Differential equations Questions and Answers

5 Find the area of the triangle with vertices A 1 2 1 B 1 3 1 and C 2 1 2 Use the result to determine the altitude from vertex C to the side AB

Type the correct answer in each box A cable car begins its trip by moving up a hill As it moves up it gains elevation at a constant rate of 50 feet minute until it reaches the peak at feet Then as the car moves down to the hills base its elevation drops at the same rate The equation that models the cable car s elevation e after t minutes is e The cable car s elevation will be 750 feet after minutes

Convert the fraction to a decimal 3 4

Find the quotient Do not round 4 1 38 1000

to find the Use lagnange multipliens 2 extreme values of f x y y x subject the constraint x y 4

PART I Show that the following solutions are the solutions to the differential equations indicated 1 sin y cos x y sin x cos y 2 y x ex e y y x 3 y n 1 x y 1 x dx dx 4 Arctan y Arctan x dy 1 y dx 1 x 0

Approximate 2 x sin x dx by computing the Left and Right sum using the partition 0 6 4 T 3 2 Your answers should be accurate to at least 4 decimal places Left Sum Right Sum You may include a formula as an answer

Find the normal osculating and rectifying planes of R t cos t sin t t at the point x 2

3 49 Consider the following LP text Maximize z 3 x 1 2 x 2 3 x 3 subject to begin array c 2 x 1 x 2 x 3 leq 2 11 3 x 1 4 x 2 2 x 3 geq 8 x 1 x 2 x 3 geq 0 end array The optimal simplex tableau at the end of Phase I is Explain why the nonbasic variables and X1 X3 X4 X5 can never assume positive values at the end of Phase II Hence conclude that their columns can be dropped before we start Phase II In essence the removal of these variables reduces the constraint equations of the problem to x 2 meaning that it is not necessary to carry out Phase II in this problem

point amina occupies the part of the rectangle 0 x 1 0 y 4 and the density at each point is given by the function p x y 7x 5y 5 What is the total mass Where is the center of mass

ng polar coordinates evaluate the integral R sin x y dA where R is the region 16 x y 25

The assets and liabilities of a salesperson are listed below Home Value 264 500 Retirement Portfolio 473 906 Credit Card Balances 58 236 Bank Loan 35 275 187 632 22 750 139 664 Home Furnishings 23 100 Mortgage Car Value Investments What is the value of the liquid assets O 139 664 O 264 500

Find the equation of the circle shown in the figure x 4 y 4 16 6 4 X 2 y 6 4 2 2 2 X

Suppose a gauge at the outflow of a reservoir measures the flow rate of water in units of ft3 hr The total amount of water that flows out of the reservoir is the area under the flow rate curve Consider the flow rate function shown in the figure to the right Complete parts a through d below Water flow rate f hr 7500 6000 4500 3000 1500 0 Ay a Find the amount of water that flows out of the reservoir over the interval 0 4 The amount of water that flows out of the reservoir over the interval 0 4 is Simplify your answer Type an integer or a decimal 0 2 ft ft A Time hr 00 8 110

Q3 a Name the following equation then solve it x 2 y prime prime 20 y 0 b Name the equation given below and show how you can reduce it to a first order but of linear type y prime 2 x y 5 y 3

50 40 30 20 10 10 10 10 20 30 40 Lester is a lifeguard and spots a drowning child 40 meters along the shore and 50 meters from the the child Lester runs along the shore for a while and then jumps into the water and swims from t directly to the child Lester can run at a rate of 3 meters per second and swim at a rate of 0 9 me second How far along the shore should Lester run before jumping into the water in order to save child Round your answer to three decimal places

et y 3z and the change in y Ay when z 1 and Az 0 4 ind the differential dy when z 1 and dz Question Help Video 0 4

Given the equation below find dz dy dz y 36x 0 3xy y 40 Now find the equation of the tangent line to the curve at 1 1 Write your answer in mz b format

