Differential equations Questions and Answers
![T 13 Exponential growth and decay word problems TYQ A colony of 100 bacteria doubles in size every 60 hours What will the population be 180 hours from now](https://media.kunduz.com/media/sug-question-candidate/20230421164546080950-5550994.jpg?w=256)
Calculus
Differential equationsT 13 Exponential growth and decay word problems TYQ A colony of 100 bacteria doubles in size every 60 hours What will the population be 180 hours from now
![The graph of y f x is shown in blue Draw the graph of y f x shown in red by determining how a stretch or compression will transform the location of the indicated blue points Drag the movable red points to the desired coordinates You may click on a point to verify its coordinates](https://media.kunduz.com/media/sug-question-candidate/20230421154517326864-5550837.jpg?w=256)
Calculus
Differential equationsThe graph of y f x is shown in blue Draw the graph of y f x shown in red by determining how a stretch or compression will transform the location of the indicated blue points Drag the movable red points to the desired coordinates You may click on a point to verify its coordinates
![and G The difference between angles](https://media.kunduz.com/media/sug-question-candidate/20230421152442707044-3653173.jpg?w=256)
![Solve using the square root method 3 x 6 2 3 144](https://media.kunduz.com/media/sug-question-candidate/20230421051224100610-5548870.jpg?w=256)
![ind a formula for the linear function depicted in the following graph Answer 10 ta Av Hop S Enable Zoom Pan X](https://media.kunduz.com/media/sug-question-candidate/20230421033336540877-5506401.jpg?w=256)
Calculus
Differential equationsind a formula for the linear function depicted in the following graph Answer 10 ta Av Hop S Enable Zoom Pan X
![3 Consider the finite difference formula 1 f x f x 2h 8f x h 8f x h f x 2h 0 h 12h Derive this formula by using Taylor s theorem Hint set n 4 in Taylor s expan sion](https://media.kunduz.com/media/sug-question-candidate/20230420144435496811-5548146.jpg?w=256)
Calculus
Differential equations3 Consider the finite difference formula 1 f x f x 2h 8f x h 8f x h f x 2h 0 h 12h Derive this formula by using Taylor s theorem Hint set n 4 in Taylor s expan sion
![One milliliter of liquid has the equivalent V](https://media.kunduz.com/media/sug-question-candidate/20230420043437032328-3653173.jpg?w=256)
![Which of the following matrices is in reduced row echelon form Select one O a O b O C O d 105 0 5 0 0 1 10 1 0 1 00 0 0 0 1 1 2 0 0 00 1 0 00 0001 0 0 0 05 07](https://media.kunduz.com/media/sug-question-candidate/20230420041630593090-5327513.jpg?w=256)
Calculus
Differential equationsWhich of the following matrices is in reduced row echelon form Select one O a O b O C O d 105 0 5 0 0 1 10 1 0 1 00 0 0 0 1 1 2 0 0 00 1 0 00 0001 0 0 0 05 07
![Solve the given initial value problem for y f x dy 1 y where y 6 when x 2 2 x 1 4 O y ln x 1 6 O y ln x 1 6 O y In d 1 6 4](https://media.kunduz.com/media/sug-question-candidate/20230420001408582966-4090272.jpg?w=256)
Calculus
Differential equationsSolve the given initial value problem for y f x dy 1 y where y 6 when x 2 2 x 1 4 O y ln x 1 6 O y ln x 1 6 O y In d 1 6 4
![Solve the given initial value problem for y f x dy e4 x where y 0 when x 4 y x 4 4e4 x y x 1 e4 x y x 4 4e4 x y x 1 e4 X](https://media.kunduz.com/media/sug-question-candidate/20230420001336821700-4090272.jpg?w=256)
Calculus
Differential equationsSolve the given initial value problem for y f x dy e4 x where y 0 when x 4 y x 4 4e4 x y x 1 e4 x y x 4 4e4 x y x 1 e4 X
![e initial height Question 2 A soccer ball is kicked across a field Determine how long the soccer ball](https://media.kunduz.com/media/sug-question-candidate/20230419234508798373-4984822.jpg?w=256)
Calculus
Differential equationse initial height Question 2 A soccer ball is kicked across a field Determine how long the soccer ball
