Calculus Questions

f x b Check to see that your answer to part a is reasonable by comparing the graphs of f and f A graphing calculator is recommended 4 f 4 2 2 y 100 50 50 100 y 100 50 50 100 2 4 4 f X S 4 4 2 2 y 100 50 50 100 y 100 50 50 100 2 2 4 4 X X

The graph of fis given State the numbers at which f is not differentiable Enter your answers as a comma separated list y X 4 2

The graph of fis given State the numbers at which is not differentiable Enter your answers as a comma separated li than 0 2 4 e

Let f t Vt For a 0 find f a f a

Individual pathways PRACTER 1 4 0 11 14 MOLDATE 2 5 6 6 12 15 To answer questions online and to receive inmediate corrective feedback and fully worked solutions for all questions go to your learnON title at www jecplan com m Fluency Show that the following numbers are rational by writing each of them in fraction form 15 Show that the following numbers are rational by writing each of them in fraction form vinh 0 0002 10 Show that the following numbers are rational by writing each of them in fraction form 0 1 36 3 612 d 0 08 Write the first 8 digits of each of the following recurring decimals b 031 051 d 6011 Jacaranda Maths Queit 9 MARTER 3 7 10 13 16 17 0 5 we Write the first X digits of each of the following recurring decimals 7 024 89124 c 5 1234 4 5 1234 Write the following fractions as recurring decimals using doc notation Use a calculator Understanding From the following list of numbers 15 B 11 Write the following fractions as recurring decimals using dot notation Use a calculator 23 46 N 990 write down the natural numbers write down the rational numbers a 6 7 15 d a Write these numbers in order from smallest to largest 2 1 2 12 2 121 2121 212 K7 2 918 99 5 183 3 002 173 99 46 900 b write down the integers d write down the recurring decimals

Problem 3 Suppose that a linear system of equations in unknowns x y and has the following augmented matrix 1 2 3 2 3 3 2 4 2 Problem 3 3 Use Gauss Jordan elimination to solve the system for x y and z Enter the values of x y and z here separated by commas

Problem 5 Determine whether the following linear system has one solution no solutions or infinitely many solutions x14x2 6x3 6x4 3 4x16x27x3 5x4 4 8x1 2x2 3x3 9x4 16 A No solution B One solution C Infinitely many solutions

Relatively paine numbers Co poime Two numbers a b are said to be retatively poine it and only if the GGD of a ie 1 ie GCD a b 1 2018 2019 1 Ex 8 15 1

must be used to obtain the particular solution desired here Since tan 10 0 and tan 1 c 0 7 Thus the solution of the initial value problem is 77 tan x tan y Let us look at another example Example 3 Solved Solution Let us write the given equation as 1 ky my2 dx dy 1 Let Now we will try to decompose the coefficient of dy into partial fractions 1 B y k my A yk my where A and B are constants to be determined From 6 we get 1 A k my By or 4 which gives 1 Ak and 1 M 1 ky my2 where k 0 and m 0 are real constants and R E Bk m 6

Problem 9 Consider the following matrices Problem 9 a Problem 9 b 4 033 13 15 10 B 1 4 2 4 2 3 a Let D BTC Find d21 i e find the entry in row 2 column 1 of the matrix D b Find tr 2BB c Let E B CA Find e32 i e find the entry in row 3 column 2 of the matrix E

Problem 10 Let A be a 3 10 matrix and let B be m n Suppose that BTAT is a 21 x 3 matrix What is the value of m Problem 10

Problem 4 Use Gauss Jordan Elimination to solve the following system of equations 2x 2y 8z 2 x 5y 5z 2 6x 14y 26z 0 A x 3 51 B x 14 1 C x 3 5 D x 7 5 E x x 14 St F X y 4 y 3 1 z t 2 t y t z t 2 1 19 G x 3 5t H no solution I x y 2 1 x 1 51 4 y 3 1 2 t 8 y 2 z t t

blem 11 Find matrices A x and b that express the below system of linear equations as a single matrix equation Ax b 7x17x2 x3 6x4 2 2x1 4x2 8x3 5x4 2 3x1 2x2 4x3 7x4 1 C A A A 2 60 0 0 0 4 B A 2 E A 7 7 1 4 8 5 x 3 2 4 7 7 7 1 2 4 8 3 2 4 7 2 3 7 1 4 8 2 4 X X2 X X2 X3 b b 6 X1 8 0 b x3 D A 2 7 7 1 7 F A 2 3 3 2 4 7 7 1 6 60 4 8 5 X x3 3 2 4 7 8 7 4 2 5 4 1 8 0 X X3 X4

