Calculus Notes 5.1: Approximating and Computing Area. How would you find the area of the field with that road as a boundary?
|
|
- Jasmin Watkins
- 6 years ago
- Views:
Transcription
1 Calculus Notes 5.1: Approximating and Computing Area. How would you find the area of the field with that road as a boundary? Say a road runner runs for 2 seconds at 5 m/s, then another second at 15 m/s, 3 more seconds at 10 m/s, and last 2 more seconds at 5 m/s. How far does it travel total?
2 Calculus Notes 5.1: Approximating and Computing Area. )
3 Calculus Notes 5.1: Approximating and Computing Area. 2 R f ( ) = [ ] Example 1: Calculate R 4 and for x x on the interval 1,3. 6 Then calculate,, and M. R 5 5 L 5 Ĝ Which one is an overestimate? Underestimate?
4 Calculus Notes 5.1: Approximating and Computing Area. x = b a N
5 Calculus Notes 5.1: Approximating and Computing Area. L8 8 1 f ( ) [ ] Example 2: Calculate R 8,, and M for x =x on the interval 2,4. ;
6 Calculus Notes 5.1: Approximating and Computing Area. {
7 Calculus Notes 5.1: Approximating and Computing Area. { x = b a N
8 Calculus Notes 5.1: Approximating and Computing Area. {
9 Calculus Notes 5.1: Approximating and Computing Area. Example 1: Use linearity and formulas [3] [5] to rewrite and evaluate the sums. a n b. j( j 1) j= 0 n= 51 {
10 Calculus Notes 5.1: Approximating and Computing Area. Example 2: Calculate the limit for the given function and interval. Verify your answer by using geometry. a. limr f x 9x 0,2 N N ( ) = [ ] b. limln f( x) = x + 2 [ 0,4] N 1 2 )
11 Calculus Notes 5.1: Approximating and Computing Area. R N Example 3: Find a formula for and compute the area under the graph as a limit. 1 a. f x x 2 0,4 2 ( ) = + [ ] ( ) 2 = [ ] b. f x x 1,5 s
12 Calculus Notes 5.2: The Definite Integral From 5.1: When you add up the rectangles, do they all have to be the same width? s For Riemann sum approximations, we relax the requirements that the rectangles have to have equal width.
13 Calculus Notes 5.2: The Definite Integral s
14 Calculus Notes 5.2: The Definite Integral R f P C f( x) = 7 + 2cos 3 ( ) Example 1: Calculate,,, where on P : x0 = 0< x1 = 1< x2 = 1.7< x3 = 3.1< x4 = 4 C : { 0.3,1.1,2.5,3.4 } What is the norm P? s x x [ 0,4]
15 Calculus Notes 5.2: The Definite Integral *
16 Calculus Notes 5.2: The Definite Integral
17 Calculus Notes 5.2: The Definite Integral *
18 Calculus Notes 5.2: The Definite Integral Example 2: Calculate 0 5 ( ) 3 x dx and 5 3 xdx 0 *
19 Calculus Notes 5.2: The Definite Integral *
20 Calculus Notes 5.2: The Definite Integral Example 3: Calculate 4 7 ( 2 + ) x x dx *
21 Calculus Notes 5.3: The Fundamental Theorem of Calculus, Part I Recall from 5.2 Reading Example 5: xdx + xdx = xdx Note that 1 F x x 3 3 ( ) = xdx = ( 3 3 ) xdx = xdx xdx = 7 4 = ( 279 ) = is an antiderivative of 2 x So we can re-write it as: xdx F ( 7) F( 4) =
22 Calculus Notes 5.2: The Definite Integral Example 1: evaluate the integral using FTC I: 0 9 a. 2dx 1 1 ( ) b. 5u u u du 姠 3π 4 c. sin θ d θ π 4
23 Calculus Notes 5.2: The Definite Integral Example 2: Show that the area of the shaded parabolic arch in figure 8 is equal to four-thirds the area of the triangle shown. A 2 4 A+B 2 ( )( ) y = x a b x B 姠
24 Calculus Notes 5.4: The Fundamental Theorem of Calculus, Part II From 5.3 From 5.4 姠
25 Calculus Notes 5.4: The Fundamental Theorem of Calculus, Part II Proof of FTC II: First, use the additive property of the definite integral to write the change in A(x) over [x,x+h]. x+ h x x h A ( x + h) A( x)= ( t) dt f ( t) f dt = f ( t) + a a x In other words, A(x+h)-A(x) is equal to the area of the thin section between the 姠 graph and the x-axis from x to x+h. dt