Help Entering Answers 1 point Find the three angles of the triangle with the given vertices P 1 1 1 Q 1 5 2 and 1 2 3 R 6 2 7 P ZQ ZR Use exact answers If you don t get this in 3 tries you can see a similar example online However try to use this as a last resort or after you have already solved the problem There are no See

Use Green s Theorem to evaluate fyr dx x dy where C is the left half x 0 of x y 25

Use the chain rule to find and where First the pieces 885868 dx dy 2x y s 2 8 And putting it all together dz ds 168s 162 158t 162 dy dx z x xy y x 2s 3t y 8s 7t x 2y 3 7

For the given position vectors r t compute the tangent velocity vector r t for the given value of t A Let r t cos 5t sin 5t Then 0 B Let r t 1 Then r 5 0 C Let r t esti e stj tk Then r 2 5e 10 4 i 5e 10 j 0 01 k

SNA empting re is no like 1 quare root of a ria was trying to of arst ng R 6 G ytical geometry that 1 as well as the B vs Geresiae Cardase Ledevita Ferrad PE1 4574 0825 SEE 0 O y Opt N d 1 the principal square root 0 Manala TRE fi to represent the a b Today the understanding of tudies of such topics as force stresses c 25 J KINA PRENE Submit Quiz e 1 K Quizzes Phoenix 2 Chobharr Unith O 0 of 5 questions saved hathematician was the first to call the number imaginary mathematician was the first to use the symbol i MUST be defined fhxs xu zt chi m Frut

1 point Set up the integral for the area enclosed by the entirety of the three leaf rose r 5 sin 30 Volume

Consider the curve r e t cos 5t e t sin 5t e Compute the arclength function s t with initial point t 0

Match the functions with the graphs labeled A G As always you may click on the thumbnail image to produce a larger image in a new window sometimes exactly on top of the old one Just take your time process of elimination will help with ones that are not obvious 1 f x y sin y 2 f x y 1 1 x y 3 f x y 3x y 4 f x y sin x sin y e x y 5 f x y x 32 6 f x y cos x y 1 x y 7 f x y x y F

Find the arclength of the curve r t 7 2t et e t 0 t 1

Evaluate the function at the specified points h x y z 1 yz 5 4 6 3 2 5 x4 At 5 4 6 At 3 2 5

Which equation has x 6 as the solution O log 36 2 log 2x 9 3 O log 216 x log 2x 3 2

Find the distance from the point 4 1 0 to the plane 4x 4y 3z 6

Which of the following is the inverse of y 3 o y 3 2 3 O y log X OY 3 O y log x

Find the solution of the following initial value problems f x 12x 7 f 0 4 The solution to the initial value problem is f x

Make use of Laplace transform to solve the following differential equation that satisfy the given initial conditions y 10y 9y 51 y 0 1 y 0 2

Question 2 i 10 marks Find a ER such that y x x is a solution of the following differentia equation x y 10xy 18y 0 x 0 00 ii 10 marks Find the general solution of the following differential equation x y 10xy 18y x x 0

Find the maximum and minimum values of f x y 4 y 10 x subject to the constraint x4 y 25 using the Lagrange s multiplier method

Textbook Videos An inverted pyramid is being filled with water at a constant rate of 55 cubic centimeters per second The pyramid at the top has the shape of a square with sides of length 5 cm and the height is 13 cm Find the rate at which the water level is rising when the water level is 7 cm h cm sec b b

Given the function f 1 1 e Determine the open interval s where the function is concave up Determine the open interval s where the function is concave down Determine any points of inflection