![el Lines ind the value of x Then find the meas](https://media.kunduz.com/media/sug-question-candidate/20230419204110610666-3653173.jpg?w=256)
![Lines 1 and 12 are parallel lines Determine the measures 412](https://media.kunduz.com/media/sug-question-candidate/20230419195435677712-3653173.jpg?w=256)
![8 For the tangent line to the curve at the origin for r 2sin20 a 0 0 0 37 c 8 0 2 b d 0 0 0 0 0 0 EINE](https://media.kunduz.com/media/sug-question-candidate/20230419120638473459-4710940.jpg?w=256)
Calculus
Differential equations8 For the tangent line to the curve at the origin for r 2sin20 a 0 0 0 37 c 8 0 2 b d 0 0 0 0 0 0 EINE
![5 Find the equation for the ellipse with foci 11 0 and semimajor axis a 6 a 25 y 1 36 c 2 1 36 25 b d 6 5 5 1 1 y 1 6](https://media.kunduz.com/media/sug-question-candidate/20230419120551637027-4710940.jpg?w=256)
Calculus
Differential equations5 Find the equation for the ellipse with foci 11 0 and semimajor axis a 6 a 25 y 1 36 c 2 1 36 25 b d 6 5 5 1 1 y 1 6
![Find the function p that satisfies the following conditions assume x 0 30 231 p x p 4 8](https://media.kunduz.com/media/sug-question-candidate/20230419043622921477-5502565.jpg?w=256)
Calculus
Differential equationsFind the function p that satisfies the following conditions assume x 0 30 231 p x p 4 8
![A particle moves on a straight line and has acceleration a t 36t 6 Its position at time t 0 is s 0 10 and its velocity at time t 0 is v 0 3 What is its position at time t 6](https://media.kunduz.com/media/sug-question-candidate/20230419043806964818-5502565.jpg?w=256)
Calculus
Differential equationsA particle moves on a straight line and has acceleration a t 36t 6 Its position at time t 0 is s 0 10 and its velocity at time t 0 is v 0 3 What is its position at time t 6
![Given f x then f x f x Note that your answer should not contain a general constant e with f 0 9 f 0 5 C](https://media.kunduz.com/media/sug-question-candidate/20230419043727391052-5502565.jpg?w=256)
Calculus
Differential equationsGiven f x then f x f x Note that your answer should not contain a general constant e with f 0 9 f 0 5 C
![Using known series determine the Taylor series for the function f x 1 1 x 2 x In 2 x x sin2 x at x 0](https://media.kunduz.com/media/sug-question-candidate/20230419032841153118-5544664.jpg?w=256)
Calculus
Differential equationsUsing known series determine the Taylor series for the function f x 1 1 x 2 x In 2 x x sin2 x at x 0
![Procedure For the differential equation y y 6y 0 y and y2 y lead to the system the substitutions y y y2 y 6y y2 Use the procedure to solve y y 42y 0 NOTE Use c and ce for the constants of integration Y1](https://media.kunduz.com/media/sug-question-candidate/20230419024929491896-5546812.jpg?w=256)
Calculus
Differential equationsProcedure For the differential equation y y 6y 0 y and y2 y lead to the system the substitutions y y y2 y 6y y2 Use the procedure to solve y y 42y 0 NOTE Use c and ce for the constants of integration Y1
![15 If f x 3 9x x which of the following is true about the end behavior of f x As z f z and as z f x As I f x and as a f x 0 As I f 1 and as a f x As I f x and as o f x None of the above](https://media.kunduz.com/media/sug-question-candidate/20230418192124809161-3500550.jpg?w=256)
Calculus
Differential equations15 If f x 3 9x x which of the following is true about the end behavior of f x As z f z and as z f x As I f x and as a f x 0 As I f 1 and as a f x As I f x and as o f x None of the above
![Find the difference of the complex numbers in the complex plane 4 3 31](https://media.kunduz.com/media/sug-question-candidate/20230418164610690681-3899453.jpg?w=256)
Calculus
Differential equationsFind the difference of the complex numbers in the complex plane 4 3 31
![Find the missing values by solving the parallelogram shown in the figure The lengths of the diagonals are given by c and d Round your answers to two decimal places C 3 01 X a 5 b 6 a 0 b d 10 63 x 0 30 150 0](https://media.kunduz.com/media/sug-question-candidate/20230418160853372401-3899453.jpg?w=256)