26 Calculus Notes 5.4: The Fundamental Theorem of Calculus, Part II Proof of FTC II continued: To simplify the rest of the proof, we assume that f(x) is increasing. Then, if h>0, this thin section lies between the two rectangles of heights f(x) and f(x+h). We have limf h 0+ ( x + h) = f ( x) So the Squeeze Theorem gives us: So we have: hf ( x) Area of small rectangle 姠 A( x + h) A( x) hf ( x + h) Area of section Area of big rectangle Now divide by h to squeeze the difference quotient between f(x) and f(x+h) ( x + h) A( x) A f ( x) f + h because f(x) is continuous, and limf h 0 ( x h) ( x) = f ( x) A similar argument shows that for h<0:
27 Calculus Notes 5.4: The Fundamental Theorem of Calculus, Part II Proof of FTC II continued: Equation 1 Equation 2 A ( x) A ( x) = f ( x) Equation 1 and Equation 2 show that exists and. Y
28 Calculus Notes 5.4: The Fundamental Theorem of Calculus, Part II 쥠
29 Calculus Notes 5.4: The Fundamental Theorem of Calculus, Part II F ( 1) F ( 0) π F 4 Example 1: Find,, and, where F ( x) = x tantdt 1 E
30 Calculus Notes 5.4: The Fundamental Theorem of Calculus, Part II Example 2: Find formulas for the functions represented by the integrals. a. x 2 ( t t) dt b. 0 x e t dt 쥠 c. x dt 2 t
31 Calculus Notes 5.4: The Fundamental Theorem of Calculus, Part II Example 3: Calculate the derivative. a. d dθ 0 ( cot u) 1 du b. 2 d x t dx t dt 䢐
32 Calculus Notes 5.4: The Fundamental Theorem of Calculus, Part II Example 4: Explain why the following statements are true. Assume f(x) is differentiable. (a) If c is an inflection point of A(x), then f ( c) = 0 (b) A(x) is concave up if f(x) is increasing. 쥠 (c) A(x) is concave down if f(x) is decreasing.
33 6 0 Calculus Notes 5.5 & 5.6: Net or Total Change as the Integral of a Rate and Substitution Method. v ( t) dt = 6 ( 2 t 3t) 6 2t 3dt 0 = = 1 ( ) 2 2 ( 6) 36 ( ) ( 0) 30 ( ) = = 18m 6 0 쥠 ( t) dt = Position function: s( t) Velocity function: s ( t) = v( t) Acceleration function: s t = v t = at ( ) ( ) ( ) Example 1: Find the total displacement and total distance traveled in the interval [0,6] by a particle moving in a straight line with velocity ( t ) 6 0 v 2t 3dt= 0 = 2t 3m/s t = v v ( t) ( t) 3 < 0fort < 2 3 > 0fort > 2 v =2t 3m/ s 3 2 = tdt + 2t 3dt t 3dt 1.5 2t 3dt ( 3 ) ( 3 ) 5. t t t = t 0 ( 18 ( 225. )) ( ( 225) 0) =. = 22.5
34 Calculus Notes 5.5 & 5.6: Net or Total Change as the Integral of a Rate and Substitution Method. Example 2: The number of cars per hour passing an observation point along a highway is called the traffic flow rate q(t) (in cars per hour). t ( ) a. Which quantity is represented by the integral qt t ( ) b. The flow rate is recorded at 15-minute intervals between 7:00 and 9:00 AM. Estimate the number of cars using the highway during this 2-hour period. 䢐 2 2 qtdt t1 the observation point during the time interval [ t, ] t 1 dt The integral represents the total number of cars that passed 1 t 2
35 Calculus Notes 5.5 & 5.6: Net or Total Change as the Integral of a Rate and Substitution Method. Find the integral of the following: a. ( 2 x x + ) dx b. 2xcos( x ) c. 1+ 2xdx ۵
36 Calculus Notes 5.5 & 5.6: Net or Total Change as the Integral of a Rate and Substitution Method. Find the integral of the following: d. x dx ( ) 2 e. 2 x + 9 cosx dx f. 1+ 2x 2 dx ۵
37 Calculus Notes 5.5 & 5.6: Net or Total Change as the Integral of a Rate and Substitution Method. du = du dx dx so ۶ so This equation is called the Change of Variables Formula.
