Indefinite Integration Questions and Answers

O medical equipment industry manufactures X ray machines The unit cost C the cost in dollars to make each X ray machine depends on the number of machines made If x machines are made then the unit cost is given by the function C x 0 7x 210x 35 616 What is the minimum unit cost Do not round your answer

Remember presentation counts You must follow the submission instructions and also make sure your work is easy to follow with each step having all information needed to draw your conclusions and nothing not needed Your conclusions should be clearly stated at each step Sentence answers should be clear and concise 1 Consider the finite state machine whose next state function and output function are given by V a 81 82 86 82 85 NA 83 84 86 84 83 86 NS NG 84 8 and W 81 YN YN 83 YY 84 85 YN a Apply the minimization process to the above machine as follows Determine P and provide a brief reason why states are in the same block of the partiton Determine the remaining partitions in the process being clear at each step to give com plete information but to not include information that is not needed to make your con clusion for each comparison of states Clearly state why you have terminated the process b Let y be a minimal distinguishing string for s4 and se Without determining y what is and how do you know this e Determine a minimal distinguishing string for 82 and ss Be sure to give reasons for your answer d Determine a minimal distinguishing string for s3 and se Be sure to give reasons for your answer e Draw the state diagram for the original machine with starting state Your diagram should have no crossing arrows f Draw the state diagram for the minimized machine Your diagram should have no crossing arrows

Question 2 et f x 8 x x if x 5 2x 9 if 25 etermine the following limits Enter D lim f x 5

1 Watch the video on Ocean Acidification 2 Examine the NOAA Data Viewer for Ocean pH 3 Close the information window that opens in the center of the screen 4 Observe the change in pH on the map as you move the slider in the Ocean pH box from left to right 5 Use the data from global climate models and climate observations over time that relate to the effect of climate change on ocean acidification Aragonite is a name for calcium carbonate the substance that marine organisms make their skeletons and shells out of 1000 a Atmospheric CO Atmospheric CO atm 8 Surface saturation state 800 600 400 200 160 120 80 b Global Ocean pH c Southern Ocean Saturation Aragonite Saturation 2000 2020 8 1 7 9 7 7 2060 2080 2100 Surface pH Year Data Source IPCC Changes in global average surface pH and saturation state with respect to aragonite in the Southern Ocean under various climate model scenarios Time series of a atmospheric CO2 for the six illustrative SRES scenarios b projected global average surface pH and c projected average saturation state in the Southern Ocean from the BERN2 5D EMIC Plattner et al 2001 dentify and describe relationships within the lata sots including changes over time on

a Determine the convergence of the integnal 1 x 11x S 2 by comparing it to the integtal S 2 1 x2 dx dx x d int 2 infty frac 1 x 2 11 int 2 infty frac 1 x 2 d x Since int 2 infty frac 1 x 2 d x quad int 2 infty frac 1 x 2 11 x b If the integral converges find the number to which it converges int 2 infty frac 1 x 2 11 x dx

Determine the end behavior of the graph of the function h x 15x 8x 5x 9x 7 As x 00 h x and as x 0 h x 1 8 X 88 5

Evaluate the improper integral 100 100 SS 00 O A O B O C D dy dx x y 800 3 1 In 2 400 3 1 In 2 4000 3 1 In 2 8000 3 1 In 2

Provide an LU M L factorization of M no row swaps during row reduction 6 48 12 2 12 11 4 3

Below is the graph of the function y f x 11 1 8 1 0 Use Simpson s Rule with 6 subintervals to estimate the area under the curve from 0 to x 6