oblem 1 Use Gauss Jordan elimination to find the reduced row echelon form of the following matrix 132 2 1 A 0 0 1 E 0 0 6 1 2 3 1 5 0 23 0 1 0 0 1 0 0 1 0 1 00 WNIO WE 23 25 15 2 23 3 100 1 B 0 0 1 F 0 0 0 0 1 O 0 1 0 1 0 H2N 32 11 13 28 15 41 1 0 0 233 C 0 0 1 1 0 99 O 0 G 0 1 0 0 1 0 a NIO W NW NIO WIN 0 1 13 2 41 D H 002 5 1 0 0 0 1 10 0 1 00 0 10 0 15 1

roblem 2 Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the following reduced row echelon form Solve the system 0 0 0 6 0 0 3 0 1 0 3 0 0 1 5 0 0 A x 6 35 9t B x 9 6s 3t C x 6 3s 9t D no solution x2 S inconsistent x3 9 3t X4 4 5t x5 1 x2 5 x3 3s 91 X4 5s 4t x5 1 9 4 00 0 F x 9 0 x3 9 X4 4 X5 0 G x 9 6s 3t X2 S x3 9 31 X4 4 5t X5 1 x2 S x3 3s 9t X4 5s 4t X5 t E x 6 3s 9t X2 5 3s 9t X4 5s 4t x5 1

Problem 8 Let A and B be square matrces of the same size Which of the following statements are always true i tr 47 tr 4 ii tr cA c tr 4 where c is a scalar iii tr A B tr 4 tr B A ii and iii only B i and ii only C iii only D all of them E ii only F 1 only G none of them H i and iii only

Problem 6 Suppose that you were to try to find a parabola y ax bx c that passes through the x y pairs 4 15 3 2 and 2 1 To obtain the coefficients a b and c you would try to solve a system of linear equations whose augmented matrix is which of the following Problem 6 A 225 4 1 4 3 1 1 2 1 16 E 9 4 Select v 4 1 3 1 2 1 15 2 1 225 4 1 1 16 15 2 1 3 1 1 2 B 9 4 F 225 4 1 15 1 16 2 1 9 1 1 4 16 C 9 4 4 1 15 1 2 3 2 1 1 D 225 4 15 1 4 2 1 3 1 1 2

Thus o satisfies condition of GCD a b Further het diviles c divides b and 8 also b 8 c divides 8 thie continuing C divides ni Since a b 8 process we finally have that This shows that I satisfiel condition of GCD Hexe GCD a b

For the first order system above let u t 0 t0 0 and x 0 5 Use Simulink to solve the resulting system using an integrator block and rewriting the equation as dx t dt 1 2 1 cos t x t Compare your Simulink result to the closed form solution

Honda Motor Company has discovered a problem in the exhaust system of one of its automobile lines and has voluntarily agreed to make the necessary modifications to conform to government safety requirements Standard procedure is for the firm to pay a flat fee to dealers for each modification completed Honda is trying to establish a fair amount of compensation to pay dealers and has decided to choose a number of randomly selected mechanics and observe their performance and learning rate Based on the results of this test Honda decided to pay a 60 fee for each repair 3 hours x 20 per flat rate hour Southwest Honda Inc is considering whether to participate in the program If they can average three hours or less on each repair what Honda is willing to pay them for it will make sense for them to agree They performed their own test repairs to help in their decision process Six mechanics working independently have completed two modifications each On average the mechanics took 10 hours to do the first unit and 8 00 hours to do the second The dealership expects to perform 360 such modifications which works out to 60 modifications per mechanic including the first two units already built Use Exhibit 6 5 On average how many hours will it take Southwest Honda to perform each modification using all six mechanics as planned Round your answer to 1 decimal place Answer is complete but not entirely correct

A crate with mass 32 5 kg initially at rest on a warehouse floor is acted on by a net horizontal force of 140 N How far does the crate travel in 10 0 s