38 Calculus Notes 5.5 & 5.6: Net or Total Change as the Integral of a Rate and Substitution Method. Example 3a: x x + 6x + dx 1 كا ۵
39 Calculus Notes 5.5 & 5.6: Net or Total Change as the Integral of a Rate and Substitution Method. Example 3b: 2 x x + 6x + dx 1 쥠
40 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay Calculate 1 x dt t bdt = lnx So = lnt = lnb lna= ln b b a a t a x e dx= e x ۶ كا
41 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay 12 dt Example 1: Evaluate the definite integral 2 3t+ 4 쿐
42 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay 1 5 dx Example 2: Evaluate the definite integral x ٧
43 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay Example 3: Evaluate the indefinite integral xdx 4 x + 1 퀠
44 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay Example 4: Evaluate the definite integral x dx ٧
45 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay 큰
46 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay
47 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay Example 5: In the laboratory, the number of Escherichia coli bacteria grows exponentially with growth constant k= 0.41( hours). 1 Assume that 1000 bacteria are present at time t=0. a. Find the formula for the number of bacteria P(t) at time t. b. How large is the population after 5 hours? 타 c. When will the population reach 10,000?
48 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay y = 3y Example 6: Find all solutions of. Which solution satisfies? ( ) y 0 = 9 ઠ٧ Example 7: The isotope radon-222 decays exponentially with a half-life of days. How long will it take for 80% of the isotope to decay?
49 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay Example 8: Pharmacologists have shown that penicillin leaves a person s bloodstream at a rate proportional to the amount present. a. Express this statement as a differential equation. b. Find the decay constant if 50 mg of penicillin remains in the bloodstream 7 hours after an initial injection of 450 mg. 뿐 ۶ c. Under the hypothesis (b), at what time was 200 mg of penicillin present?
50 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay Example 9: A computer virus nicknamed the Sapphire Worm spread throughout the internet on January 25, Studies suggest that during the first few minutes, the population of infected computer hosts increased exponentially with growth constant k= a. What was the doubling time for this virus? b. If the virus began in 4 computers, how many hosts were infected after 2 minutes? 3 minutes? 1 s Example 10: In 1940, a remarkable gallery of prehistoric animal paintings was discovered in the Lascaux cave in Dordogne, France. A charcoal sample from the cave walls had a C to C ratio equal to 15% of that found in the atmosphere. Approximately how old are the paintings? ٩ كا
51 Calculus Notes 5.7 & 5.8: Further Transcendental Functions & Exponential Growth and Decay Example 11: Is it better to receive $2000 today or $2200 in 2 years? Consider r=0.03 and r=0.07. Example 12: Chief Operating Officer Ryan Martinez must decide whether to upgrade his company s computer system. The upgrade 준٩ costs $400,000 and will save $150,000 a year for each of the next 3 years. Is this a good investment if r=7%?
= N. Example 2: Calculate R 8,, and M for x =x on the interval 2,4. Which one is an overestimate? Underestimate?
How would you find the area of the field with that road as a boundary? R R L 5 5 5 R f ( = [ ] Eample : Calculate and for on the interval,. 6 Then calculate,, and M. Say a road runner runs for seconds
More informationGoal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) RECTANGULAR APPROXIMATION METHODS
AP Calculus 5. Areas and Distances Goal: Approximate the area under a curve using the Rectangular Approximation Method (RAM) Exercise : Calculate the area between the x-axis and the graph of y = 3 2x.
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More information1. The accumulated net change function or area-so-far function
Name: Section: Names of collaborators: Main Points: 1. The accumulated net change function ( area-so-far function) 2. Connection to antiderivative functions: the Fundamental Theorem of Calculus 3. Evaluating
More informationWe can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).
