MATH SS124 Sec 39 Concepts summary with examples


 Eleanor Webb
 2 years ago
 Views:
Transcription
1 This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples by yourself, you my sk me if you hve ny question, you cn either sk me directly fter clss, or come to my office, it s C527 WH, during office hours, you my emil me before you come. * Function Nottion nd Intercepts We write y = f (x) to express the fct tht y is function of x. The independent vrible is x, the dependent vrible is y, nd f is the nme of the function. The grph of function hs n intercept where it crosses the horizontl or verticl xis. The horizontl intercept is the vlue of x such tht f (x) = 0. The verticl intercept is the vlue of y when x = 0, which is given by f (0). Exmple 1 The following figure shows the mount of nicotine, N = f (t), in mg, in person s bloodstrem s function of the time, t, in hours, since the person finished smoking cigrette. N (mg) t (hours) () Estimte f (3) nd interpret it in terms of nicotine. (b) About how mny hours hve pssed before the nicotine level is down to 0.1 mg? (c) Wht is the verticl intercept? Wht does it represent in terms of nicotine? (d) If this function hd horizontl intercept, wht would it represent? * Incresing nd Decresing Functions Pge 1 of 15
2 A function f is incresing if the vlues of f (x) increses s x increses. A function f is decresing if the vlues of f (x) decreses s x increses. The grph of n incresing function climbs s we move from left to right. The grph of n decresing function descends s we move from left to right. * Slope nd Rte of Chnge Given two points (x 1, y 1 ) nd (x 2, y 2 ) on the grph of liner function y = f (x). The slope is given by Slope = Rise Run = y x = y 2 y 1. x 2 x 1 * Liner Functions in Generl A liner function hs the form y = f (x) = b + mx. Its grph is line such tht () m is the slope, or the rte of chnge of y with respect to x. (b) b is the verticl intercept or vlues of y when x is zero. If m is positive, then f is incresing. If m is negtive, then f is decresing. The eqution of line of slope m through the point (x 0, y 0 ) is (.k. point slope form) y y 0 = m(x x 0 ). Exmple 2 World grin production ws 1241 million tons in 1975 nd 2048 million tons in 2005, nd hs been incresing t n pproximtely constnt rte. () Find liner function for world grin production, P, in million tons, s function of t, then number of yers since (b) Interpret the slope in terms of grin production. (c) Interpret the verticl intercept in terms of grin production. * Recognizing Dt from Liner Function Vlues of x nd y in tble could come from liner function y = b + mx if the difference in yvlues re constnt for equl differences in x. Pge 2 of 15
3 Exmple 3 Which of the following tbles of vlues could represent liner function? For ech tble tht could represent liner function, find formul for tht function. () (b) (c) x f (x) x g(x) t h(t) * Averge Rte of Chnge For function y = f (x), the Averge Rte of Chnge (AROC) of f between x = nd x = b is given by AROC = y x The units of AROC re units of y per unit of x. f (b) f () =. b Exmple 4 Using n intervl of width 0.01, wht is the formul for the verge rte of chnge(aroc) of p(x) = 0.85 x t x = 8? * Concvity The grph of function is concve up if it bends upwrd s we move left to right, or the AROC is incresing from left to right. The grph of function is concve down if it bends downwrd s we move left to right, or the AROC is decresing from left to right. A line is neither concve up nor concve down. Exmple 5 The following tble gives vlues of g(t). Is g incresing or decresing? Is the grph of g concve up or concve down? t g(t) Pge 3 of 15
4 * The Cost Function The cost function, C(q), gives the totl cost of producing quntity q of some good. The totl costs = Fixed Costs + Vrible Costs, where Fixed Costs re incurred even if nothing is produced nd Vrible Costs depend on how mny units re produced. If C(q) is liner cost function, Fixed costs re represented by the verticl intercept. Mrginl cost is represented by the slope. * The Revenue Function The revenue function, R(q), gives the totl revenue received from firm from selling quntity, q, of some good. If the good sells for price of p per unit, then which is exctly the sme s Revenue = Price Quntity, R = pq. If the price does not depend on the quntity sold, so p