Evaluate the integral 26 36 24 26 3 A Which trig substitution is correct for this integral Oz 6 tan 0 Oz 26 sin 0 01 6 sin 0 01 36 tan 0 0r 36 sec 0 I 6 sec 0 1113 108 dr B Which integral do you obtain after substituting for and simplifyin Note to enter 8 type the word theta 3 cos 1 cos 0 cos Os 0 do C What is the value of the above integral in terms of 0 C D What is the value of the original integral in terms of x

sec 0 O A 25 24 sin 0 7 25 24 cos 0 tan 0 Values of the remaining trigonometric functions of 0 from the given information cot 8 25 7 24 tan 0 0 25 7 sec 0 25 24 24 7 csc 8 3 OB sin 0 cos 0 tan 0 7 25 csc 0 I sec 8 24 25 25 7 cot 0 24 7 25 24 7 24 O C sin 0 cos 0 tan 0 7 24 csc 0 sec 0 cot 0 7 25 24 25 25 7 25 24 24 7 D sin 0 7 25 cos 8 24 25 tan 0 24 7 25 csc 8 sec 8 cot 0 7 25 24 7 24

1 4 Find the linearization L x of the function at a 1 f x x 3x a 1 2 f x ln x 3 f x cos x a 2 4 f x x 4 a 1 a 16

9 Find if y sin x X U CSX y ay x sinx cos x da X SIAX cos x

b It is given that g x is a periodic function with period 1 i e g x 1 g x for any x Using fundamental theorem of calculus show that i 4 S 0 g x dx 4 1 S g x dx 1 3 ii 9 3x dx 1 2 1 g x dx 0

Small changes in numbers in an integrand can result in very different methods required for integration Consider the integral below Assume b 0 but b could be positive or negative in the answers below 12 1 x214x b dr 1 What value of b would result in a u substitution method where the power rule would be used 2 What is one possible value of b where a u substitution method is used resulting in an arctangent 3 What is one possible value of b that would result in a partial fraction decomposition method being used 4 Evaluate the integral with the b value that results in a partial fraction decomposition method Upload your handwritten work for finding the values of b in questions 1 3 along with the work for evaluating the integral in the box below Show detailed work for evaluating the integral and finding the decomposition

Solve the following inequality 5 x 1 24x

Find the integral 3x 5x 4 dx x 2x x 0111 C In C a Ob In In x 1 2 x 1 0 C 1 2 1 1 1 C C O d C 0 1 C 1x4 C x 1 2

Suppose lim f x 12 lim g x 1 and lim f x lim g x 2 300 Xx I 00 200 lim 2 1 I 100 10g x 8 x 2g x 8

quations Solve the system of equations by the substitu inconsistent or dependent y x 6 x y 3

The table below shows the operating costs in thousands of dollars of a small business from 2000 to 2005 Which model would fit this data best quadratic quartic or logarithmic Using the model write a best fit equation then estimate the operation costs in 2008 Year Costs 2000 2 3 2001 2 6 2002 3 1 2003 3 4 2004 2005 4 5 6

2 An improper integral on an infinite interval Calculate the integral below if it converges If it does not converge enter diverges for your answer 3x e dx

1 8 Find the average value of the function on the given interval 1 f x 3x 8x 1 2 2 f x x 0 4 3 g x 4 f z 5 g t 6 f x 7 h x 3 cos x T 2 2 e z z 9 1 4 9 1 1 2 x 0 2 1 1 3 x 3 cos x sin x 0

A randomized study compared two different systems for tracking baggage at an airport The treatment group using System A reported a mean of 18 5 lost bags per day The treatment group using System B reported a mean of 16 6 lost bags per day To determine if the results are significant the data are re randomized and the differences of the means are shown in the dot plot What is the best conclusion to make based on the data The difference of the means is significant because the re randomizations show that it is outside the range of what could happen by chance The difference of the means is not significant because O the re randomizations show that it is within the range of what could happen by chance The difference of the means is significant because the O re randomizations show that it is within the range of Difference of means

or each of the following integrals find an appropriate trigonometric substitution of the form x f t to simplify the integral 4x 8 2 dx a b X x2 1 522 7 x sqrt 7 5 tan t d C c T 4x sqrt 8 4 sec t x s 126 dx 24x 34 dx x 8x 64x dx