4.4 Indefinite Integrals and the Net Change Theorem Because of the relation given by the Fundamental Theorem of Calculus between antiderivatives and integrals, the notation f(x) dx is traditionally used
More informationPart 4: Exponential and Logarithmic Functions
Part 4: Exponential and Logarithmic Functions Chapter 5 I. Exponential Functions (5.1) II. The Natural Exponential Function (5.2) III. Logarithmic Functions (5.3) IV. Properties of Logarithms (5.4) V.
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 informationScience One Integral Calculus
Science One Integral Calculus January 018 Happy New Year! Differential Calculus central idea: The Derivative What is the derivative f (x) of a function f(x)? Differential Calculus central idea: The Derivative
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 information= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds?
Mathematics 115 Professor Alan H. Stein April 18, 005 SOLUTIONS 1. Define what is meant by an antiderivative or indefinite integral of a function f(x). Solution: An antiderivative or indefinite integral
More informationSolving Exponential Equations (Applied Problems) Class Work
Solving Exponential Equations (Applied Problems) Class Work Objective: You will be able to solve problems involving exponential situations. Quick Review: Solve each equation for the variable. A. 2 = 4e
More informationLogarithmic Functions
Metropolitan Community College The Natural Logarithmic Function The natural logarithmic function is defined on (0, ) as ln x = x 1 1 t dt. Example 1. Evaluate ln 1. Example 1. Evaluate ln 1. Solution.
More informationAP Calculus Exam Format and Calculator Tips:
AP Calculus Exam Format and Calculator Tips: Exam Format: The exam is 3 hours and 15 minutes long and has two sections multiple choice and free response. A graphing calculator is required for parts of
More informationAP Calculus AB Winter Break Packet Happy Holidays!
AP Calculus AB Winter Break Packet 04 Happy Holidays! Section I NO CALCULATORS MAY BE USED IN THIS PART OF THE EXAMINATION. Directions: Solve each of the following problems. After examining the form of
More informationUNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test
UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following
More informationExploring Substitution
I. Introduction Exploring Substitution Math Fall 08 Lab We use the Fundamental Theorem of Calculus, Part to evaluate a definite integral. If f is continuous on [a, b] b and F is any antiderivative of f
More informationMath 180, Final Exam, Fall 2007 Problem 1 Solution
Problem Solution. Differentiate with respect to x. Write your answers showing the use of the appropriate techniques. Do not simplify. (a) x 27 x 2/3 (b) (x 2 2x + 2)e x (c) ln(x 2 + 4) (a) Use the Power
More informationCalculus 1 Exam 1 MAT 250, Spring 2011 D. Ivanšić. Name: Show all your work!
Calculus 1 Exam 1 MAT 250, Spring 2011 D. Ivanšić Name: Show all your work! 1. (16pts) Use the graph of the function to answer the following. Justify your answer if a limit does not exist. lim x 2 f(x)
More informationSpring 2015 Sample Final Exam
Math 1151 Spring 2015 Sample Final Exam Final Exam on 4/30/14 Name (Print): Time Limit on Final: 105 Minutes Go on carmen.osu.edu to see where your final exam will be. NOTE: This exam is much longer than
More informationTopics and Concepts. 1. Limits
Topics and Concepts 1. Limits (a) Evaluating its (Know: it exists if and only if the it from the left is the same as the it from the right) (b) Infinite its (give rise to vertical asymptotes) (c) Limits
More informationAnswer Key for AP Calculus AB Practice Exam, Section I
Answer Key for AP Calculus AB Practice Exam, Section I Multiple-Choice Questions Question # Key B B 3 A 4 E C 6 D 7 E 8 C 9 E A A C 3 D 4 A A 6 B 7 A 8 B 9 C D E B 3 A 4 A E 6 A 7 A 8 A 76 E 77 A 78 D
More informationMATH Calculus of One Variable, Part I Spring 2019 Textbook: Calculus. Early Transcendentals. by Briggs, Cochran, Gillett, Schulz.