is constnt, the grph of revenue s function of q is line through the origin, with slope equl to the price p. The mrginl revenue is lso represented by the slope. * The Profit Function Let π denote the profit, then Profit = Revenue Cost. π = R C. The brekeven point is the point where the profit is zero, or equivlently, revenue equls cost. If the profit function is liner function, then the mrginl profit is represented by the slope. Exmple 6 A compny tht mkes Adirondck chirs hs fixed costs of $5000 nd vrible costs of $30 per chir. The compny sells the chirs for $50 ech. () Find formul for the cost function. (b) Find formul for the revenue function. (c) Find the brekeven point. * The Generl Exponentil Function Pge 4 of 15
5 We sy tht P is n exponentil function of t with bse if P = P 0 t, where P 0 is the initil quntity (when t = 0) nd is the fctor by which P chnges when t increses by 1. If > 1, we hve exponentil growth; if 0 < < 1, we hve exponentil decy. The fctor is given by = 1 + r, where r is the deciml representtion of the percent rte of chnge; r my be positive (for growth) or negtive (for decy). * Comprison Between Liner nd Exponentil Functions A liner function hs constnt rte of chnge. An exponentil function hs constnt percent, or reltive, rte of chnge. Exmple 7 The nnul net sles for chocolte compny in 2008 ws 5.1 billion dollrs. In ech of the following cses, write formul for the nnul net sles, S, of this compny s function of t, where t represents the number of yers fter () The nnul net sles increses by 1.2 billion dollrs per yer. (b) The nnul net sles decreses by 0.4 billion dollrs per yer. (c) The nnul net sles increses by 4.3% per yer. (d) The nnul net sles decreses by 1% per yer. * Recognizing Dt from n Exponentil Function Vlues of t nd P in tble could come from n exponentil function P = P 0 t if rtios of P vlues re constnt for eqully spced t vlues. Exmple 8 Which of the following functions in the following tble could be liner, exponentil, or neither? Find formuls for those functions. x f (x) g(x) h(x) Pge 5 of 15
6 * The Fmilies of Exponentil Functions nd Number e The formul P = P 0 t gives fmily of exponentil functions with prmeters P 0 (the initil quntity) nd (the bse). If > 1, then the function is incresing. If 0 < < 1, then the functions is decresing. The lrger is, the fster the function grows; the closer is to 0, the fster the functions decys. The most commonly used bse is the number e = , which is clled the nturl bse. Properties of the Nturl logrithm ln (AB) = ln A + ln B (Product Rule) ( ) A ln = ln A ln B (Quotient Rule) B ln ( A p ) = p ln A (Power Rule) ln e x = x e ln x = x In ddition, ln 1 = 0 nd ln e = 1. Exmple 9 Solve 7 3 t = 5 2 t for t using nturl logrithms. * Exponentil Functions with Bse e Writing = e k, so k = ln, ny exponentil function cn be written in two forms P = P 0 t or P = P 0 e kt. If > 1, we hve exponentil growth; if 0 < < 1, we hve exponentil decy. If k > 0, we hve exponentil growth; if k < 0, we hve exponentil decy. k is clled the continuous growth or decy rte. * Doubling Time nd HlfLife The doubling time of n exponentilly incresing quntity is the time required for the quntity to double. The hlflife of n exponentilly decying quntity is the time required for the quntity to be reduced by fctor of one hlf. Pge 6 of 15
7 Exmple 10 $5, 000 is deposited into n ccount tht doubles in vlue every 2.5 yers. () Determine the continuous growth rte of the ccount. (b) Use your work from prt () to determine how long it will tke for the ccount to rech vlue of $49, 000. Exmple 11 $5, 000 is deposited into n ccount pying 2.15% interest compunded nnully. () Determine formul for the blnce, P fter t yers. (b) How long will it tke the blnce in the ccount to triple? The derivtive of function t the point A is equl to the slope of the line tngent to the curve t A. Exmple 12 Estimte the derivtive of f (x) t x = 0 grphiclly. y x Exmple 13 Given the grph of f (x) below. Sketch the grph of f (x). y 3 y x x * Definition of the Derivtive Using Averge Rtes Pge 7 of 15