If the graph of the antiderivative g x of f x 1 1 xlnx in g x is 2 3 If In 4 If sinx 3 cos x dx sin x cos X If the graph of anti derivative g x of then the term independent of x in g x is X 1 If Sex x 1 of a is 5 11 5x In x 1 x dx If 1 x dx 6 1 1 In a X kx 2 1 cos 0 2 7 1 cos0 9 7 where c is constant of integration and a b Z de 8 a In sin x cos x then k is b x dx 1 43x 1160 1 B 12 If S 4x 1 6 A 3x x 1 x 2 dx A x In x 1 x Bx c then 1 1 2x x 1 x 5 passes through e 4 then c then a b ab is equal to tane bla c then la bl equals intersect the line x 2 at a point with The value of integral x x x8 2x91 10 dx is f x 1 10 c where c is consta f 2 equals Sext x x 2x5 e dx f x c where c is constant of integration then 1 is greatest integer function dx ex 3A Cong vo Exar d where d is constant of integration and A B C Z t 58 X 1 x 1 x A B equal to c where c is constant of

1 2 3 4 If the graph of the antiderivative g x of f x in g x is If 6 If 7 If sinx 3 cos x sinx cos x If the graph of anti derivative g x of then the term independent of x in g x is In X 1 1 10 If Se dx of a is x dx 4x 1 A dx 1 x 10 In X 5 ffx in x x dx A x In x 1 x Bx c then 1 x 1 cos 8 27 1 cos 0 9 7 where c is constant of integration and a b Z de kva x kx In sinx cosx then kis 2 B 1 xinx 1 dx a 3x x 1 x 2 1 1 2x x 1 x 5 tane ba c then la b equals passes through e c then a b ab is equal to 11 10 8 The value of integral x x x 2x 0 dx is f x C where c is consta 11 f 2 equals intersect the line x 2 at a point with ord 9 Se x x 2x e dx f x c where c is constant of integration then f 1 is greatest integer function dx ex Ear d where d is constant of integration and A B C Z th 1 1 x 3A 5B A B X 1 x equal to x8 a C where c is constant c c where ke Z and c is const

4 9pts Write down the 3 x 3 matrix having eigenvalue 1 with eigenspace E span 0 0 and eigenvalue 2 with eigenspace E2 span 101 020 Ja C 0

Given the autonomous system dx dy dt dt find the critical points b solve the phase plane equation and classify the critical points analyzing trajectories of the syst find the general solution to the system and determine its stability analyzing integral curves of system a C I x y

1 Construct a clamped cubic spline for the data f 8 3 17 565 and f 8 6 18 505 Round your calculations to 3 decimal places

7 5 6 8 3 A 98 B 102 C 108 D 110 1 2 4 ell m l m If the measure of 25 3x and the meas of Z2 2x what is the measure of 283

ve the general solution of the following differential equation y 2 x 2y x 3y x tan x S 15

1 Find the general solution to the following second order linear homoge neous differential equations a t y 3ty y 0 t 0 y t b ty y 4t y 0 t 0 y cos t c t 1 y ty y 0 t 1 y et d y 6y 9y 0 y 0 0 y 0 2 do not use the eigenvalue stuff

Solve the initial value problems b 2v sin x cos x y sin x dx sin x 2v cos x dy 0 v 0

The temperature in degrees Celsius C of the water in a pond is a differential function of time The table below shows the water temperature as recorded on various days for a 15 day period r days 13 15 w C 31 24 0 27 2 31 6 35 Sw t dt 10 36 a Use the data from the table to find an approximation for w 8 Show the computations that lead to your answer Using correct units interpret the meaning of w 8 in the context of the problem b Do the data in the table support the conclusion that at some time during the 15 day period w t 0 If so within what time interval Give a reason to support your answer c Do the data in your table support the conclusion that the water temperature at some point in the 15 day period is equal to 25 C If so within what smallest time interval Justify your conclusion d Use Trapezoidal sums with five subintervals and values from the table to approximate 15

Find the indefinite integral antiderivative sin 9x dx Preview My Answers Submit Answers

23 Find 2u v when u 4 3 and v 1 2 a 7 8 b 3 5 C 2 7 d 1 9

2 What is the standard form of the equation of the conic given below 4x 5y2 16x 30y 9 0 a C x 4 y 3 4 11 C b y 3 x 2 1 4 y 3 6 5 x 2 1 x 4 y 3 11 4 1 1