MATH 1060 - Calculus of One Variable, Part I Spring 2019 Textbook: Calculus. Early Transcendentals. by Briggs, Cochran, Gillett, Schulz. 3 rd Edition Testable Skills Unit 3 Important Students should expect
More information4.1 Analysis of functions I: Increase, decrease and concavity
4.1 Analysis of functions I: Increase, decrease and concavity Definition Let f be defined on an interval and let x 1 and x 2 denote points in that interval. a) f is said to be increasing on the interval
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 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 informationMath 111 Calculus I Fall 2005 Practice Problems For Final December 5, 2005
Math 111 Calculus I Fall 2005 Practice Problems For Final December 5, 2005 As always, the standard disclaimers apply In particular, I make no claims that all the material which will be on the exam is represented
More informationSOLUTIONS FOR PRACTICE FINAL EXAM
SOLUTIONS FOR PRACTICE FINAL EXAM ANDREW J. BLUMBERG. Solutions () Short answer questions: (a) State the mean value theorem. Proof. The mean value theorem says that if f is continuous on (a, b) and differentiable
More informationReview for Final. The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter 4. Below are the topics to study:
Review for Final The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter 4. Below are the topics to study: Chapter 2 Find the exact answer to a limit question by using the
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 information1.1. BASIC ANTI-DIFFERENTIATION 21 + C.
.. BASIC ANTI-DIFFERENTIATION and so e x cos xdx = ex sin x + e x cos x + C. We end this section with a possibly surprising complication that exists for anti-di erentiation; a type of complication which
More informationSummer Term I Kostadinov. MA124 Calculus II Boston University. Evaluate the definite integrals. sin(ln(x)) x
student: Exam I Problem : Evaluate the indefinite integrals 2e x + cos(x) dx 8x 3 + 5 4 x dx Problem 2: Evaluate the definite integrals 4 3 x + x dx π/2 π/6 sin(x) dx Problem 3: Evaluate the definite integrals
More informationAP Calculus BC Chapter 4 (A) 12 (B) 40 (C) 46 (D) 55 (E) 66
AP Calculus BC Chapter 4 REVIEW 4.1 4.4 Name Date Period NO CALCULATOR IS ALLOWED FOR THIS PORTION OF THE REVIEW. 1. 4 d dt (3t 2 + 2t 1) dt = 2 (A) 12 (B) 4 (C) 46 (D) 55 (E) 66 2. The velocity of a particle
More informationMath Exam I - Spring 2008
Math 13 - Exam I - Spring 8 This exam contains 15 multiple choice questions and hand graded questions. The multiple choice questions are worth 5 points each and the hand graded questions are worth a total
More informationQuick Review Sheet for A.P. Calculus Exam
Quick Review Sheet for A.P. Calculus Exam Name AP Calculus AB/BC Limits Date Period 1. Definition: 2. Steps in Evaluating Limits: - Substitute, Factor, and Simplify 3. Limits as x approaches infinity If
More informationMATH 2300 review problems for Exam 1 ANSWERS
MATH review problems for Exam ANSWERS. Evaluate the integral sin x cos x dx in each of the following ways: This one is self-explanatory; we leave it to you. (a) Integrate by parts, with u = sin x and dv
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 informationBusiness and Life Calculus
Business and Life Calculus George Voutsadakis Mathematics and Computer Science Lake Superior State University LSSU Math 2 George Voutsadakis (LSSU) Calculus For Business and Life Sciences Fall 203 / 55
More information4.4 AREAS, INTEGRALS AND ANTIDERIVATIVES
1 4.4 AREAS, INTEGRALS AND ANTIDERIVATIVES This section explores properties of functions defined as areas and examines some of the connections among areas, integrals and antiderivatives. In order to focus
More informationChapter 5 Review. 1. [No Calculator] Evaluate using the FTOC (the evaluation part) 2. [No Calculator] Evaluate using geometry
AP Calculus Chapter Review Name: Block:. [No Calculator] Evaluate using the FTOC (the evaluation part) a) 7 8 4 7 d b) 9 4 7 d. [No Calculator] Evaluate using geometry a) d c) 6 8 d. [No Calculator] Evaluate
More informationy=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions
AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)
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 informationSec 4.1 Limits, Informally. When we calculated f (x), we first started with the difference quotient. f(x + h) f(x) h
1 Sec 4.1 Limits, Informally When we calculated f (x), we first started with the difference quotient f(x + h) f(x) h and made h small. In other words, f (x) is the number f(x+h) f(x) approaches as h gets
More informationAB 1: Find lim. x a.
