Elements of Calculus
Elements of Calculus MA 121
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This 12 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 121 at North Carolina State University taught by Robert Watson in Fall. Since its upload, it has received 11 views. For similar materials see /class/223685/ma-121-north-carolina-state-university in Mathematics (M) at North Carolina State University.
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Date Created: 10/15/15
MAl 21 Elements of Calculus 18 April 2007 Exam 4 Review Solutions Some Solutions to Review 4 Rest to follow Sun 15 APR Instructions Show all work for full credit Refer to any work done on separate pages 1 Definitions and Concepts a Give a function t which satisfies the relationship 1 Another way to interpret the relationship is f x 1 then x x C the anti derivative b Why does it not matter if I forget the C when using de nite U quot Here is an example of what will happen Let s integrate i x dx with x 3x2 The definite integral is computed as Fb 7 Ha where Fx is the antiderivative of x The antiderivative of 3x2 is Fx x3 C Now HZ 7 HO 23 C 7 03 C 8 C 7 C 8 See how the C s can celed out That should always happen So it is the convention to just drop the C altogether Sometimes this can get you into trouble But not in anything we re doing in this class c Find the error in my proof that 0 1 Let x 2x and gx 2x 1 Then f x 2 and g x 2 Antidifferentiating x 2x C and gx 2x C Then x gx So it follows 2x 2x l Subtrating 2x from both sides of the equation I find 0 l C is not a variable We don t know that the values of C in x and gx are the same d What does it mean for an improper integral to converge Diverge An improper integral has one of the limits of integration at ioo So to compute LOO xjdx I write this instead as 131quot I x dx If this limit exists the integral converges If not it diverges 9 Give an example of a function fx such that the area bounded by the xaxis x x 0 and x 1 can be determined exactly using one rectangle as opposed to an infinite series of rectangles Let x 3 Then the region itself is a rectangle f Give an example of a function fx such that I x 3x2 2x I want a function such that its antiderivative is 3x2 2x So I take the derivative of 3x2 2x This is 6x 2 2 Find f x dx for each of the following a x 3x2 5xl Using the power rule for antiderivatives add one to power then divide by new power lgetFx x3 gx2x C b x fdx441dx44mlxlc l c x lln3x5 Here is a substitution problem Let u ln3x the inside function Then if So we see the following flln3xjj5dxfu5dx qudu lgu6 C 1EUTLBXJJ s C 9x2 3 d39 X1 33 Let u 3x3 3x Then 9x2 3 So we see the following 39X Zj3 cdxfdxfllldu lnlul C WM 3le c 9 x Notice x X Then using the power rule for antiderivatives Fx x C 9quot Find Ii x dx for each of the following a x x2 x x x2 x so Fx l x3 15x2 Then HZ 7 HO g 2 14 b x xexz To find the antiderivative of x fxexz dx 1 will need to use substitution Let u x2 the inside function Then if 2x Note that I can rewrite the integral as 1x271 x2 1 duu 1 nilulxz like dxijfbce dxijfae dxijfe duije e Solget Fx ljexzt Then H217 FOj1j64 7 4 Find the area of the region bounded by the xaxis and the curve x 702 4 First I notice that the limits of integration have not been given to me but I know that the region will be bounded at the points where the curve crosses the x axis So first I solve x 0 This gives me x i2 If x 702 4 then by the power rule for antiderivatives Fx 7x3 4x FKZJiFPZJ 8 8163 5 Find the volume of the solid constructed by rotating the above region about the x axis We derived a very handy formula in class for this The volume is given by V I 7Tfx2dx From the last problem I know the limits of integration are i2 And 71 is a constant so I can pull it out of the integral So I have v n szzJ 4120 zxugxzwadx WEXS 7 x3 16x1l32 w321 811627 32 78J1672m 7512 157T 6 Find the area of the region bounded by the curves x and gx x2 First I need to know my limits of integration I know the region is bounded by the points where the two curves meet so I will find their intersections If x2 then clearly x 0 is a solution Also by squaring both sides I get x x4 Then 1 x3 So x l is a solution So I know the limits of integration