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

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

and G The difference between angles

Solve using the square root method 3 x 6 2 3 144

ind 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

One milliliter of liquid has the equivalent V

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

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

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

e 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

Lines 1 and 12 are parallel lines Determine the measures 412

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

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

Find 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

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

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

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

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

Find 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

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

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

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

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

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

Write the constraint below as a linear inequalit

Solve the system of linear equations by graphing x 6 y 5

Determine 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

the 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

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

Find 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

dA Solve the initial value problem 5e t A 0 2 dt

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

The 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

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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

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

2 3 equation in exponential form 125 3 4

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

Find the derivative of the function y x e r 5 y

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

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

Solve the following DE 2x y y x 2y