AB 1: Find lim x a f ( x) AB 1 Answer: Step 1: Find f ( a). If you get a zero in the denominator, Step 2: Factor numerator and denominator of f ( x). Do any cancellations and go back to Step 1. If you
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 informationPurdue University Study Guide for MA Credit Exam
Purdue University Study Guide for MA 16010 Credit Exam Students who pass the credit exam will gain credit in MA16010. The credit exam is a two-hour long exam with multiple choice questions. No books or
More information5.3 Interpretations of the Definite Integral Student Notes
5. Interpretations of the Definite Integral Student Notes The Total Change Theorem: The integral of a rate of change is the total change: a b F This theorem is used in many applications. xdx Fb Fa Example
More informationIntegration by Substitution
Integration by Substitution Dr. Philippe B. Laval Kennesaw State University Abstract This handout contains material on a very important integration method called integration by substitution. Substitution
More informationAB CALCULUS SEMESTER A REVIEW Show all work on separate paper. (b) lim. lim. (f) x a. for each of the following functions: (b) y = 3x 4 x + 2
AB CALCULUS Page 1 of 6 NAME DATE 1. Evaluate each it: AB CALCULUS Show all work on separate paper. x 3 x 9 x 5x + 6 x 0 5x 3sin x x 7 x 3 x 3 5x (d) 5x 3 x +1 x x 4 (e) x x 9 3x 4 6x (f) h 0 sin( π 6
More informationC3 PAPER JUNE 2014 *P43164A0232* 1. The curve C has equation y = f (x) where + 1. (a) Show that 9 f (x) = (3)
PMT C3 papers from 2014 and 2013 C3 PAPER JUNE 2014 1. The curve C has equation y = f (x) where 4x + 1 f( x) =, x 2 x > 2 (a) Show that 9 f (x) = ( x ) 2 2 Given that P is a point on C such that f (x)
More informationMath RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5
Math 201-203-RE - Calculus II Antiderivatives and the Indefinite Integral Page 1 of 5 What is the Antiderivative? In a derivative problem, a function f(x) is given and you find the derivative f (x) using
More informationMath 106 Answers to Exam 3a Fall 2015
Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical
More informationAdd Math (4047/02) Year t years $P
Add Math (4047/0) Requirement : Answer all questions Total marks : 100 Duration : hour 30 minutes 1. The price, $P, of a company share on 1 st January has been increasing each year from 1995 to 015. The
More informationSection 4.4 The Fundamental Theorem of Calculus
Section 4.4 The Fundamental Theorem of Calculus The First Fundamental Theorem of Calculus If f is continuous on the interval [a,b] and F is any function that satisfies F '() = f() throughout this interval
More informationPractice Problems for Test 3 - MTH 141 Fall 2003
Practice Problems for Test - MTH 4 Fall 00 Sections 4., 4., 4., 4.5, 4.6, 4.7, 5., 5., 5., 5.4, 6. NOTE: This is a selection of problems for you to test your skills. The idea is that you get a taste of
More information5 t + t2 4. (ii) f(x) = ln(x 2 1). (iii) f(x) = e 2x 2e x + 3 4
Study Guide for Final Exam 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its expression to be well-defined. Some examples of the conditions are: What
More informationWed. Sept 28th: 1.3 New Functions from Old Functions: o vertical and horizontal shifts o vertical and horizontal stretching and reflecting o
Homework: Appendix A: 1, 2, 3, 5, 6, 7, 8, 11, 13-33(odd), 34, 37, 38, 44, 45, 49, 51, 56. Appendix B: 3, 6, 7, 9, 11, 14, 16-21, 24, 29, 33, 36, 37, 42. Appendix D: 1, 2, 4, 9, 11-20, 23, 26, 28, 29,
More informationMATH 135 Calculus 1 Solutions/Answers for Exam 3 Practice Problems November 18, 2016
MATH 35 Calculus Solutions/Answers for Exam 3 Practice Problems November 8, 206 I. Find the indicated derivative(s) and simplify. (A) ( y = ln(x) x 7 4 ) x Solution: By the product rule and the derivative
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 informationMath 1310 Lab 10. (Sections )
Math 131 Lab 1. (Sections 5.1-5.3) Name/Unid: Lab section: 1. (Properties of the integral) Use the properties of the integral in section 5.2 for answering the following question. (a) Knowing that 2 2f(x)
More informationFall 2009 Math 113 Final Exam Solutions. f(x) = 1 + ex 1 e x?