Calculus
Differential equationsFind the missing values by solving the parallelogram shown in the figure The lengths of the diagonals are given by c and d Round your answers to two decimal places C 3 01 X a 5 b 6 a 0 b d 10 63 x 0 30 150 0
![nge in the radius during that time 230 A spherical balloon is being inflated at a constant rate If the volume of the balloon changes from 36 in 3 to 2887 in between time t 30 and t 60 seconds find the net change in the radius of the balloon during that time](https://media.kunduz.com/media/sug-question-candidate/20230418043147873452-4565383.jpg?w=256)
Calculus
Differential equationsnge in the radius during that time 230 A spherical balloon is being inflated at a constant rate If the volume of the balloon changes from 36 in 3 to 2887 in between time t 30 and t 60 seconds find the net change in the radius of the balloon during that time
![A company finds that the rate at which the quantity of a product that consumers demand changes with respect to price is given by the marginal demand function D x 5000 where x is the price per unit in dollars Find the demand function if it is known that 1004 units of the product are demanded by consumers when the price is 5 per unit Dod](https://media.kunduz.com/media/sug-question-candidate/20230418030202761707-3967093.jpg?w=256)
Calculus
Differential equationsA company finds that the rate at which the quantity of a product that consumers demand changes with respect to price is given by the marginal demand function D x 5000 where x is the price per unit in dollars Find the demand function if it is known that 1004 units of the product are demanded by consumers when the price is 5 per unit Dod
![Determine whether there is a relative maximum a relative minimum a saddle point or insufficient information to determine the nature of the function f x y at the critical point xo Yo fxx xor Yo 25 fyy xor Yo 7 fxy xo Yo 10 relative minimum insufficient information O saddle point relative maximum](https://media.kunduz.com/media/sug-question-candidate/20230418021101613880-5544318.jpg?w=256)
Calculus
Differential equationsDetermine whether there is a relative maximum a relative minimum a saddle point or insufficient information to determine the nature of the function f x y at the critical point xo Yo fxx xor Yo 25 fyy xor Yo 7 fxy xo Yo 10 relative minimum insufficient information O saddle point relative maximum
![The following sum is a right Riemann sum for the definite integral where b and f x 2 40 7 49 n The limit of these Riom 2 7n 49 44 2 40 TH 7 12 77 n n n f x f x dx](https://media.kunduz.com/media/sug-question-candidate/20230418013546231808-5502565.jpg?w=256)
Calculus
Differential equationsThe following sum is a right Riemann sum for the definite integral where b and f x 2 40 7 49 n The limit of these Riom 2 7n 49 44 2 40 TH 7 12 77 n n n f x f x dx
![Consider the integral L 2 dz a Find the Riemann sum for this integral using right endpoints and n 4 b Find the Riemann sum for this same integral using left endpoints and n 4](https://media.kunduz.com/media/sug-question-candidate/20230418013459170826-5502565.jpg?w=256)
Calculus
Differential equationsConsider the integral L 2 dz a Find the Riemann sum for this integral using right endpoints and n 4 b Find the Riemann sum for this same integral using left endpoints and n 4
![Write the constraint below as a linear inequalit](https://media.kunduz.com/media/sug-question-candidate/20230417182701917151-5422331.jpg?w=256)
![Solve the system of linear equations by graphing x 6 y 5](https://media.kunduz.com/media/sug-question-candidate/20230417175025360320-3653173.jpg?w=256)
![Determine a function that satisfies the differential equation d y dx2 4y b Explain how you found your CO](https://media.kunduz.com/media/sug-question-candidate/20230416014038292833-5535586.jpg?w=256)
Calculus
Differential equationsDetermine a function that satisfies the differential equation d y dx2 4y b Explain how you found your CO
