
12 Sigma Notation V Example Suppose 10 i=1 a i = 3 and 10 i=1 b i = 5. Then n (2a i b i ) = 2 i=1 10 i=1 a i 10 i=1 b i = = 1. Example (telescoping sums) Suppose a finite sequence a 1,..., a n+1 is given. Then n (a i+1 a i ) = (a 2 a 1 ) (a n+1 a n ) = a n+1 a 1. i=1
13 Sums of powers I Example (special sums) n i = i=1 n j 2 = j=1 n(n + 1) 2 n(n + 1)(2n + 1) 6
14 Sums of powers II We can apply these formulas and linearity to work out some more complicated sums. 10 i=1 (i + 1)(i 3) = = = 10 i=1 10 (i 2 2i 3) i 2 2 i= i=1 i 10 i= = 245.
15 Archimedes area I Recall the area between the x-axis, the line x = 2, and y = x 2. Divide [0, 2] into n equal parts of length h. We got n 1 A(r 1 ) + + A(r n 1 ) = h (ih) 2 i=1 3 (n 1)n(2n 1) = h A(R). 6 Using h = 2/n, we can now let n tend to infinity! lim A(r 8(n 1)n(2n 1) 1) + + A(r n ) = lim n n 6n 3 = 8 3.
16 Handshake numbers and little Carl-Friedrich Gauss I Suppose 101 people are in a room and everybody shakes hands with everybody else. Let h be the number of handshakes taking place. Then h = = 5050.
17 Definite Integral I Definition (Riemann Sum) Consider a function f (x) defined on a closed interval [a, b]. Suppose we have a partition P of [a, b], given by a = x 0 < x 1 <... < x n 1 < x n = b, let x i := x i x i 1. In each subinterval [x i 1, x i ], pick an arbitrary sample point x i. Then R P (f ) := n f ( x i ) x i i=1 is called a Riemann sum for f corresponding to P. Riemann sums are approximations to the area between the graph of f (x) and the x-axis. Choosing suitable sample points gives
18 Definite Integral II approximations from below (red rectangles) or above (yellow rectangles). For most examples, we use the regular partition which just divides [a, b] into n equal intervals of constant size h = (b a)/n. Note that the i-th interval in this partition is [a + (i 1)h, a + ih]. Definition The norm of a partition P is the maximal size of one of its intervals. It is written P. For the regular partition E, we have E = h = (b a)/n.
19 Definite Integral III Definition (Definite Integral) Let f be a function defined on [a, b]. If the limit exists, then b a lim R P(f ) P 0 f (x) dx := lim P 0 R P(f ) is called the definite integral of f from a to b. The variable x in the integral is a dummy variable (Las Vegas - what happens in here, stays in here). It could be p, y, z or whatever x 2 dx = y 2 dy = z 2 dz
20 Definite Integral IV The endpoints a, b are often called limits of the integral (they are not really limits!). If f (x) is nonnegative, the definite integral is the area between the graph of f and the x-axis. If f (x) is negative, the definite integral is the negative of the area. If f is both negative and positive, the contributions may cancel each other out. b a f (x) dx = Aup A down
21 Definite Integral V Example Find the total area and the definite integral between the lines y = x + 1, x = 4, x = 3 and the x-axis. Solution We partition [ 4, 1] into n subintervals of length h. The i-th subinterval will be [ 4 + (i 1)h, 4 + ih] with
22 Definite Integral VI h = ( 1 ( 4))/n = 3/n. Then choose x i = 4 + ih. The corresponding Riemann sum is R P (x + 1) = n ( 4 + ih + 1)h = 3nh + h 2 1 n(n + 1) 2 i=1 Substitute h = 3/n and calculate the limit as n to get 1 4 x + 1 dx = = 9 2. Do the same thing with [ 1, 3] to get h = 4/n and 3 1 x + 1 dx = lim n n ( 1 + ih + 1)h i=1 = lim 1 n h2 n(n + 1) = 8. 2
23 Definite Integral VII So the total area is = 12.5 whereas 3 4 x + 1 dx = = 3.5. The next two sections are about the Fundamental Theorem of Calculus. In many cases it allows to calculate these areas much easier and faster, without doing any limits. It is very similar to doing derivatives using rules of differentiation instead of doing limits. But unlike differentiation, this does not work in all cases. Eg for f (x) = sin(x)/x, the Fundamental Theorem of Calculus does not help and approximations are the best we can do. So Riemann sums (and some more sophisticated methods but similarly involving lots of Σ-notation, see section 4.6) are still useful to know.