. What are the domain and range of the function Fall 9 Math 3 Final Exam Solutions f(x) = + ex e x? Answer: The function is well-defined everywhere except when the denominator is zero, which happens when
More informationPRELIM 2 REVIEW QUESTIONS Math 1910 Section 205/209
PRELIM 2 REVIEW QUESTIONS Math 9 Section 25/29 () Calculate the following integrals. (a) (b) x 2 dx SOLUTION: This is just the area under a semicircle of radius, so π/2. sin 2 (x) cos (x) dx SOLUTION:
More informationDistance And Velocity
Unit #8 - The Integral Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Distance And Velocity. The graph below shows the velocity, v, of an object (in meters/sec). Estimate
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2017, WEEK 14 JoungDong Kim Week 14 Section 5.4, 5.5, 6.1, Indefinite Integrals and the Net Change Theorem, The Substitution Rule, Areas Between Curves. Section
More informationBC Exam 2 - Part I 28 questions No Calculator Allowed. C. 1 x n D. e x n E. 0
1. If f x ( ) = ln e A. n x x n BC Exam - Part I 8 questions No Calculator Allowed, and n is a constant, then f ( x) = B. x n e C. 1 x n D. e x n E.. Let f be the function defined below. Which of the following
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 informationLSU AP Calculus Practice Test Day
LSU AP Calculus Practice Test Day AP Calculus AB 2018 Practice Exam Section I Part A AP CALCULUS AB: PRACTICE EXAM SECTION I: PART A NO CALCULATORS ALLOWED. YOU HAVE 60 MINUTES. 1. If y = ( 1 + x 5) 3
More informationCalculus I Announcements
Slide 1 Calculus I Announcements Read sections 4.2,4.3,4.4,4.1 and 5.3 Do the homework from sections 4.2,4.3,4.4,4.1 and 5.3 Exam 3 is Thursday, November 12th See inside for a possible exam question. Slide
More informationSince the exponential function is a continuous function, it suffices to find the limit: lim sin x ln x. sin x ln x = ln x
Math 180 Written Homework Assignment #9 Due Tuesday, November 18th at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180 students,
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.9 Antiderivatives In this section, we will learn about: Antiderivatives and how they are useful in solving certain scientific problems.
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 informationSection 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a
More informationNote: Final Exam is at 10:45 on Tuesday, 5/3/11 (This is the Final Exam time reserved for our labs). From Practice Test I
MA Practice Final Answers in Red 4/8/ and 4/9/ Name Note: Final Exam is at :45 on Tuesday, 5// (This is the Final Exam time reserved for our labs). From Practice Test I Consider the integral 5 x dx. Sketch
More informationLecture : The Indefinite Integral MTH 124
Up to this point we have investigated the definite integral of a function over an interval. In particular we have done the following. Approximated integrals using left and right Riemann sums. Defined the
More informationPH1105 Lecture Notes on Calculus. Main Text: Calculus for the Life Sciences by. Bittenger, Brand and Quintanilla
PH1105 Lecture Notes on Calculus Joe Ó hógáin E-mail: johog@maths.tcd.ie Main Text: Calculus for the Life Sciences by Other Text: Calculus by Bittenger, Brand and Quintanilla Anton, Bivens and Davis 1
More informationCALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): f t dt
CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS d d d d t dt 6 cos t dt Second Fundamental Theorem of Calculus: d f tdt d a d d 4 t dt d d a f t dt d d 6 cos t dt Second Fundamental
More informationApplied Calculus I. Review Solutions. Qaisar Latif. October 25, 2016
Applied Calculus I Review Solutions Qaisar Latif October 25, 2016 2 Homework 1 Problem 1. The number of people living with HIV infections increased worldwide approximately exponentially from 2.5 million
More informationParticle Motion Problems
Particle Motion Problems Particle motion problems deal with particles that are moving along the x or y axis. Thus, we are speaking of horizontal or vertical movement. The position, velocity, or acceleration
More informationScience One Math. January
Science One Math January 10 2018 (last time) The Fundamental Theorem of Calculus (FTC) Let f be continuous on an interval I containing a. 1. Define F(x) = f t dt with F (x) = f(x). on I. Then F is differentiable
More informationy=5 y=1+x 2 AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions
AP Calculus Chapter 5 Testbank Part I. Multiple-Choice Questions. Which of the following integrals correctly corresponds to the area of the shaded region in the figure to the right? (A) (B) (C) (D) (E)