8 For ny function f, we define the derivtive function, f, by f f (x + h) f (x) (x) = lim, h 0 h provided the limit on the right hnd side exists. The function f is sid to be differentible t ny point x t which the derivtive function is defined. We write lim f (x) x c to represent the number pproched by f (x) s x pproches c. Exmple 14 () lim (b) lim x e 0.2x (x 2)(3x 2 ) = x 2 x 2 4 * Wht Does the Second Derivtive Tell Us? f > 0 on n intervl mens f is incresing, so the grph of f is concve up there. f < 0 on n intervl mens f is decresing, so the grph of f is concve down there. Exmple 15 For ech function given in the following tbles, do the signs of the first nd second derivtives of the function pper to be positive or negtive over the given intervl? x () f (x) (b) (c) (d) x g(x) x h(x) x w(x) * Mrginl Anlysis The cost function, C(q), gives the totl cost of producing quntity q of some good. Define Mrginl Cost = MC(q) = C (q), which gives us tht Mrginl Cost C(q + 1) C(q). Pge 8 of 15
9 The revenue function, R(q), gives the totl revenue received from firm from selling quntity, q, of some good. Define Mrginl Revenue = MR(q) = R (q), which gives us tht Mrginl Revenue R(q + 1) R(q). Exmple 16 A compny s cost of producing q liters of chemicl is C(q) dollrs; this quntity cn be sold for R(q) dollrs. Suppose C(2000) = 5930 nd R(2000) = () Wht is the profit t production level of 2000? (b) If MC(2000) = 2.1 nd MR(2000) = 2.5, wht is the pproximte chnge in profit if q is incresed from 2000 to 2001? Should the compny increse or decrese production from q = 2000? (c) If MC(2000) = 4.77 nd MR(2000) = 4.32, should the compny increse or decrese production from q = 2000? * For derivtives of functions, refer to the derivtive worksheet, mke sure you cn remember those useful derivtive formuls by the end of tht worksheet. Exmple 17 Determine the derivtive of P(x) = (3x + 9)7 x + 10e x +7 x 4 * Using the Derivtive Formuls Exmple 18 Find n eqution for the tngent line t x = 1 to the grph of y = x 3 + 2x 2 5x + 7. Sketch the grph of the curve nd its tngent line on the sme xes. Exmple 19 Let h(x) = f (g(x)) nd k(x) = g( f (x)). Use the following figure to estimte () h (1) nd (b) k (2). Pge 9 of 15
10 8 y f (x) g(x) x * Testing For Locl Mxim nd Minim First Derivtive Test for Locl Mxim nd Minim Suppose p is criticl point of continuous function f. Then, s we go from left to right: If f chnges from decresing to incresing t p, then f hs locl minimum t p. If f chnges from incresing to decresing t p, then f hs locl mximum t p. Second Derivtive Test for Locl Mxim nd Minim Suppose p is criticl point of continuous function f, nd f (p) = 0. If f is concve up t p ( f (p) > 0), then f hs locl minimum t p. If f is concve down t p ( f (p) < 0), then f hs locl mximum t p. If the second derivtive test filed, then you need to use first derivtive test to determine the mximum nd minimum. * Concvity nd Inflection Points A point t which the grph of function f chnges concvity is clled n inflection point of f. If p is n inflection point of f, then either f (p) = 0 or f is undefined t p. Exmple 20 Find the criticl points nd inflection points of f (x) = x 3 9x 2 48x + 52, nd then find those locl minimums nd mximums. Pge 10 of 15
11 Exmple 21 Mike is building fence for grden, three sides of the enclosure will be mde up beutiful lttice fencing mteril nd the fourth side will be the wll of his house.given tht he hs 260 feet of fencing mteril vilble wht is the lrgest re he cn enclose? * Left nd RightHnd Sums nd Definite Integrls Let f (t) be function tht is continuous for t b. We divide the intervl [, b] into n equl subdivisions, ech of width t, so t = b n. Let t 0, t 1, t 2,, t n be endpoints of the subdivisions. For lefthnd sum, we use the vlues of the function from the left end of the intervl. For righthnd sum, we use the vlues of the function from the right end of the intervl. Actully, we hve Lefthnd sum = Righthnd sum = n 1 i=0 f (t i ) t = f (t 0 ) t + f (t 1 ) t + + f (t n 1 ) t n f (t i ) t = f (t 1 ) t + f (t 2 ) t + + f (t n ) t i=1 The definite integrl of f from to b, written f (t)dt, is the limit of the lefthnd or righthnd sums with n subdivisions of [, b] s n gets rbitrrily lrge. In other words, f (t)dt = lim n (Lefthnd sum) = lim n ( ) n 1 f (t i ) t i=0 nd f (t)dt = lim n (Righthnd sum) = lim n ( n i=1 f (t i ) t ). Ech of these sums is clled Riemnn sum, f is clled the integrnd, nd nd b re clled the limits of integrtion. Exmple 22 Vlues for function f (t) re in the following tble. Estimte 30 f (t)dt by constructing 20 left nd righthnd sums with n = 5. Pge 11 of 15