24 Definite Integral VIII MATLAB demo of Riemann sum with 10 rectangles for sin(x)/x. Shown on top is the approximation , below is a slider where in MATLAB you could change the number of rectangles. Theorem (A - Integrability) If a function f is bounded on an interval [a, b] and if it is continuous there except at a finite number of points, then f is integrable on [a, b]. Example
25 Definite Integral IX This function is not integrable on [0, 1]. { 1 for x > 0 f (x) = x 2 13 for x = 0 Definition (Limits upside down) For a < b, define a b b f (x) dx := f (x) dx. a This definition makes the following theorem work for numbers a, b, c in any order.
26 Definite Integral X Theorem (B - Interval Additivity Property) If f is integrable on an interval and a, b, c are points in that interval, then b c b f (x) dx = f (x) dx + f (x) dx. a a c
27 FFTC I When we work out integrals 1 4 t + 1 dt, 3 4 t + 1 dt, 10 4 t + 1 dt,... we see that we should do this evaluation with a variable upper boundary. Let F (x) := x 4 t + 1 dt and we find after doing our usual Riemann Sum and the limit n F (x) = 1 2 ((x + 1)2 9)
28 FFTC II The First Fundamental Theorem of Calculus does one better - it allows to calculate F (x) defined similarly for many integrands f (x), not just x + 1. Theorem (A - First Fundamental Theorem of Calculus) Let f (x) be continuous on the closed interval [a, b]. Then d dx x a f (t) dt = f (x). In other words... In the situation of Theorem A, consider the function defined by F (x) := x a f (t) dt
29 FFTC III Then F (x) is an antiderivative of f (x)!... and this is the reason behind writing f (x) dx for antiderivatives. We verify the theorem with the example f (x) = x + 1. It gives F (x) = 1 2 (x + 1)2 + C, and this agrees with the result we found above up to the constant C. What is the correct value of C? Here is a trick! F (a) = a a f (x) dx = 0 whatever f (x) may be.
30 FFTC IV In the example above, a = 4, and from F ( 4) = 0 we deduce 0 = 1 2 ( 4 + 1)2 + C so C = 9/2, which agrees with the result found using Riemann sums. This is much easier, faster... plain wonderful Theorem (B - Comparison Property) If f, g are integrable functions on [a, b] such that f (x) g(x) for all x in [a, b], then b a f (x) dx b a g(x) dx
31 FFTC V The Property Comparison Theorem (C - Boundedness Property) If f is integrable on [a, b] and m f (x) M for all x in [a, b], then m(b a) b a f (x) dx M(b a)
32 FFTC VI For the first inequality, take the constant function g(x) = m on [a, b] and apply Theorem B. m(b a) = b a m dx b a f (x) dx. With the constant function h(x) = M, Theorem B gives the second inequality in Theorem C. The textbook has a nice picture about the geometrical meaning of this fact. Theorem (D - Linearity of the Definite Integral) Suppose f (x), g(x) are integrable functions on [a, b] and that k is a constant. Then kf (x) and f (x) + g(x) are integrable and b a kf (x) dx = k b a f (x) dx, b b b
33 FFTC VII - use linearity of the summation operator and of limits.
34 Recall the definition So we have to calculate Proof of FFTC I F (x) := x a f (t) dt. d 1 1 F (x) = lim (F (x + h) F (x)) = lim dx h 0 h h 0 h x+h x f (t) dt. Let m, M be the minimum resp. maximum values of f (t) on the interval [x, x + h]. Note that m, M depend on h. From Theorem C, we get with b a = (x + h) x = h m 1 h x+h x f (t) dt M.
35 Proof of FFTC II Since f (t) is continuous, both m and M must tend to f (x) as h tends to zero. By the Squeeze Theorem for limits, 1 lim h 0 h which concludes the proof. x+h x f (t) dt = f (x)
36 Example Find Examples for FFTC I d dx 4 x tan 2 u cos u du. Solution Flip the limits of the integral, changing the sign. Then apply the FFTC. Example Find d x 2 3t 1 dt. dx 1
37 Examples for FFTC II Solution Attention - the upper limit here is x 2, not x! Define then we are looking for F (u) := u 1 3t 1 dt, d dx F (x 2 ) = d du F (u) d dx u where u = x 2. We get (3u 1) 2x = 2x(3x 2 1).
38 SFTC I The Second Fundamental Theorem of Calculus is a rewording of the First geared towards evaluating definite integrals. Theorem (Second Fundamental Theorem of Calculus) Let f (x) be integrable on [a, b], and let F (t) be any antiderivative of f (x). Then b a f (x) dx = F (b) F (a). Consider both sides of the equation as functions of b. Differentiate both sides to get f (b). So both sides are antiderivatives of f (x), and they differ only by a constant, b a f (x) dx = F (b) F (a) + C.
39 SFTC II Let b = a and you get 0 = C. Example Find b a x r dx. Solution An antiderivative of x r is 1 r+1 x r+1, so b a x r dx = 1 r + 1 (br+1 a r+1 ). Definition (Bracket) There is a shorthand for F (b) F (a) which is useful when F (t) is complicated - define [F (t)] b a := F (b) F (a).