K The velocity of a parachutist during free fall is f t 82 1 e 0 12t meters per second Answer the following questions using the graph Recall that acceleration is the derivative of velocity b What is the approximate acceleration when t 0 10 m sec2 c Approximately when is the parachutist s velocity 51 m sec 8 seconds d Approximately when is the acceleration 5 m sec CHE Ay 80 70 60 50 40 30 20 10 0 0 10 20 30 S

escribe all absolute extreme points of P r s m n c d GALER Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The absolute maximum point is There is no absolute minimum point Type an ordered pair B The absolute minimum point is is no absolute maximum point Type an ordered pair There OC The absolute maximum point is the absolute minimum point is Type ordered pairs OD There are no absolute extreme points and

A cup of coffee has its temperature F in degrees Fahrenheit at time t given by the function F t 80 95 e 0 05t where time is measured in minutes Use central difference with h 0 01 to estimate the value of F 10 Round to the nearest tenth 2 9 What are the units on the value of F 10 degF per min What is the practical meaning of the value of F 10 The coffee has decreased by 10 degrees over 2 9 minutes The coffee s temperature is decreasing at about 2 9 degrees per minute The coffee s temperature has decreased by about 2 9 degrees over ten minutes Part 1 of 2 Which do you expect to be greater F 10 or F 20 F 10 S Write a sentence that describes the behavior of the function y How do you think its graph will look Why Part 2 of 2 F t on the time interval 0 t 30

13500 The value V of a particular automobile in dollars depends on the number of miles m the car has been driven according to the function V h m Suppose that h 35000 14300 and h 45000 13900 What is the average rate of change Round to the nearest thousandth of h on the interval 35000 45000 and what are the units on this value Average rate of change dollars per mile In addition to the information given in a say that h 55000 13500 Determine the best possible estimate of h 45000 Round your answer to the nearest hundredth Hint Which value do you expect to be greater h 45000 or h 55000 h 45000 and h 55000 are equal Write a sentence to describe the long term behavior of the function V h m plus another sentence to describe the long term behavior of h m Provide your discussion in practical terms regarding the value of the car and the rate at which that value is changing

The temperature change T in Fahrenheit degrees in a patient that is generated by a dose q in milliliters of a drug is given by the function T f q What does it mean to say f 45 1 5 If a patient takes a dose of 45mL of a drug the patient will experience a body temperature change of 1 5 degrees F If a patient takes a does of 1 5 mL of a drug the patient will experience a body temperature change of 45 degrees F Part 1 of 2 Olf a patient takes a degrees of a drug the patient will experience a body temperature change of T ml A person s sensitivity s to the drug is defined by the function s q f q What are the units of sensitivity degrees Fahrenheit per milliliter Part 2 of 2 Suppose that f 45 0 02 Write a complete the following sentence to explain the meaning of this value The sentence will include information given in the first question For a patient taking a expect a temperature change that is about ml dose adding 1 more ml to the dose leads us to degrees less than

The velocity of a ball that has been tossed vertically in the air is given by v t 48 32t where v is measured in feet per second ft s and t is measured in seconds The ball is in the air from t 0 until t 3 When is the ball s velocity greatest Determine the value of v 1 v 1

A laboratory study investigating the relationship between diet and weight in adult humans found that the weight of a subject W in pounds was a function W f c of the average number of Calories C consumed by the subject in a day In the statement f 1500 195 Hint What are the units of 1500 cal What are the units of 195 lb In the statement f 1900 0 Hint What are the units of 1900 cal What are the units of 0 lb cal In the statement f 225 2500 Hint What are the units of 225 lb What are the units of 2500 cal dW de What are the units of f c lb cal Suppose that Sam reads about f in this study and draws the following conclusion If Marissa increases her average calorie intake from 2800 to 2850 calories per day then her weight will increase by approximately 0 8 pounds Fill in the blanks below so that the equation supports her conclusion

Select the FIRST correct reason why the given series converges NOTE Make sure that you can provide the proper arguments to support your answers A Convergent geometric series B Convergent p series C Integral test D Comparison with a convergent p series E Converges by limit comparison test F Converges by alternating series test n n 1 1 1 n 8 n 2 n4 8 sin 2n 5 6 n 2 n 1 n cos NT In 3n 3 n ln n

