Calculus Questions

a Find the inverse of the function below on the given interval and write it in the form y f x b Verify the relationships f f x x and f f x x f x x 36 for x 0 a f x

Find the following derivative x 1 Inx d dx dx 6 x 1 Inx

3 d 1 Use x e to explain why In x for x 0 dx X First explain why x must be greater than zero What is the range of the function f y e A 0 00 B 0 00 C 0 00 D 1 00 Which of the following must be true about the value of x if x e A If y is negative then x is also negative B xze C X20 D X 0 explain why In x dx X Begin by solving the equation x e for y

d Show that Inkx d Compute the derivative of the natural log of kx Begin by letting u kx and substituting u into the equation In kx d d dx In x given that x 0 and k 0 is a real number dx

Find the inverse of the function f x x 5 The inverse function of f x x 5 is f x 165465 2010 2018 ALDE LE BUR CONSER DEPRESS Il es CERVE Gestal PAR 1000 S MEREN SELINS PRYLAR

K 1 Use x e to explain why Inx for x 0 X d dx First explain why x must be greater than zero What is the range of the function f y OA 0 00 OB 0 00 O C 00 00 OD 1 00

K Use the following table to determine the indicated derivatives or state that the derivative cannot be determined 6 0 1 5 3 4 3 6 8 x f x 7 1 3 f x 2 3 2 3 2 a f 4 b f 1 c 3 d f 3 a Choose the correct answer below and if necessary fill in the answer box within your choice A 4 3 Type an integer or a fraction OB 4 cannot be determined b Choose the correct answer below and if necessary fill in the answer box within your choice OA 1 cannot be determined OB 1 Type an integer or a fraction

K a Find the inverse of the function below on the given interval and write it in the form y f x b Verify the relationships f f x x and f f x x f x x 6 A OB OA 5 B 9 X 6 6 EX 5 x x 6 X Type integers or simplified fractions Verify the relationship f f x x Select the correct choice below and fill in the answer box es to complete your choice 5 9 9 f f x 6 x x 6 f1 f x x 6 HELEID X simplified fractions

The data in the table below represents the percentage of patients who have survived after diagnosis of advanced stage breast cancer at 6 month intervals of time Time after Diagnosis years 0 5 1 1 5 2 2 5 3 3 5 Percentage Surviving 95 2 82 6 74 0 58 7 46 4 40 2 33 8 a Using a graphing utility build an exponential model from the data y 0 0x Type integers or decimals rounded to four decimal places as needed

Find all local and global max min and any inflection points for the function g x 2 cos x sin 2x for t Interval 0 27

A chemist has a 100 gram sample of a radioactive material He records the amount of radioactive material every week for 6 weeks and obtains the following data Complete parts a f Week x 0 1 2 3 4 5 6 Weight y in grams 100 0 86 0 72 0 64 1 53 1 45 4 38 8 a Treating week x as the independent variat use a graphing utility to fit an exponential funct to the data y ab Round to four decimal places as needed

A radioactive material decays according to the function A t Age grams of the radioactive material a What is the decay rate of the radioactive material b Graph the function using a graphing utility c How much radioactive material is left after 20 years d When will only 600 grams of the radioactive material be left e What is the half life of the radioactive material a The decay rate of the radioactive material is 2 44 Type an integer or a decimal b Graph the function Choose the correct graph below The graphing window is 0 1600 100 by 0 1 0 25 A e 0 0244t where Ao is the initial amount present and A is the amount present at time t in years Assume that a scientist has a sample of 8 OB OF O C OD OU

The population of a southern city follows the exponential law Use this information to answer parts a and b Point a If N is the population of the city and t is the time in years express N as a function of t N t Noekt Kt Type an expression using t as the variable and in terms of e b If the population doubled in size over 16 months and the current population is 60 000 what will the population be 5 years from now The population will be approximately 807259 people Do not round until the final answer Then round to the nearest whole number as needed

After the release of radioactive material into the atmosphere from a nuclear power plant in a country in 1982 the hay in that country was contaminated by a radioactive isotope ha the hay to cows when 15 of the radioactive isotope remains how long did the farmers need to wait to use this hay The farmers needed to wait approximately days for it to be safe to feed the hay to the cows Round to one decimal place as needed

The size P of a certain insect population at time t in days obeys the function P t 700e0 04t a Determine the number of insects at t 0 days b What is the growth rate of the insect population c Graph the function using a graphing utility d What is the population after 10 days e When will the insect population reach 840 f When will the insect population double b The growth rate of the insect population is 4 c Graph P t 700e0 04t Choose the correct graph below The graphing window is 0 10 1 by 0 1400 100 OA B d The population after 10 days is approximately 1044 insect s Do not round until the final answer Then round to the nearest whole number as needed e It will take about 4 6 day s for the insect population to reach 840 Do not round until the final answer Then round to the nearest tenth as needed f It will take about day s for the insect population to double Do not round until the final answer Then round to the nearest tenth as needed c D o 57