40 Example Find π 0 sin(x) dx. SFTC III Solution An antiderivative of sin(x) is cos(x). So π 0 sin(x) dx = [ cos(t)] π 0 = cos(π) ( cos(0)) = = 2. Check for sign errors - since sin(x) 0, the integral cannot be negative! Example Determine 2 2 3x 2 x + 4 dx.
41 SFTC IV Solution First find an antiderivative. 3x 2 x + 4 dx = x x 2 + 4x + C. Then evaluate the antiderivative between 2 and 2. 2 [ 3x 2 x + 4 dx = x 3 1 ] x 2 + 4x 2 = ( ) ( ( 2) 3 + 4( 2) ) = 32. Note that the even power of x in the antiderivative cancels out. It also does not matter what C we choose, so we chose C = 0. Revision
42 SFTC V To evaluate a definite integral b a f (x) dx first find an antiderivative F (x) of f (x). then use the SFTC, ie take [F (x)] b a. If the integration is easy, you can do both steps in one go. For hard integrals, it sometimes helps to use
43 The Substitution Method I This is again the Chain Rule in reverse gear. Suppose you can rewrite the integrand f (x) as f (x) = g(u(x))u (x) and G(u) is an antiderivative of g(u) then f (x) dx = g(u(x))u (x) dx = g(u) du + C = G(u(x)) + C. In case g(u) = u r we already know this as the Generalized Power Rule! A good way to memorize the Substitution Method is u (x) dx = du
44 The Substitution Method II which dovetails nicely with the Leibniz notation u (x) = du dx. The problem is to find g(u) and u(x). Look out for layered parentheses to find g(u(x))! Look out for factors which could be u (x)! Example (1)
45 The Substitution Method III Find sin( x) x dx Solution Use u(x) = x, expand fraction by factor 2/2. Theorem (B - Substitution Method for Definite Integrals) For the appropriate limits, we memorize x from a to b u from u(a) to u(b) so for definite integrals the substitution method says b a g(u(x))u (x) dx = u(b) u(a) g(u) du
46 Example Find 9 The Substitution Method IV 4 sin( x) x Solution Use u = x, evaluate the antiderivative 2 sin(u) found in Example 1 between u = 2 and u = 3. Change the limits according to x 4 9 u(x) 2 3 so 9 4 sin( x) x dx = dx. sin u du = 2[ cos u] 3 2 = 2(cos 3 cos 2).
47 The Substitution Method V You could also evaluate 2 cos( x) between x = 4 and 9 for the same result. Example (2) Find π/2 0 x sin 3 (x 2 ) cos(x 2 ) dx Solution Use u(x) = sin(x 2 ), introduce factor 2/2. Example (3)
48 The Substitution Method VI Find 1 0 x + 1 (x 2 + 2x + 6) 2 dx Solution Use u(x) = x 2 + 2x + 6, introduce factor 2/2. You should practice with the worked examples until one-line hints like the ones above are all you need to reproduce the whole computation. Example (My Own) Find π 0 sin(sin x) cos x dx.
49 The Substitution Method VII Solution Use u(x) = sin x, g(u) = sin u. Change the limits according to x 0 π u(x) 0 0 so the new limits are sin 0 = 0 and sin π = 0, the integral is zero!
50 MVT for integrals I Theorem (A - Mean Value Thm for Integrals) If f is continuous on [a, b], then there is a number c in (a, b) such that Define for t in [a, b] b a f (x) dx = f (c)(b a). F (t) := t a f (x) dx and apply the Mean Value Theorem to F (t). This says that there exists c in (a, b) such that F (b) F (a) = F (c)(b a)
51 MVT for integrals II and using the SFTC, then F (c) = f (c) we get the stated equality. Definition (Average) The number f (c) in Theorem A is called the average value of f (x) in [a, b]. Example If f (t) is the temperature at time t during a certain day, then there is at least one moment when the temperature equals the average temperature.
52 Symmetry, Periodicity I Theorem (B - Symmetry) If f is an even function, then a a If f is an odd function, then a Example f (x) dx = 2 a a 0 f (x) dx = 0. f (x) dx.
53 Symmetry, Periodicity II Evaluate 2 x sin 4 (x) + x 3 x 4 dx. 2 Solution The first two summands are odd functions, so the integral over these is zero. The integral equals 2 2 x sin 4 (x) + x 3 x 4 dx = = 2 5 Example Find the area of a semicircle of radius x 4 dx [ x 5 ] 2 0 = 64 5.