A particle is moving along the curve f a As it passes throught the point 5 3 its instantaneous rate of change is 6 We also know that if you draw the tangent line to this curve at the point 4 8 it will be a horizontal ta this information find the value of f 5 f 4 f 5 012 018 017 013 09

K Find the poor of the rectangle Express the perimeter using the same unit of measure that appear on the green sis The part of the rectangle is

A garden is in the shape of a rectangle 43 feet long and 20 feet wide If fencing costs 6 a foot what will it cost to place fencing around the garden OA 378 OB 756 OC 5 160 OD 1 512

Consider the series 1 an where 8 5n 6 5n 5 In this problem you must attempt to use the Ratio Test to decide whether the series converges Compute an an 1 L lim n 8 an Enter the numerical value of the limit L if it converges INF if it diverges to infinity MINF if it diverges to negative infinity or DIV if it diverges but not to infinity or negative infinity L Which of the following statements is true A The Ratio Test says that the series converges absolutely B The Ratio Test says that the series diverges C The Ratio Test says that the series converges conditionally D The Ratio Test is inconclusive but the series converges absolutely by another test or tests E The Ratio Test is inconclusive but the series diverges by another test or tests F The Ratio Test is inconclusive but the series converges conditionally by another test or tests Enter the letter for your choice here Activ Go to

For each of the following series tell whether or not you can apply the Alternating Series Test If you can apply this test enter D if the series diverges or C if the series converges If you can t apply this test even if you know how the series behaves by some other test enter N NOTE Make sure that you can provide the proper arguments to support your answers N 3 4 5 6 8888WWW n 1 n 1 n 1 n 1 n 1 1 cos nn n5 1 en n 1 n n 1 n nn 1 n en 1 n 5 Activ Go to

For the following alternating series 0 6 0 6 5 3 5 n 1 an 0 6 0 6 7 7 how many terms do you have to go for your approximation your partial sum to be within 0 000000110 7 from the conver value of that series

The graphs of two functions f and g are given below 2 7 3 1 2 f Evaluate each limit if the limit does not exist say why a lim f x g x b lim f x g x c lim f x g x d lim f x g x 3 2 g x x 1 f x h lim e lim f x g x f lim g x g lim f x cos x 4 0 2 14 2 2

The bell tower of the cathedral in Pisa Italy leans 5 6 from the vertical A tourist stands 104 m from its base with the tower leaning directly toward her She measures the angle of elevation to the top of the tower to be 29 5 Find the length of the tower to the nearest meter

Sketch the triangle LA 30 2C 75 b 14 B 75 14 30 75 C B 14 30 A B Solve the triangle using the Law of Sines Round side lengths to one decimal place a C 30 14 14 C 75 A 75 30 B C

Solve the triangle using the Law of Sines Assume b and c 15 and 2C 64 Round the length to two decimal places a LA LB A O b C B

Solve the triangle using the Law of Sines Assume c 80 ZA 49 and LB 25 Round lengths to two decimal places a b 20 A b C a B

Use the Law of Sines to find the indicated side x Assume c 25 7 Round your answer to one decimal place A 52 70 B

a Find the point on the graph of y ex where the tangent line has slope 5 b Plot the graph of y ex and the tangent line whose slope is 5 a The point on the graph where the tangent line has slope 5 is In 5 5 Type an ordered pair Type an exact answer b Choose the correct graph below O A 40 OB 10 40 O C 10 10 Ay O D 10 C

Differentiate y In In dx x 3 4 ex 2 x 3 4 ex 2

Use logarithmic differentiation to differentiate the following function f x x 2 5 3x 4 4 f x 0

Part 1 of 2 The coordinates of the relative extreme point are Type an ordered pair Type an exact answer Find the coordinates of the relative extreme point of the given function and determine if the point is a relative maximum point or a relative minimum point f x e 9x O Points 0 of 1 E

Solve the given equation for x In x 1 In x 5 1 X Type an exact answer

Differentiate y Im x 3 x 5 x 4 lm x 3 x 5 x 4 0

Solve the following equation for x e5x 13 Type an exact answer X

Differentiate the following function 4 In 5x x y d 4 dx ln 5x In 176 5x Q X

256 a Find the point on the graph of y ex where the tangent line has slope 5 b Plot the graph of y ex and the tangent line whose slope is 5 a The point on the graph where the tangent line has slope 5 is Type an ordered pair Type an exact answer