2 Give three examples of Ordinary Differential Equations of second order with constant coefficients one example for each case not solved in the lectures and solve these equations 25 points 3 Test for exactness If exact solve If not find an integrating factor and solve Find the general and particular solution 4 y m dx 5 x n dy 0 with initial condition y m 1 n where m and n are last two digits of your student ID number 4 Find the eigenvalues and eigenvectors for the matrix 353 46 0 25 points 25 points

The Hubble Space Telescope HST has excellent resolving power because there is no atmospheric distortion of the light Its 2 40 m diameter primary mirror can collect light from distant galaxies that formed early in the history of the universe How far apart can two star clusters be from each other if they are 14 0 billion lightyears away from Earth and are barely resolved by the HST using visible light with a wavelength of 400 nm

The population of a colony of mosquitoes obeys the law of uninhibited growth Use this information to answer parts a through c a If N is the population of the colony and t is the time in days express N as a function of t Consider No is the original amount at t 0 and k 0 is a constant that represents the growth rate N t Noekt Type an expression using t as the variable and in terms of e b The population of a colony of mosquitoes obeys the law of uninhibited growth If there are 1000 mosquitoes initially and there are 1800 after 1 day what is the size of the colony after 4 days Approximately mosquitoes Do not round until the final answer Then round to the nearest whole number as needed

The size P of a certain insect population at time t in days obeys the function P t 700e0 04t a Determine the number of insects at t 0 days b What is the growth rate of the insect population c Graph the function using a graphing utility d What is the population after 10 days e When will the insect population reach 840 f When will the insect population double

A piece of charcoal is found to contain 39 of the element B that it originally had a When did the tree from which the charcoal came die Use 5500 years as the half life of the element B b Using a graphing utility graph the relation between the percentage of the element B remaining and time c Using INTERSECT determine the time that elapses until half of the the element B remains d Verify the answer found in part a a The tree died years ago Round to the nearest whole number as needed HEI

After the release of radioactive material into the atmosphere from a nuclear power plant in a country in 1990 the hay that country was contaminated by a radioactive isotope half life 8 days If it is safe to feed the hay to cows when 9 the radioactive isotope remains how long did the farmers need to wait to use this hay The farmers needed to wait approximately Round to one decimal place as needed www days for it to be safe to feed the hay to the cows

The half life of radium is 1690 years If 30 grams are present now how much will be pres grams Do not round until the final answer Then round to the nearest thousandth as needed

A radioactive material decays according to the function A t Ae 0 0244t where A is the initial amount present and A is the amount present at time t in years Assume that a scientist has a sample of 500 grams of the radioactive material a What is the decay rate of the radioactive material b Graph the function using a graphing utility c How much radioactive material is left after 40 years d When will only 300 grams of the radioactive material be left e What is the half life of the radioactive material

How long does it take for an investment to double in value if it is invested at 2 compounded quarterly Compounded continuously At 2 compounded quarterly the investment doubles in about 34 74 years Round to two decimal places as needed At 2 compounded continuously the investment doubles in about Round to two decimal places as needed years www

If the rate of inflation averages r per annum over n years the amount A that SP will purchase after n years is A P 1 r where r is expressed as a decimal If the inflation rate averages 3 1 how much will 2 000 purchase in 5 years In 5 years 2 000 will purchase Round to the nearest cent as needed

Points 0 of 4 After 20 years Trish will have 34898 19 Round to the nearest cent as needed After 20 years Sean will have Round to the nearest cent as needed Sa Trish invests 6 000 in her IRA in a bond trust that pays 9 interest compounded semiannually Sean invests 6 000 in his IRA in a certificate of deposit that pays 8 8 compounded continuously Who has m money after 20 years Trish or Sean

O Points 0 of 2 How many years will it take for an initial investment of 20 000 to grow to 70 000 Assume a rate of interest of 2 compounded continuously t will take about 27 20 years for the investment to grow to 70 000 Round to two decimal places as needed

For each set of values calculate the missing variable using the ideal gas law n Set 1 Set 2 Set 3 Set 4 P 2 61 atm 302 kPa P3 782 torr V 1 69 L 2382 mL 3 0 0250 m V4 T 36 1 C T2 288 K 303 K n n1 3 23 mol 1 08 mol 5 26 mol mol T2

Suppose f x 0 for all x andf x dx 18 Let R be the region in the first quadrant bounded by the coordinates axes y f x 2 and x 3 Find the volume of the solid generated by 0 revolving R around the y axis Set up the integral that gives the volume of the solid Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers OA f x dx GILD OB x dx Sav