54 Symmetry, Periodicity III Solution First note that a semicircle of radius 3 is the area between the graph of 9 x 2 and the x-axis. So we may write A = x 2 dx. First, use the symmetry ( 9 x 2 is even), so we get twice the area of a quarter circle. The we need substitution. Substitutions like u(x) = 9 x 2 do not help here, but we can apply substitution in reverse! Use x = 3 sin u (1)
55 Symmetry, Periodicity IV This gives dx = 3 cos u du and new limits u = 0, u = π/2. A = 2 = 2 = π/2 9 x 2 dx 0 π/ sin 2 u3 cos u du cos 2 u du With the trig formula cos 2 u = (1 + cos 2u)/2 we can integrate this as A = 9 π/ cos 2u du = 9 [u + sin(2u)/2] π/2 0 = 9 2 π.
56 Symmetry, Periodicity V I know that you know this area is 9π/2! but we have rediscovered it using an integral. This is a nice way of checking that something you learned in high school is actually true - or at least consistent with the rest of Calculus. Theorem (C - Periodicity) If f is periodic with period p, then b+p a+p f (x) dx = b a f (x) dx. Example
57 Symmetry, Periodicity VI Evaluate 2π 0 sin x dx. Solution Split this integral by integrating from 0 to π, then from π to 2π. The periodicity theorem gives then that both integrals are equal, because sin(x + π) = sin(x).
58 Conclusion I 1. Suppose an object moves along a line with position x(t) at time t. Then the velocity at time t is v(t) = x (t). Conversely, if we know the velocity v(t) and the position at a given time t 0, then x(t) = t t 0 v(s) ds + x(t 0 ). So initial position x(0) and the velocity function v(t) determine position at any time t. 2. The technology projects are excellent! Even if we cannot do this as a homework, glance through it and see where Riemann sums really become exciting - namely as soon as you cannot find antiderivatives.
59 Conclusion II 3. In Calculus II, integrals will be used to work out the area of many different regions in the plane resp. volumes in space.
Integrals. 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 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 informationMA 137 Calculus 1 with Life Science Applications. (Section 6.1)
MA 137 Calculus 1 with Life Science Applications (Section 6.1) Alberto Corso alberto.corso@uky.edu Department of Mathematics University of Kentucky December 2, 2015 1/17 Sigma (Σ) Notation In approximating
More informationAP Calculus AB. Integration. Table of Contents
AP Calculus AB Integration 2015 11 24 www.njctl.org Table of Contents click on the topic to go to that section Riemann Sums Trapezoid Approximation Area Under a Curve (The Definite Integral) Antiderivatives
More informationAP Calculus AB Integration
Slide 1 / 175 Slide 2 / 175 AP Calculus AB Integration 2015-11-24 www.njctl.org Slide 3 / 175 Table of Contents click on the topic to go to that section Riemann Sums Trapezoid Approximation Area Under
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e)...
Math, Exam III November 6, 7 The Honor Code is in effect for this examination. All work is to be your own. No calculators. The exam lasts for hour and min. Be sure that your name is on every page in case
More informationAP Calculus AB. Slide 1 / 175. Slide 2 / 175. Slide 3 / 175. Integration. Table of Contents
Slide 1 / 175 Slide 2 / 175 AP Calculus AB Integration 2015-11-24 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 175 Riemann Sums Trapezoid Approximation Area Under
More informationChapter 4 Integration
Chapter 4 Integration SECTION 4.1 Antiderivatives and Indefinite Integration Calculus: Chapter 4 Section 4.1 Antiderivative A function F is an antiderivative of f on an interval I if F '( x) f ( x) for
More informationProject 1: Riemann Sums
MS 00 Integral Calculus and Differential Equations 1 Project 1: Riemann Sums In this project you prove some summation identities and then apply them to calculate various integrals from first principles.
More informationWe saw in Section 5.1 that a limit of the form. arises when we compute an area.
INTEGRALS 5 INTEGRALS Equation 1 We saw in Section 5.1 that a limit of the form n lim f ( x *) x n i 1 i lim[ f ( x *) x f ( x *) x... f ( x *) x] n 1 2 arises when we compute an area. n We also saw that
More information7.1 Indefinite Integrals Calculus
7.1 Indefinite Integrals Calculus Learning Objectives A student will be able to: Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve differential
More informationM155 Exam 2 Concept Review
M155 Exam 2 Concept Review Mark Blumstein DERIVATIVES Product Rule Used to take the derivative of a product of two functions u and v. u v + uv Quotient Rule Used to take a derivative of the quotient of
More informationdy = f( x) dx = F ( x)+c = f ( x) dy = f( x) dx
Antiderivatives and The Integral Antiderivatives Objective: Use indefinite integral notation for antiderivatives. Use basic integration rules to find antiderivatives. Another important question in calculus
More informationThe Integral of a Function. The Indefinite Integral
The Integral of a Function. The Indefinite Integral Undoing a derivative: Antiderivative=Indefinite Integral Definition: A function is called an antiderivative of a function on same interval,, if differentiation
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 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 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 information() Chapter 8 November 9, / 1
Example 1: An easy area problem Find the area of the region in the xy-plane bounded above by the graph of f(x) = 2, below by the x-axis, on the left by the line x = 1 and on the right by the line x = 5.