are 0 and l Next I observe that on the interval 01 the curve x is the higher curve and gx x2 is the lower curve So the area is given by 0 x 7 gx dx A I x 7 gxdx A I ixzdx 3 A xz7 gx31ll 7 Consider the following supply function 5x and demand function Dx where price is a function of quantity 5x x272x2 Dx 72x 11 a Find the equillibrium point This is x such that 5x Dx So I solve x2 7 2x 2 72x 11 to get x2 9 or x i3 Now x can t be negative by a restricted domain argument with respect to the application so lonly consider the case when x 3 When x 3 then 53 DB 5 The equillibrium point is 35 b Find the consumer s surplus at this point The consumer s surplus is given by I Dx dx 7 QP where Q is the quantity and P is the price First Q x 3 at the equilibrium point So P DB 5 Then I have surplus L Dxdx 7 QP fg72x11dx7 315 7x2llxllg 715 79 331715 9 c Find the producer s surplus at this point The producer s surplus is given by QP 7 L Sxdx where Q is the quantity and P is the price Recall Q x 3 at the equilibrium point Alos P 53 5 Then I have surplus QP 7 I Sxdx 315 7 3x2 72x2dx 15 7 lgxg 7x2 ijlg 15 7 6 9 8 Find the area under the piecewise defined function from x 0 to x 99 l ifxgEO X1 2 ifxgt50 I start by attempting to compute 9 xjdx I have one complication though x has two meanings depending on the value of x But I know I can split the integral So Iwill do that in such a way that it is clear what I need to antidifferentiate First 9 x dx I x dx IE x dx For the left integral it is clear I should use 1 for x as x is always less than or equal to 50 For the right integral it is clear I should use 2 for x as x is always greater than 50 So I have JZquot xde 12 xjdx 1 xjdx 130 1 dx 1de 013quot 2x133 501141987100 148 9 Find the amount of continuous money ow in which 5000 per year is being invested at 6 com pounded continuously for 45 years I can compute the amount of continuous money ow by calculating JSS 500063906t dt Do you remember why that is funds 135 5000e06tdt 5000135 eOtht 5000ampe390645 7 5000ampe06t135 m 115664431 I m rich 10 In 1897 the world s consumption of Watsoniuml 99 was 55 000 ft3 The amount used has steadily increased at an average rate of 10 per year What is the consumption rate in 2007 What was the total amount of Watsoniuml 99 mn nmed7 First I need to establish my model Clearly we have an exponential growth situation so I use wt Cekt where wt is the amount of Watsoniuml 99 consumed at year t C the initial amount is 55000 k is 10 So I have wt 550006391t Then w110 is the amount consumed 110 years since 1897 or in 2007 w110 m 3 293 077 794 The total amount consumed in the 110 year time span is Iglowmdt This is given a5 total fgmwmdt f1 55000e tdt 550001O e tdt 55000e39 l2l 55000e1110 7 e10 m 32930227943 11 Find xjdx for each of the following c xjdx 7 um bldx 7 ham 1 x 131 lnxH dx E1nb470dx 00 The improper integral diverges 00 7 139 b 1 1 flxjdx 13 J1 72dquot 7 139 1b 13quot l1d lt 71W 71m ibaoo 1 The improper integral converges C x e x b 0 xjdx 111 f1 e xdx 7 lim 7 7X13 ibaoo e l1dx lim 7 7b 71 ib m 6 6 dx 7 1m 71 l babe ebedx l The irnproper integral converges 12 Find 211 quiji where x 2x on the closed interval 35 This is the limit definition of the definite integral equivalent to 2xdx The anti derivative of x which we ll call Rx is x2 Then HS 7 H3 25 7 9 l6 MA 121 Elements of Calculus 18 April 2007 Exam 4 Review You should also be able to do WebAssign 10 It is not due until after the exam but some problems from the assignment show up here lwould work on 10 before Wednesday This is a review examination The actual exam will not be as long It is intended to help you review important concepts and give you an idea of what kind of questions may be asked but not the questions themselvesl Inst 1 N 901 ructions Show all work for full credit Refer to any work done on separate pages Definitions and Concepts df a 7 1 b Why does it not matter if I forget the C when using de nite integration a Give a function t which satisfies the relationship c Find the error in my proof that 0 1 Let x 2x and gx 2x 1 Then f x x 2x C and gx 2x C Then x gx from both sides of the equation I find 0 1 and g x 2 Antidifferentiating 2 So it follows 2x 2x 1 Subtrating 2x d What does it