12 t f (t) Exmple 23 Given the grph of y = f (t) in the below. Estimte 6 f (t)dt by constructing left nd 0 righthnd sums with t = 2. Drw the corresponding rectngles for both left nd righthnd sums. y t * The Definite Integrl s n Are: When f (x) is Positive When f (x) is positive nd < b: Are under grph of f between nd b = f (x)dx. When f (x) is positive for some xvlues nd negtive for others, nd < b: f (x)dx is the sum of the res bove the xxis, counted positively, nd the res below the xxis, counted negtively. * Are Between Two Curves Given functions f (x) nd g(x) for x b, then Are between grphs of f (x)nd g(x)for x b = f (x) g(x) dx. Exmple 24 Use definite integrl to find the re enclosed by y = 2 + 8x 3x 2 nd y = x. * The Nottion nd Units for the Definite Integrl Pge 12 of 15
13 The unit of mesurement for f (x)dx is the product of the units for f (x) nd the units for x. If f (t) is rte of chnge of quntity, then the Totl chnge in quntity between t = nd t = b is given by f (t)dt. Exmple 25 A cup of coffee t 90 is put into 20 room when t = 0. The coffees s temperture is chnging t rte of r(t) = 7(0.9 t ) per minute, with t in minutes. Estimte the coffee s temperture when t = 10. * The Fundmentl Theorem of Clculus The Fundmentl Theorem of Clculus If F (t) is continuous for t b, then F (t)dt = F(b) F(). In words: The definite integrl of the derivtive of function gives the totl chnge in the function. By The Fundmentl Theorem of Clculus, if you re sked to use ntiderivtives(without using clcultor) to estimte n integrl f (t)dt, first find the ntiderivtive F(t) of f (t), nd then F(b) F() Exmple 26 The grph of derivtive f (x) is shown in the following figure. 1 y x Fill in the tble of vlues for f (x) given tht f (3) = 2. x f (x) 2 Pge 13 of 15
14 Exmple 27 Consider the grph of f (x), NOT f (x), given below: f (x) x 12 () List ll vlues of x where the function f (x) hs n inflection point. (b) List ll vlues of x where the function f (x) hs locl minimum. (c) At x = 1 the function f (x) is incresing, decresing or neither? (d) At x = 2 the function f (x) is concve up, concve down, or neither? (e) Sort the vlues of f ( 2), f (7), nd f (9). Pge 14 of 15
15 * Mrginl Cost nd Chnge in Totl Cost If C (q) is mrginl cost function nd C(0) is the fixed cost, Cost to increse production from units to b units = C(b) C() = Totl vrible cost to produce b units = 0 C (q)dq C (q)dq Totl cost of producing b units = Fixed cost + Totl vrible cost = C(0) + 0 C (q)dq Exmple 28 A business hs mrginl cost given by C (x) = 5 ln(2x 2 + 3) + 10 nd mrginl revenue given by R (x) = 0.2x where x is the number of goods, nd cost nd revenue re in dollrs. () Explin the mening of R (30), no clcultion is necessry, include units. (b) Determine the chnge in profit when production is chnged from 5 to 35 items. Pge 15 of 15
5 Accumulated Change: The Definite Integral
5 Accumulted Chnge: The Definite Integrl 5.1 Distnce nd Accumulted Chnge * How To Mesure Distnce Trveled nd Visulize Distnce on the Velocity Grph Distnce = Velocity Time Exmple 1 Suppose tht you trvel
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationMath 116 Calculus II
Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More information1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x
I. Dierentition. ) Rules. *product rule, quotient rule, chin rule MATH 34B FINAL REVIEW. Find the derivtive of the following functions. ) f(x) = 2 + 3x x 3 b) f(x) = (5 2x) 8 c) f(x) = e2x 4x 7 +x+2 d)
More information1 The Definite Integral As Area
1 The Definite Integrl As Are * The Definite Integrl s n Are: When f () is Positive When f () is positive nd < b: Are under grph of f between nd b = f ()d. Emple 1 Find the re under the grph of y = 3 +
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationAP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review
AP Clculus AB Unit 5 (Ch. 6): The Definite Integrl: Dy Nme o Are Approximtions Riemnn Sums: LRAM, MRAM, RRAM Chpter 6 Review Trpezoidl Rule: T = h ( y + y + y +!+ y + y 0 n n) **Know how to find rectngle
More informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
More informationSection 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40
Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationSection Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?