More informationMATH 2413 TEST ON CHAPTER 4 ANSWER ALL QUESTIONS. TIME 1.5 HRS.
MATH 1 TEST ON CHAPTER ANSWER ALL QUESTIONS. TIME 1. HRS. M1c Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Use the summation formulas to rewrite the
More information5.5. The Substitution Rule
INTEGRALS 5 INTEGRALS 5.5 The Substitution Rule In this section, we will learn: To substitute a new variable in place of an existing expression in a function, making integration easier. INTRODUCTION Due
More informationCalculus I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationFIRST YEAR CALCULUS W W L CHEN
FIRST YER CLCULUS W W L CHEN c W W L Chen, 994, 28. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 2. (a) (b) (c) (d) (e) (a) (b) (c) (d) (e) (a) (b) (c) (d) (e)...
Math 55, Exam III November 5, The Honor Code is in effect for this examination. All work is to be your own. No calculators. The exam lasts for hour and 5 min. Be sure that your name is on every page in
More informationScience One Integral Calculus. January 8, 2018
Science One Integral Calculus January 8, 2018 Last time a definition of area Key ideas Divide region into n vertical strips Approximate each strip by a rectangle Sum area of rectangles Take limit for n
More informationFinal Exam Review Exercise Set A, Math 1551, Fall 2017
Final Exam Review Exercise Set A, Math 1551, Fall 2017 This review set gives a list of topics that we explored throughout this course, as well as a few practice problems at the end of the document. A complete
More informationMATH 250 TOPIC 13 INTEGRATION. 13B. Constant, Sum, and Difference Rules
Math 5 Integration Topic 3 Page MATH 5 TOPIC 3 INTEGRATION 3A. Integration of Common Functions Practice Problems 3B. Constant, Sum, and Difference Rules Practice Problems 3C. Substitution Practice Problems
More informationThe total differential
The total differential The total differential of the function of two variables The total differential gives the full information about rates of change of the function in the -direction and in the -direction.
More informationArea. A(2) = sin(0) π 2 + sin(π/2)π 2 = π For 3 subintervals we will find
Area In order to quantify the size of a -dimensional object, we use area. Since we measure area in square units, we can think of the area of an object as the number of such squares it fills up. Using this
More informationMath Final Exam Review
Math - Final Exam Review. Find dx x + 6x +. Name: Solution: We complete the square to see if this function has a nice form. Note we have: x + 6x + (x + + dx x + 6x + dx (x + + Note that this looks a lot
More informationCalculus II Lecture Notes
Calculus II Lecture Notes David M. McClendon Department of Mathematics Ferris State University 206 edition Contents Contents 2 Review of Calculus I 5. Limits..................................... 7.2 Derivatives...................................3
More information4.3. Riemann Sums. Riemann Sums. Riemann Sums and Definite Integrals. Objectives
4.3 Riemann Sums and Definite Integrals Objectives Understand the definition of a Riemann sum. Evaluate a definite integral using limits & Riemann Sums. Evaluate a definite integral using geometric formulas
More information5.3 Definite Integrals and Antiderivatives
5.3 Definite Integrals and Antiderivatives Objective SWBAT use properties of definite integrals, average value of a function, mean value theorem for definite integrals, and connect differential and integral
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 informationAdvanced Calculus Questions
Advanced Calculus Questions What is here? This is a(n evolving) collection of challenging calculus problems. Be warned - some of these questions will go beyond the scope of this course. Particularly difficult
More informationSteps for finding area using Summation
Steps for finding area using Summation 1) Identify a o and a 0 = starting point of the given interval [a, b] where n = # of rectangles 2) Find the c i 's Right: Left: 3) Plug each c i into given f(x) >
More informationMath Practice Exam 2 - solutions
C Roettger, Fall 205 Math 66 - Practice Exam 2 - solutions State clearly what your result is. Show your work (in particular, integrand and limits of integrals, all substitutions, names of tests used, with
More informationPuzzle 1 Puzzle 2 Puzzle 3 Puzzle 4 Puzzle 5 /10 /10 /10 /10 /10
MATH-65 Puzzle Collection 6 Nov 8 :pm-:pm Name:... 3 :pm Wumaier :pm Njus 5 :pm Wumaier 6 :pm Njus 7 :pm Wumaier 8 :pm Njus This puzzle collection is closed book and closed notes. NO calculators are allowed
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 informationExample. Evaluate. 3x 2 4 x dx.