Let Cy be the circle of radius 8 inches centered at the origin in the xy coordinate system Compute the areas of the shaded regions in the pictures below the inner circle in the second picture is the unit c Round your answers to two decimal places A 25 133 A A3 A in y x C8 in in y 1 4 x 1 A3 y x C8 A2 y x C8

If f is continuous the Find and 32 f t dt 14 1 2 intergrul E4 F E5 dt f

a c Evaluate tan xsec xdx 1 1 sec x sec x C 5 1 1 5 tan x tan x C 5 b d 1 1 secx secx C 4 6 1 6 4 tan x tan x C 4 6

a c Evaluate 1 7 3x 6dx In x 3 In x 2 C In x 3 In x 2 C b d In x 3 In x 2 C In x 21 In x 3 C

Compute the indefinite integral of the following fur r t t 5t 8t 5 17 Show all steps

3 points Find a larger integral that converges or a smaller integral that diverges 5 1 x 1 Se da sin a da dr

Suppose P t represents the population of a certain mosquito colony where t is measured in days The current population of the colony is known to be 560 mosquitos that is P 0 560 If P 0 106 mosquitos per day estimate the size of the population in 11 days Mosquitos round to nearest whole number

K Fill in the blank with the appropriate word phra When multiplying two binomials a method tha the method When multiplying two binomials a method tha method

8 1 2 x x x 1 da

13 36 Values of Trigonometric Functions Find the exact value of the trigonometric function 13 cos 150 16 sin 30 19 csc 630 22 sec 120 25 sin 3TT 2 14 sin 240 17 cot 120 20 cot 210 23 tan 750 4TT 26 cos 3 15 tan 330 18 csc 300 21 cos 570 24 cos 660 27 tan 4T 3

41 46 Expressing One Trigonometric Function in Terms of Another Write the first trigonometric function in terms of the second for in the given quadrant 41 tan 8 cos 0 0 in Quadrant III

Find the time it takes for 7 300 to double when invested at an annual interest rate of 8 compounded continuously years Find the time it takes for 730 000 to double when invested at an annual interest rate of 8 compounded continuously years

You know that 15 f t dt 16 0 I f 5 20 f 15 23 Evaluate es e 3ln x dx

3 Bacteria population A bacteria culture initially contains 100 cells and grows at a rate proportional to its size After an hour the population has increased to 420 a Find an expression for the number of bacteria after t hours b Find the number of bacteria after 3 hours c Find the rate of growth after 3 hours d When will the population reach 10 000

60 6 4 7 Exercises Evaluate the following integrals 1 2 6 x x 1 x 3 2x 1 x x 8 da da

1 2 3 sin sin z dr I COS cos 3x dx sin 1 3 dr sin r cos z dz I

Evaluate the following integrals I sin 1 2 sin z dr cos 3x dx

A graph of a function is given Use the graph to find the indicated values y 0 11 a g 1 b g 3 c 9 5 4 Orq g X

A table of values for a one to one function is given Find the indicated value f f 6 4 X f x 1 0 2 2 3 4 4 6 5 1 6 3

What is the sum of the infinite geometric series 3 3 3 2 4 8 93 16 3 032 D 4 O 6 0100 3 3 16

The depth in feet of water at a dock changes with the rise and fall of tides The depth is modeled by the function D t 4 sin T t 3T 2 where t is the number of hours after midnight Find the rate at which the depth is changing at 1 a m Round your answer to 4 decimal places ft hr

Evaluate the indefinite integral 5 2 dz

Evaluate the indefinite integral 4x 4 10 dz dx 4x

Find the following integral Note that you can check your answer your answer by differentiation S 7e 2 Vy dy

2 Evaluate the following indefinite integrals as infinite series a Example of an elliptic integral VI 1 x4dx b Fresnel integral si c Exponential integral de ex dx sin x dx

A central angle is drawn in a circle of radius r as in the figure below Use the following as necessary r and 0 0 A T a The length of the arc subtended by 0 is s b The area of the sector with central angle 0 is A