More informationMATH 1207 R02 MIDTERM EXAM 2 SOLUTION
MATH 7 R MIDTERM EXAM SOLUTION FALL 6 - MOON Name: Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () (5 pts) Find
More informationMA 113 Calculus I Fall 2016 Exam 3 Tuesday, November 15, True/False 1 T F 2 T F 3 T F 4 T F 5 T F. Name: Section:
MA 113 Calculus I Fall 2016 Exam 3 Tuesday, November 15, 2016 Name: Section: Last 4 digits of student ID #: This exam has five true/false questions (two points each), ten multiple choice questions (five
More informationA.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the
A.P. Calculus BC Summer Assignment 2018 I am so excited you are taking Calculus BC! For your summer assignment, I would like you to complete the attached packet of problems, and turn it in on Monday, August
More informationApplied Calculus I. Lecture 29
Applied Calculus I Lecture 29 Integrals of trigonometric functions We shall continue learning substitutions by considering integrals involving trigonometric functions. Integrals of trigonometric functions
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 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 informationUnit #6 Basic Integration and Applications Homework Packet
Unit #6 Basic Integration and Applications Homework Packet For problems, find the indefinite integrals below.. x 3 3. x 3x 3. x x 3x 4. 3 / x x 5. x 6. 3x x3 x 3 x w w 7. y 3 y dy 8. dw Daily Lessons and
More informationMath 121 Test 3 - Review 1. Use differentials to approximate the following. Compare your answer to that of a calculator
Math Test - Review Use differentials to approximate the following. Compare your answer to that of a calculator.. 99.. 8. 6. Consider the graph of the equation f(x) = x x a. Find f (x) and f (x). b. Find
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 informationFinal Exam 12/11/ (16 pts) Find derivatives for each of the following: (a) f(x) = 3 1+ x e + e π [Do not simplify your answer.
Math 105 Final Exam 1/11/1 Name Read directions carefully and show all your work. Partial credit will be assigned based upon the correctness, completeness, and clarity of your answers. Correct answers
More informationAP Calculus AB. Review for Test: Applications of Integration
Name Review for Test: Applications of Integration AP Calculus AB Test Topics: Mean Value Theorem for Integrals (section 4.4) Average Value of a Function (manipulation of MVT for Integrals) (section 4.4)
More informationINTERMEDIATE VALUE THEOREM
THE BIG 7 S INTERMEDIATE VALUE If f is a continuous function on a closed interval [a, b], and if k is any number between f(a) and f(b), where f(a) f(b), then there exists a number c in (a, b) such that
More informationIntegrated Calculus II Exam 1 Solutions 2/6/4
Integrated Calculus II Exam Solutions /6/ Question Determine the following integrals: te t dt. We integrate by parts: u = t, du = dt, dv = e t dt, v = dv = e t dt = e t, te t dt = udv = uv vdu = te t (
More informationL. Function Analysis. ). If f ( x) switches from decreasing to increasing at c, there is a relative minimum at ( c, f ( c)
L. Function Analysis What you are finding: You have a function f ( x). You want to find intervals where f ( x) is increasing and decreasing, concave up and concave down. You also want to find values of
More informationScience One Integral Calculus. January 9, 2019
Science One Integral Calculus January 9, 2019 Recap: What have we learned so far? The definite integral is defined as a limit of Riemann sums Riemann sums can be constructed using any point in a subinterval
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 informationMath 1431 Final Exam Review
Math 1431 Final Exam Review Comprehensive exam. I recommend you study all past reviews and practice exams as well. Know all rules/formulas. Make a reservation for the final exam. If you miss it, go back
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 informationSection 4.2 Logarithmic Functions & Applications
34 Section 4.2 Logarithmic Functions & Applications Recall that exponential functions are one-to-one since every horizontal line passes through at most one point on the graph of y = b x. So, an exponential
More informationMath 131 Exam II "Sample Questions"
Math 11 Exam II "Sample Questions" This is a compilation of exam II questions from old exams (written by various instructors) They cover chapters and The solutions can be found at the end of the document
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 information