![Integrate assuming x 0 S xin x In 5x dx x n Type an exact answer x In 5x dx](https://media.kunduz.com/media/sug-question-candidate/20230415194947809969-4364004.jpg?w=256)
Calculus
Differential equationsIntegrate assuming x 0 S xin x In 5x dx x n Type an exact answer x In 5x dx
![the recurrence relation for the power series solution about x 0 of the differential equation y y 0 is](https://media.kunduz.com/media/sug-question-candidate/20230415173731555800-5521628.jpg?w=256)
Calculus
Differential equationsthe recurrence relation for the power series solution about x 0 of the differential equation y y 0 is
![6 Find at least first four non zero terms with the help of recurrence relation of two linearly independent solutions in a power series expansion about r 0 for a general solution to the given differential equation](https://media.kunduz.com/media/sug-question-candidate/20230415124448491304-5538341.jpg?w=256)
Calculus
Differential equations6 Find at least first four non zero terms with the help of recurrence relation of two linearly independent solutions in a power series expansion about r 0 for a general solution to the given differential equation
![For each of the following part make sure to show and justify your steps dy a Find the general solution to the equation dx b Find the solution which goes though the point 2 4 4y 12 c Does this equation have any equilibrium solutions If so find them and classify as stal or unstable](https://media.kunduz.com/media/sug-question-candidate/20230415132818818926-4880803.jpg?w=256)
Calculus
Differential equationsFor each of the following part make sure to show and justify your steps dy a Find the general solution to the equation dx b Find the solution which goes though the point 2 4 4y 12 c Does this equation have any equilibrium solutions If so find them and classify as stal or unstable
![Find polar equations of the tangent lines to the curve at the pole r 2cos 30](https://media.kunduz.com/media/sug-question-candidate/20230415103302853731-4710940.jpg?w=256)
Calculus
Differential equationsFind polar equations of the tangent lines to the curve at the pole r 2cos 30
![The price per barrel of crude oil in the period 1980 2013 in constant 2014 dollars can be approximated by P t 0 27t 8 6t 93 dollars 0 t 33 where t is time in years since the start of 1980 Find P t and P 31 P t P 31 per year What does the second answer tell you about the price of crude oil Hint See Example 2 The price of a barrel of crude oil was increasing at a rate of per year in](https://media.kunduz.com/media/sug-question-candidate/20230414171528175719-4768444.jpg?w=256)
Calculus
Differential equationsThe price per barrel of crude oil in the period 1980 2013 in constant 2014 dollars can be approximated by P t 0 27t 8 6t 93 dollars 0 t 33 where t is time in years since the start of 1980 Find P t and P 31 P t P 31 per year What does the second answer tell you about the price of crude oil Hint See Example 2 The price of a barrel of crude oil was increasing at a rate of per year in
![dA Solve the initial value problem 5e t A 0 2 dt](https://media.kunduz.com/media/sug-question-candidate/20230414151103586370-3764988.jpg?w=256)
![In Problems 1 8 state the order of the given ordinary differ ential equation Determine whether the equation is linear or nonlinear by matching it with 6 Le 1 1 x y 4xy 5y cos x dy y 0 dx 1 6y 0 du dr 2 x 3 15y 4 d u dr 4 dx 5 dx u cos r u 1 dy dx](https://media.kunduz.com/media/sug-question-candidate/20230414050640350686-5535700.jpg?w=256)
Calculus
Differential equationsIn Problems 1 8 state the order of the given ordinary differ ential equation Determine whether the equation is linear or nonlinear by matching it with 6 Le 1 1 x y 4xy 5y cos x dy y 0 dx 1 6y 0 du dr 2 x 3 15y 4 d u dr 4 dx 5 dx u cos r u 1 dy dx
![The region in the first quadrant bounded by y 4x 3x y 7 and the y axis is rotate about the line x 3](https://media.kunduz.com/media/sug-question-candidate/20230414024025567629-5202343.jpg?w=256)