3x 2 4 x 3 + 4 dx. Solution: We need a new technique to integrate this function. Notice that if we let u x 3 + 4, and we compute the differential du of u, we get: du 3x 2 dx Going back to our integral,
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 informationReview Sheet for Exam 1 SOLUTIONS
Math b Review Sheet for Eam SOLUTIONS The first Math b midterm will be Tuesday, February 8th, 7 9 p.m. Location: Schwartz Auditorium Room ) The eam will cover: Section 3.6: Inverse Trig Appendi F: Sigma
More informationAP Calculus BC : The Fundamental Theorem of Calculus
AP Calculus BC 415 5.3: The Fundamental Theorem of Calculus Tuesday, November 5, 008 Homework Answers 6. (a) approimately 0.5 (b) approimately 1 (c) approimately 1.75 38. 4 40. 5 50. 17 Introduction In
More informationSample Final Questions: Solutions Math 21B, Winter y ( y 1)(1 + y)) = A y + B
Sample Final Questions: Solutions Math 2B, Winter 23. Evaluate the following integrals: tan a) y y dy; b) x dx; c) 3 x 2 + x dx. a) We use partial fractions: y y 3 = y y ) + y)) = A y + B y + C y +. Putting
More informationFinal Exam SOLUTIONS MAT 131 Fall 2011
1. Compute the following its. (a) Final Exam SOLUTIONS MAT 131 Fall 11 x + 1 x 1 x 1 The numerator is always positive, whereas the denominator is negative for numbers slightly smaller than 1. Also, as
More informationDistance and Velocity
Distance and Velocity - Unit #8 : Goals: The Integral Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite integral and
More informationMAC Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below.
MAC 23. Find the x-value that maximizes the area of the shaded rectangle inscribed in a right triangle below. (x, y) y = 3 x + 4 a. x = 6 b. x = 4 c. x = 2 d. x = 5 e. x = 3 2. Consider the area of the
More information1 1 1 V r h V r 24 r
February egional 8 ) f x x x f x x f ' ' 9 6 ) ) ) 6x x 6 7 N x x x N ' x N ' 6 6 x x x x x x 8 7 9 9 9 V r h V r r 6 y taking the derivative of the volume with respect to time, we can find the rate at
More informationIntegration. 5.1 Antiderivatives and Indefinite Integration. Suppose that f(x) = 5x 4. Can we find a function F (x) whose derivative is f(x)?
5 Integration 5. Antiderivatives and Indefinite Integration Suppose that f() = 5 4. Can we find a function F () whose derivative is f()? Definition. A function F is an antiderivative of f on an interval
More informationCHAPTER 6 VECTOR CALCULUS. We ve spent a lot of time so far just looking at all the different ways you can graph
CHAPTER 6 VECTOR CALCULUS We ve spent a lot of time so far just looking at all the different ways you can graph things and describe things in three dimensions, and it certainly seems like there is a lot
More informationChapter 6 Section Antiderivatives and Indefinite Integrals
Chapter 6 Section 6.1 - Antiderivatives and Indefinite Integrals Objectives: The student will be able to formulate problems involving antiderivatives. The student will be able to use the formulas and properties
More informationLearning Objectives for Math 166
Learning Objectives for Math 166 Chapter 6 Applications of Definite Integrals Section 6.1: Volumes Using Cross-Sections Draw and label both 2-dimensional perspectives and 3-dimensional sketches of the
More informationMain topics for the First Midterm
Main topics for the First Midterm Midterm 2 will cover Sections 7.7-7.9, 8.1-8.5, 9.1-9.2, 11.1-11.2. This is roughly the material from the first five homeworks and and three quizzes. In particular, I
More information1 + lim. n n+1. f(x) = x + 1, x 1. and we check that f is increasing, instead. Using the quotient rule, we easily find that. 1 (x + 1) 1 x (x + 1) 2 =
Chapter 5 Sequences and series 5. Sequences Definition 5. (Sequence). A sequence is a function which is defined on the set N of natural numbers. Since such a function is uniquely determined by its values
More informationMath 230 Mock Final Exam Detailed Solution
Name: Math 30 Mock Final Exam Detailed Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and
More informationINTEGRALS5 INTEGRALS
INTEGRALS5 INTEGRALS INTEGRALS 5.3 The Fundamental Theorem of Calculus In this section, we will learn about: The Fundamental Theorem of Calculus and its significance. FUNDAMENTAL THEOREM OF CALCULUS The
More informationIntegration. Tuesday, December 3, 13
4 Integration 4.3 Riemann Sums and Definite Integrals Objectives n Understand the definition of a Riemann sum. n Evaluate a definite integral using properties of definite integrals. 3 Riemann Sums 4 Riemann
More informationCalculus II Practice Test 1 Problems: , 6.5, Page 1 of 10
Calculus II Practice Test Problems: 6.-6.3, 6.5, 7.-7.3 Page of This is in no way an inclusive set of problems there can be other types of problems on the actual test. To prepare for the test: review homework,
More informationc) xy 3 = cos(7x +5y), y 0 = y3 + 7 sin(7x +5y) 3xy sin(7x +5y) d) xe y = sin(xy), y 0 = ey + y cos(xy) x(e y cos(xy)) e) y = x ln(3x + 5), y 0
Some Math 35 review problems With answers 2/6/2005 The following problems are based heavily on problems written by Professor Stephen Greenfield for his Math 35 class in spring 2005. His willingness to
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 informationSolutions to Math 41 Final Exam December 10, 2012
Solutions to Math 4 Final Exam December,. ( points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. x ln(t + ) dt (a) lim x x (5 points)
More informationExercises given in lecture on the day in parantheses.