Section 5.  Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationThe Fundamental Theorem of Calculus, Particle Motion, and Average Value
The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More information1.1 Functions. 0.1 Lines. 1.2 Linear Functions. 1.3 Rates of change. 0.2 Fractions. 0.3 Rules of exponents. 1.4 Applications of Functions to Economics
0.1 Lines Definition. Here re two forms of the eqution of line: y = mx + b y = m(x x 0 ) + y 0 ( m = slope, b = yintercept, (x 0, y 0 ) = some given point ) slopeintercept pointslope There re two importnt
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationChapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the coordinte of ech criticl vlue of g. Show
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More informationROB EBY Blinn College Mathematics Department
ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationPractice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.
Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite
More informationObjectives. Materials
Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the
More informationMath 31S. Rumbos Fall Solutions to Assignment #16
Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)
More informationSections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation
Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationPrep Session Topic: Particle Motion
Student Notes Prep Session Topic: Prticle Motion Number Line for AB Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position,
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationSample Problems for the Final of Math 121, Fall, 2005
Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Theorem Suppose f is continuous
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationAP * Calculus Review
AP * Clculus Review The Fundmentl Theorems of Clculus Techer Pcket AP* is trdemrk of the College Entrnce Emintion Bord. The College Entrnce Emintion Bord ws not involved in the production of this mteril.
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More information( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht
More information( ) Same as above but m = f x = f x  symmetric to yaxis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More informationFirst Semester Review Calculus BC
First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewiseliner function f, for 4, is shown below.
More informationStudent Session Topic: Particle Motion
Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be
More informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls
More information6.5 Numerical Approximations of Definite Integrals
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 6.5 Numericl Approximtions of Definite Integrls Sometimes the integrl of function cnnot be expressed with elementry functions, i.e., polynomil,
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More information0.1 Chapters 1: Limits and continuity
1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97
More informationcritical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0)
Decoding AB Clculus Voculry solute mx/min x f(x) (sometimes do sign digrm line lso) Edpts C.N. ccelertion rte of chnge in velocity or x''(t) = v'(t) = (t) AROC Slope of secnt line, f () f () verge vlue
More informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
More informationSummary Information and Formulae MTH109 College Algebra
Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)
More informationUnit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NONCALCULATOR SECTION
Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NONCALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)
More informationAP Calculus. Fundamental Theorem of Calculus
AP Clculus Fundmentl Theorem of Clculus Student Hndout 16 17 EDITION Click on the following link or scn the QR code to complete the evlution for the Study Session https://www.surveymonkey.com/r/s_sss Copyright
More informationReview on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5.  5.3) Remrks on the course. Slide Review: Sec. 5.5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationDistance And Velocity
Unit #8  The Integrl Some problems nd solutions selected or dpted from HughesHllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationMathematics 19A; Fall 2001; V. Ginzburg Practice Final Solutions
Mthemtics 9A; Fll 200; V Ginzburg Prctice Finl Solutions For ech of the ten questions below, stte whether the ssertion is true or flse ) Let fx) be continuous t x Then x fx) f) Answer: T b) Let f be differentible
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More informationMath 116 Final Exam April 26, 2013
Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 20172018 Tble of contents 1 Antiderivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Antiderivtive Function Definition Let f : I R be function
More informationMath Bootcamp 2012 Calculus Refresher
Mth Bootcmp 0 Clculus Refresher Exponents For ny rel number x, the powers of x re : x 0 =, x = x, x = x x, etc. Powers re lso clled exponents. Remrk: 0 0 is indeterminte. Frctionl exponents re lso clled
More informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.45.9, Sections 6.16.3, nd Sections 7.17.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
More informationB Veitch. Calculus I Study Guide
Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More information1 Functions Defined in Terms of Integrals
November 5, 8 MAT86 Week 3 Justin Ko Functions Defined in Terms of Integrls Integrls llow us to define new functions in terms of the bsic functions introduced in Week. Given continuous function f(), consider
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
More informationTopics for final
Topics for 161.01 finl.1 The tngent nd velocity problems. Estimting limits from tbles. Instntneous velocity is limit of verge velocity. Slope of tngent line is limit of slope of secnt lines.. The limit
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More information3.1 Exponential Functions and Their Graphs
. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255  Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information