mean for an improper integral to converge Diverge 9 Give an example of a function fx such that the area bounded by the xaxis x x 0 and x 1 can be determined exactly using one rectangle as opposed to an infinite series of rectangles f Give an example of a function fx such that I x 3x2 2x Find f x dx for each of the following a x 3x2 5x1 b x c x lln3x5 d x 1731 9 x J Find Ii x dx for each of the following a x x2 x b x xex2 Find the area of the region bounded by the xaxis and the curve x 7x2 4 Find the volume of the solid constructed by rotating the above region about the xaxis Find the area of the region bounded by the curves x and x x2 N 0 Consider the following supply function 5x and demand function Dx where price is a function of quantity 5x x272x2 Dx 72x 11 a Find the equillibrium point b Find the consumer s surplus at this point c Find the producer s surplus at this point Find the area under the piecewise defined function from x 0 to x 99 if x g 50 X1 if x gt 50 1 2 Find the amount of continuous money ow in which 5000 per year is being invested at 6 com pounded continuously for 45 years In 1897 the world s consumption of Watsoniuml 99 was 55 000 ft3 The amount used has steadily increased at an average rate of 10 per year What is the consumption rate in 2007 What was the total amount of Watsoniuml 99 consumed Find xjdx for each of the following x x 72 x 64 Find 1quotquot quotuse a b c 211 fx1Axi where x 2x on the closed interval 35 1 9quot 5 a Simplify X MA121 Elements of Calculus Exam 1 Review Questions September 6 2008 Definitions and Concepts a Give an example of an equation in which the value of y varies inversely with the value of x b Give an example of a function whose domain is xlx y 4 Give an example of a function whose domain is xlx 2 4 Give an example of a rational function which has a vertical asymptote at x 3 Give an example of a graph of a function which is continuous on the interval 0 5 What is the difference between continuity at a point and continuity over an interval morass If the limit as x approaches 0 from the left is the same as the limit as x approaches 0 from the right what can I say about the general limit h True or False If true give a short statement as to why If false provide a counterexample If a function is continuous at x 0 then it must be continous on the interval 71 2y 3 J 2 X3 y 213 b Simplify x zy zjg bczyg jZ d Express x 7 3 x 2 as a quadratic of the form sz Bx C What is the degree of the resulting polynomial What are the x and y intercepts e Simplify W What is the domain of the function What are the locations of any asymptotes or holes Consider the function fx x2 7 l a Compute K72 b Compute the difference quotient c What is the average rate of change as x changes from 2 to 7 lim f 0 3H d Compute x ls x continuous at x 0 Why or why not Find the equilibrium point for the given demand and supply functions Consider only values which make sense with respect to the application Dlv17v13 5v1v27v73 5 S N Consider the piecewise function defined below x2 ifX7 4 ftx1 8 if x 4 a Evaluate 4 b Evaluate im xj C Is x continuous at x 3 What about x 4 d Is x continuous on the interval 0 10 Why or why not Evaluate 1m 1 a39 xal x7 lim 1 b39 xgt1x lim xaoo x7 C N um x 716 d39 xgt4 x74 e umhx2h2xx3 39 hgt0 f um 5x2719x2 3H x273 Watson Enterprises is considering producing and selling catproof keyboards They plan to sell each keyboard for 40 dollars Materials for each keyboard cost 20 dollars and the machines to produce the keyboards cost 100 000 dollars The machines must only be purchased once a Set up the linear function Cx which gives the total cost of producing x keyboards The cost per unit is the cost of the materials plus an equal fraction of the cost of the machine So for instance if five keyboards are produced the cost per keyboard is 20 dollars plus onefifth of the cost of the machine 20 000 dollars So the cost per unit if x units are produced is given by M0 1ooooo b What is the cost per unit if 10 keyboards are produced um MW c Compute X m d What does your answer to c tell us about the cost per unit m mama Emma m w Mnme 1 IZIEZIWWNMMW m my gm 5 mm m gm L 15 g y mmmmmmmam u
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