Calculus
Differential equationsThe region in the first quadrant bounded by y 4x 3x y 7 and the y axis is rotate about the line x 3
![Expone Sometimes we don t need to model rapid growth but rather rather decline We can use the exponential decay model P t Pe kt where k is the exponential decay rate k 0 Pis the initial amount and t is the time otore that calls 85 000 each](https://media.kunduz.com/media/sug-question-candidate/20230414021138261467-4341414.jpg?w=256)
Calculus
Differential equationsExpone Sometimes we don t need to model rapid growth but rather rather decline We can use the exponential decay model P t Pe kt where k is the exponential decay rate k 0 Pis the initial amount and t is the time otore that calls 85 000 each
![hment 11 blem Problem List cher wants to fence in](https://media.kunduz.com/media/sug-question-candidate/20230414021023634371-5357555.jpg?w=256)
![A woman invests 6200 in an account that pays 6 interest per year compounded continuously a What is the amount after 2 years Round your answer to the nearest cent b How long will it take for the amount to be 8000 Round your answer to two decimal places yr](https://media.kunduz.com/media/sug-question-candidate/20230414020459758780-4787741.jpg?w=256)
Calculus
Differential equationsA woman invests 6200 in an account that pays 6 interest per year compounded continuously a What is the amount after 2 years Round your answer to the nearest cent b How long will it take for the amount to be 8000 Round your answer to two decimal places yr
![Sketch a graph of a function f that has at least all of the following properties Draw in any vertical asymptote s where appropriate lim x 6 f x 4 lim x 3 f x 18 f 0 2 f x 0 when x 2 f0 5 0 lim x 8 f x 1 lim x 8 f x 4 limx f x](https://media.kunduz.com/media/sug-question-candidate/20230413122257127851-5523402.jpg?w=256)
Calculus
Differential equationsSketch a graph of a function f that has at least all of the following properties Draw in any vertical asymptote s where appropriate lim x 6 f x 4 lim x 3 f x 18 f 0 2 f x 0 when x 2 f0 5 0 lim x 8 f x 1 lim x 8 f x 4 limx f x
![2 3 equation in exponential form 125 3 4](https://media.kunduz.com/media/sug-question-candidate/20230412213551078559-5366129.jpg?w=256)
![Find the linear approximation of the function f x 1 x at a 0 L x Use L x to approximate the numbers 0 9 and 0 99 Round your answers to four decimal places 0 9 0 99 Illustrate by graphing f and the tangent line y 2 1 2 1 2 3 1 2 3 4 WebAssign Plot 5 X 5 2 5 10 15 20](https://media.kunduz.com/media/sug-question-candidate/20230412211458481596-4768444.jpg?w=256)
Calculus
Differential equationsFind the linear approximation of the function f x 1 x at a 0 L x Use L x to approximate the numbers 0 9 and 0 99 Round your answers to four decimal places 0 9 0 99 Illustrate by graphing f and the tangent line y 2 1 2 1 2 3 1 2 3 4 WebAssign Plot 5 X 5 2 5 10 15 20
![Find the derivative of the function y x e r 5 y](https://media.kunduz.com/media/sug-question-candidate/20230412211214079371-4768444.jpg?w=256)
![aluate the integral by making an appropriate change of variables 7 Cos 5 YX dA where R is the trapezoidal region with vertices 1 0 8 0 0 8 and](https://media.kunduz.com/media/sug-question-candidate/20230412035019282164-5485101.jpg?w=256)
Calculus
Differential equationsaluate the integral by making an appropriate change of variables 7 Cos 5 YX dA where R is the trapezoidal region with vertices 1 0 8 0 0 8 and
![4 Use geometry or symmetry or both to evaluate the integral 2 2 y y sin z dA where D x y x y 1](https://media.kunduz.com/media/sug-question-candidate/20230412025155763911-5485101.jpg?w=256)
Calculus
Differential equations4 Use geometry or symmetry or both to evaluate the integral 2 2 y y sin z dA where D x y x y 1
![Solve the following DE 2x y y x 2y](https://media.kunduz.com/media/sug-question-candidate/20230412012106147345-5526651.jpg?w=256)