Differential equations Questions and Answers

3 Find and classify the equilibrium solutions for the each of of the following differentia equations a y 3 y b y 5 y c y y 3 y d y y 5y 6 e y 1 y y 2 y 3 f y y y

Find the eigenvalues and eigenfunctions for the following boundary value problem with 0 y xy 0 with y 0 0 y 5 0 Eigenvalues An 22 2n 1 100 Eigenfunctions Yn COS 2n 1 100 X

3 In the apothecary weight system one scruple is 20 grains and three scruples equals one dram Determine the number of grains in 50 drams 4 On a trip of 834 miles an automobile consumed 35 5 gallons of gasoline costing 4 23 per gallon What was the cost of gasoline per mile traveled 5 The following relationships describe an antiquated set of British liquid units Assume all equalities provided are defined 1 hogshead 7 firkin 18 pottle 1 firkin 140 pottle 1 puncheon 504 pottle 1 tun Use dimensional analysis to determine the number of tuns in 144 hogshead

The limit below represents f c for some function f and real number c 8 x 16 2 x 8 Determine the function f lim I 8 f x Determine the value of c

Suppose f x is continuous over an interval a x b and a c d b If f c 5 then check all that apply f x is increasing through a c Of does not have a minimum at x c f x is decreasing through a c f c 5 If does not have a maximum at c f c exists If f d 3 then check all that apply f x is increasing through a d f does not have a maximum at x d f does not have a minimum at x d f d 3 f x is decreasing through d f d exists Based on this we know that for some between c and x d f x 0 f has a maximum Of has a minimum If does not exist

tempt 1 0 5 points eing dumped from a conveyor belt at a rate o is such that it forms a pile in the shape of a d are always equal How fast is the height of th feet high Round the result to the nearest hur t min

Determine whether 31 is a solution of the le 31 a solution

13 15 SIX 213 17 5 5 2x 1 x x 12 x 9 Rationalize the numerator b a b a 14 2 x 3 18 1 16 x hx h 1 x 5 3 x h 3x h

0 Suppose a population of rabbits Y is given by the logistic equation 300 5 55e 84t y where Y is the number of rabbits at time t in months a Find the initial population b Find the carrying capacity That is the maximum population think limit c How long until the rabbit population grows to 100 d Find an initial value problem differential equation and an initial value where the given equation y is the solution If you understand logistic equations very little work

Find A the area of the region under the curve y 2x 9 and above the x axis for 2 x 5

For the data provided below what is the median 1 3 5 6 6 7 9 10 14 14 15 16 20 21 27 32 41 O 14 01 O 41 40

20 points Linearize the following nonlinear differential equations for the input equilib rium of u 1

Product Quotient and Trig Let f x and g x be differentiable functions such that f 5 2 f 5 6 g 5 4 and g 5 3 If h x f x g x then find h 5 Give the exact value NO DECIMALS h 5 0

6x y 9 and passing through 6 7 Express the equation in standard form

The region R in the first quadrant of the zy plane is bounded by the curves y x 0 and y 0 A solid S is formed by rotating R about the y axis The exact volume of S is 2x 12x 32

A bucket filled with sand has a total weight of 25 pounds You begin steadily lifting the bucket to the top of a building 35 feet above the ground using a thin cable of negligible weight but the bucket has a hole in it and leaks 0 1 pounds of sand for each foot you lift it The work you do lifting the bucket to the top of the building is foot pounds

f x is the horizontal axis is X Given the differential equation x t f x t List the constant or equilibrium solutions to this differential equation in increasing order and indicate whether or not these equilibria a stable semi stable or unstable

A trough full of water is 7 meters long 2 meters wide at the top and 2 meters deep with vertical cross sections parallel to the ends shaped like an isosceles triangle The amount of work required to empty the trough by pumping the water over the top edge is Recall that th Joules

3 Find each solution x t determine any stable equilibrium solutions a x 5x x b x 2 x

Knowledge check 1 winter 2U23 2 1 par a On what interval Aa 8 EB February 4 2023 at 1 19 PM on what intervals would the graph of your position be concave up and cave down on what intervals would the graph of your position be increasing and reasing ow many miles did your group travel between 10am and 2pm How is that distance relate to the graph Compute the total distance traveled on your first day driving ketch an accurate labeled graph of your position 3 7 v t

A cliff diver leaps from a 100 foot cliff into the Pacific The diver s height above the water is modeled by the function P x 16x 2 10x 100 where x is time in seconds During which interval is the diver at least halfway down

Rain water flows into an open tank at a rate of g t 200e 0 3t where g is in gallons per day and t is in days The water in the tank evaporates at a rate of 3 per day Let w t be the amount of water in the tank on day t Write a differential equation in w that models this scenario

The contour plot of z f x y is shown below Determine if the given quantity in each part is positive negative or zero Negative V 1 Negative af x Positive V 3 at the point Q Positive 2 The value of Vf j at the point P 2 1 4 1 af y 4 The value of Vfi at the point P 2 1 at the point Q 4 1 2 4 1 6 0 8 OF 0 8 1 6 2 4 30 0 4 5 30 0 5 0 1 5 15 0 0 X 20 0 10 0 25 0 1 5 Click on graph to enlarge 30 0 30 0 4 5