A.Miller M22 Fall 23 Exercises given in lecture on the day in parantheses. The ɛ δ game. lim x a f(x) = L iff Hero has a winning strategy in the following game: Devil plays: ɛ > Hero plays: δ > Devil plays:
More informationf(x) g(x) = [f (x)g(x) dx + f(x)g (x)dx
Chapter 7 is concerned with all the integrals that can t be evaluated with simple antidifferentiation. Chart of Integrals on Page 463 7.1 Integration by Parts Like with the Chain Rule substitutions with
More informationChapter 6: The Definite Integral
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 6: The Definite Integral v v Sections: v 6.1 Estimating with Finite Sums v 6.5 Trapezoidal Rule v 6.2 Definite Integrals 6.3 Definite Integrals and Antiderivatives
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 informationMath 221 Exam III (50 minutes) Friday April 19, 2002 Answers
Math Exam III (5 minutes) Friday April 9, Answers I. ( points.) Fill in the boxes as to complete the following statement: A definite integral can be approximated by a Riemann sum. More precisely, if a
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 informationMath 229 Mock Final Exam Solution
Name: Math 229 Mock Final Exam Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and that it
More information3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:
3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable
More information1 Review of di erential calculus
Review of di erential calculus This chapter presents the main elements of di erential calculus needed in probability theory. Often, students taking a course on probability theory have problems with concepts
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Infinite Series 9. Sequences a, a 2, a 3, a 4, a 5,... Sequence: A function whose domain is the set of positive integers n = 2 3 4 a n = a a 2 a 3 a 4 terms of the sequence Begin
More informationCalculus 1: Sample Questions, Final Exam
Calculus : Sample Questions, Final Eam. Evaluate the following integrals. Show your work and simplify your answers if asked. (a) Evaluate integer. Solution: e 3 e (b) Evaluate integer. Solution: π π (c)
More informationChapter 11 - Sequences and Series
Calculus and Analytic Geometry II Chapter - Sequences and Series. Sequences Definition. A sequence is a list of numbers written in a definite order, We call a n the general term of the sequence. {a, a
More informationM152: Calculus II Midterm Exam Review
M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance
More informationOld Math 220 Exams. David M. McClendon. Department of Mathematics Ferris State University
Old Math 0 Exams David M. McClendon Department of Mathematics Ferris State University Last updated to include exams from Spring 05 Contents Contents General information about these exams 4 Exams from 0
More information1.1 Definition of a Limit. 1.2 Computing Basic Limits. 1.3 Continuity. 1.4 Squeeze Theorem
1. Limits 1.1 Definition of a Limit 1.2 Computing Basic Limits 1.3 Continuity 1.4 Squeeze Theorem 1.1 Definition of a Limit The limit is the central object of calculus. It is a tool from which other fundamental
More informationIntegration. Copyright Cengage Learning. All rights reserved.
4 Integration Copyright Cengage Learning. All rights reserved. 1 4.3 Riemann Sums and Definite Integrals Copyright Cengage Learning. All rights reserved. 2 Objectives Understand the definition of a Riemann
More informationProblem set 5, Real Analysis I, Spring, otherwise. (a) Verify that f is integrable. Solution: Compute since f is even, 1 x (log 1/ x ) 2 dx 1
Problem set 5, Real Analysis I, Spring, 25. (5) Consider the function on R defined by f(x) { x (log / x ) 2 if x /2, otherwise. (a) Verify that f is integrable. Solution: Compute since f is even, R f /2
More informationMa 530 Power Series II
Ma 530 Power Series II Please note that there is material on power series at Visual Calculus. Some of this material was used as part of the presentation of the topics that follow. Operations on Power Series
More informationMAC Calculus II Spring Homework #6 Some Solutions.
MAC 2312-15931-Calculus II Spring 23 Homework #6 Some Solutions. 1. Find the centroid of the region bounded by the curves y = 2x 2 and y = 1 2x 2. Solution. It is obvious, by inspection, that the centroid
More informationIntegration. 2. The Area Problem
Integration Professor Richard Blecksmith richard@math.niu.edu Dept. of Mathematical Sciences Northern Illinois University http://math.niu.edu/ richard/math2. Two Fundamental Problems of Calculus First
More information1 5 π 2. 5 π 3. 5 π π x. 5 π 4. Figure 1: We need calculus to find the area of the shaded region.