Determine whether the following statements are true and give an explanation or counterexample a When using the shell method the axis of the cylindrical shells is parallel to the axis of revolution b If a region is revolved about the y axis then the shell method must be used c If a region is revolved about the x axis then in principle it is possible to use the disk washer method and integrate with respect to x or the shell method and integrate with res y a Choose the correct answer below www OA False The shell method produces shells that are always perpendicular to the axis of revolution B True The shell method always produces shells parallel to the y axis C True The rotation of the function about the axis creates a cylindrical shape parallel to the axis D False The rotation of the function about the axis creates a disk shape perpendicular to the axis b Choose the correct answer below OA True Any time a function is revolved about an axis the shell method must be used OB False The shell method can only be used when a region is revolved about the x axis OC True Then the axis of the cylindrical shells is parallel to the y axis O D False The disk washer method may be used by integration with respect to y Final chec

Let R be the region bounded by y x x 1 and y 0 Use the shell method to find the volume of the solid generated when R is revolved about the line y 3 Set up the integral that gives the volume of the solid using the shell method Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers O A O B TO dy LE Jod dx

1 8x2 1 2x x 1 d da

1 2x 4 3x

Jerome will be buying a used car for 10 000 in 3 years How much money should he ask his parents for now so that if he invests it at 10 compounded continuously he will have enough to buy the car Jerome should ask for Round to the nearest cent as needed www Save mou

b 4 x dx 5

a 1 dz

Find the principal needed now to get the given amount that is find the present value To get 2000 after 2 years at 7 compounded daily 3 4 The present value of 2000 is Round to the nearest cent as needed

For f x a fog 14 11 Y 5 X 4 A and g x b gof 9 find the following composite functions and state the domain of each X c fof 9 4xing your UL Select the correct choice below and fill in any answer boxes within your choice c fof x 9 The domain of fog is x x 0 x x 2 Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB The domain off og is all real numbers d gog 9x 36 b g of x 5 Simplify your answer Use integers or fractions for any numbers in the expression Select the correct choice below and fill in any answer boxes within your choice 5x 20 20 4x A The domain of g of is x x 4 Type an inequality Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OB The domain of g of is all real numbers Simplify your answer

Suppose that g x 6 2 a What is g 1 When x 1 what is the point on the graph of g b If g x 38 what is x When g x 38 what is the point on the graph of g 13 6 a g 1 13 6 The point Type integers or simplified fractions b If g x 38 x The point Type integers or simplified fractions is on the graph of g is on the graph of g

20 The equation 35n t 50 0 relates t the total cost in dollars of boarding a dog in a kennel with the number of nights n the dog is boarded and the cost of medical insurance By a Write the equation in the form y mx b b Use the grid to graph the total cost versus the number of nights of boarding c What is the total cost to board a dog for 7 nights at the kennel 235 P d What is the slope of the line 35 e What does the slope represent Per night to board the dog f What is the cost of medical insurance

Solve the following logarithmic equation log 3x 2 2 log 3x 15 Select the correct choice below and if necessary fill in the answer box to complete your choice A 1 The solution set is 27 Simplify your answer Use a comma to separate answers as needed Type an integer or a simp B The solution is the empty set

Write the expression as a sum and or difference of logarithms Express powers as factors log2 4x ww log2 4x Type an exact answer in simplified form

Suppose that In 3 a and In 11 b Use properties of logarithms to write the logarithm in terms of a and b In 33 In 33 33 ww

K x f x f x Use the following table to determine the indicated derivatives or state that the derivative cannot be determined 6 0 1 5 2 a OA 3 4 1 3 2 b 1 c 3 d f 3 3 3 7 3 4 Type an integer or a fraction O B 4 cannot be determined Part 1 of 4 6 8 2 a Choose the correct answer below and if necessary fill in the answer box within your choice ECCE O Points 0 of 1

Using synthetic division determine whether the numbers are zeros of the polynomial function 4 2 1 h x x 3x 2x 4x 8 Is 2 a zero of h x x 3x 2x 4x 8 No OYes Is 1 a zero of h x x 3x 2x 4x 8 No Yes

Suppose that G x log2 2x 2 3 a What is the domain of G b What is G 7 What point is on the graph of G c If G x 1 what is x What point is on the graph of G d What is the zero of G a The domain of G is 1 00 Type your answer in interval notation b G 7 1 The point 7 1 is on the graph of G Type an ordered pair c If G x 1 then x 1 The point is on the graph of G Type an ordered pair

Find four intervals on which f is one to one making each interval as large as possible Four intervals on which f is one to one are a 6 6 3 3 0 0 Type your answer in interval notation Use a comma to separate answers as needed CHECKED 8 7 6 5 Ay 10 V FIAB 3 2 10 2

Question 1 100 points a S 1 x da b S c 1 1 d e x dx x 9 8x2 2x x 1 dx 3x 2 2x 4x 1dx 2x 4 3x 8 da

Substitute m 2 and b 3 into the equation

Use the given graph of f x to find the intervals on which f x is increasing the intervals on which f x is decreasing and the local extrema Sketch a possible graph of y f x Af x 5