. Area In order to quantify the size of a 2-dimensional object, we use area. Since we measure area in square units, we can think of the area of an object as the number of such squares it fills up. Using
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.1 (differential equations), 7.2 (antiderivatives), and 7.3 (the definite integral +area) * Read these sections and study solved examples in your textbook! Homework: -
More informationQuestions from Larson Chapter 4 Topics. 5. Evaluate
Math. Questions from Larson Chapter 4 Topics I. Antiderivatives. Evaluate the following integrals. (a) x dx (4x 7) dx (x )(x + x ) dx x. A projectile is launched vertically with an initial velocity of
More informationProblem Worth Score Total 14
MATH 241, Fall 14 Extra Credit Preparation for Final Name: INSTRUCTIONS: Write legibly. Indicate your answer clearly. Revise and clean up solutions. Do not cross anything out. Rewrite the page, I will
More informationAP Calculus AB Summer Math Packet
Name Date Section AP Calculus AB Summer Math Packet This assignment is to be done at you leisure during the summer. It is meant to help you practice mathematical skills necessary to be successful in Calculus
More informationChapter 1: Limits and Continuity
Chapter 1: Limits and Continuity Winter 2015 Department of Mathematics Hong Kong Baptist University 1/69 1.1 Examples where limits arise Calculus has two basic procedures: differentiation and integration.
More informationSolutions to Homework 1
Solutions to Homework 1 1. Let f(x) = x 2, a = 1, b = 2, and let x = a = 1, x 1 = 1.1, x 2 = 1.2, x 3 = 1.4, x 4 = b = 2. Let P = (x,..., x 4 ), so that P is a partition of the interval [1, 2]. List the
More informationAlbertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.
Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2015 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the
More information18.02 Multivariable Calculus Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 3. Double Integrals 3A. Double
More informationThe definite integral gives the area under the curve. Simplest use of FTC1: derivative of integral is original function.
5.3: The Fundamental Theorem of Calculus EX. Given the graph of f, sketch the graph of x 0 f(t) dt. The definite integral gives the area under the curve. EX 2. Find the derivative of g(x) = x 0 + t 2 dt.
More informationIntegration in the Complex Plane (Zill & Wright Chapter 18)
Integration in the omplex Plane Zill & Wright hapter 18) 116-4-: omplex Variables Fall 11 ontents 1 ontour Integrals 1.1 Definition and Properties............................. 1. Evaluation.....................................
More informationMath 261 Final Exam - Practice Problem Solutions. 1. A function f is graphed below.
Math Final Eam - Practice Problem Solutions. A function f is graphed below. f() 8 7 7 8 (a) Find f(), f( ), f(), and f() f() = ;f( ).;f() is undefined; f() = (b) Find the domain and range of f Domain:
More informationChapter 5: Integrals
Chapter 5: Integrals Section 5.3 The Fundamental Theorem of Calculus Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus: Sec. 5.3: The Fundamental Theorem of Calculus Fundamental
More informationDepartment of Mathematical x 1 x 2 1
Contents Limits. Basic Factoring Eample....................................... One-Sided Limit........................................... 3.3 Squeeze Theorem.......................................... 4.4
More informationIndefinite Integration
Indefinite Integration 1 An antiderivative of a function y = f(x) defined on some interval (a, b) is called any function F(x) whose derivative at any point of this interval is equal to f(x): F'(x) = f(x)
More informationMATH 408N PRACTICE FINAL
05/05/2012 Bormashenko MATH 408N PRACTICE FINAL Name: TA session: Show your work for all the problems. Good luck! (1) Calculate the following limits, using whatever tools are appropriate. State which results
More information3.1 Day 1: The Derivative of a Function
A P Calculus 3.1 Day 1: The Derivative of a Function I CAN DEFINE A DERIVATIVE AND UNDERSTAND ITS NOTATION. Last chapter we learned to find the slope of a tangent line to a point on a graph by using a
More information4.10 Dirichlet problem in the circle and the Poisson kernel
220 CHAPTER 4. FOURIER SERIES AND PDES 4.10 Dirichlet problem in the circle and the Poisson kernel Note: 2 lectures,, 9.7 in [EP], 10.8 in [BD] 4.10.1 Laplace in polar coordinates Perhaps a more natural
More informationSolutions for the Practice Final - Math 23B, 2016
olutions for the Practice Final - Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy
More information(c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + 1) at x = 0.
Chapter 7 Challenge problems Example. (a) Find the equation of the tangent line for ln(x + ) at x = 0. (b) Find the equation of the parabola that is tangent to ln(x + ) at x = 0 (i.e. the parabola has
More information