A particle moves along a line At time t in seconds the particle s velocity is v t t 6t 8 in m s Find the particle s net displacement and total distance traveled over the time interval 0 5 Net displacement Total distance traveled m m To understand the terms net displacement and total distance travelled imagine a particle that moves from z 0 to z 2 then back to z 1 Its net displacement is 1 since it started at position 0 and ended at position 1 Its total distance travelled is 3 since it moved 3 units all together

Let f f x dx 7 a What is the average value of f x on the interval from x 0 to x 4 average value b If f x is even find each of the following f x dx the average of f x on the interval x 4 to x 4 c If f x is odd find each of the following Sf x dx the average of f x on the interval x 4 to x 4

Given D 2 3 2 1 D 2 3D H 0 where I 1 0 0 1 find the value of k

Use the Laplace transform to solve the following initial value problem y 4y 32y 8 1 2 y t y 0 0 y 0 0 Notation write u t c for the Heaviside step function uc t with step at t c

Find the indicated functional values f x 3x 3 2x 2 3x 4

Find the area of the surface generated by revolving the given curve about the x axi y 4 x 1sxsl

a b Find the exact numerical value of each expression sinh ln3 cosh In 2

List out all possible numerators of the rational roots of the function g x 2x 5 4x 4 8x 3 2x 2 x 8 Use commas to separate

ple gives the increase or decrease in the number of donors to a college athletics support organization for selected years Rate of Change of Donors to a College Athletics Support Organization Year 1985 1988 1991 1994 1997 2000 Rate of change donors per year 184 812 1241 1079 404 832 a Find the function of the quadratic model that gives the rate of change in number of donors where t is the number of years since 1985 with data from 0 st s 15 Round all numerical values to three decimal places rxe donors per year b Find the function of the model that gives the number of donors where it is the number of years since 1985 Use the fact that in 1990 there were 10 706 donors Round all numerical values to three decimal places F 0 donors c Estimate the number of donors in 2003 Round your answer to the nearest integer

According to the 68 95 99 7 Rule 99 7 of high school seniors had SAT scores between what and what

1 Vertex Domain 0 9 8 Range 0 118 Axis of Symmetry x 5 Zeros Y Intercept Average rate of change between second 1 and second 3 The Average rate of change 30 YA Height in meters 120 100 80 60 40 20 1 2 3 4 56 7 Time in seconds 8 9 10

2 Solve the equation du a y 0 x 3 t 0 t x given that u 0 t u 3 t 0 u x 0 5 sin 47x 3 sin 8 x 2 sin 10 x u x t M 2

Find the Fourier Series for the function f x 0 sin x if x 0 if 0

1 point Consider the two lines and L x 2t y 1 2t z 3t L x 6 2s y 2 3s z 1 5 Find the point of intersection of the two lines P

5 Even or odd or neither Prove with algebra and a sketch of the graph from calculator A f x 2x x B f x 2x

marks Ito calculus 3 marks Let X t and Y t be two stochastic processes such that dX x X t t dt ox X t t dZx dY uy Y t t dt oy Y t t dZy Xi 1 Xi Yi 1 Yi with dZx t dzy t being the increments for two distinct Wiener processes Zx t and Zy t Let X t X and Y ti Y Show that Xi 1Yi 1 XiYi Xi Yi 1 Yi Yi Xi 1 Xi Then using the definition of the Ito integral which is the limit of a discrete sum show that X s dy s XY Y s dx s dx s dY s

The expression In t 2ln x ln 6 can be simplified to In w where w

1 10 points Consider the equation cosx y x sinx y x secx y x 0 The functiony 1 x secxis a solution of this equation Obtain a second linearly independent solutiony2of the equation for 2 x 2 and write the general solution

Refer to the graph of y f x x x shown a Find the slope of the secant line joining 3 f 3 and 1 f 1 b Find the slope of the secant line joining 3 f 3 and 3 h f 3 h c Find the slope of the graph at 3 f 3 d Find the equation of the tangent line to the graph at 3 f 3 a The slope is 3 Type an integer or a simplified fraction b The slope is 5 h c The slope is 5 Type an integer or a simplified fraction d y 5x 9

Find the average value of f x cos x over the interval average value The average value is Type an exact answer using as needed TC TU 4 4 Draw a graph of the function and indicate the

h 0 8 7 2 7 6 4 N 6 8 10 6 8 6 2 5 7 5 3 e estimate distances from velocity data it is some cessary to use times to 11 12 13 that are not spaced We can still estimate distances using the iods Ati ti ti 1 For example in 1992 the space Endeavour was launched on mission STS 49 in install a new perigee kick motor in an Intelsat com ions satellite The table provided by NASA gives stimate the total amount of infection needed to develop ymptoms of measles N f t 1000 In Copyright 2021 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02 200 203 ight 2021 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part Due to electronic rights some third party content may be suppressed from the eBook and or eChapter s view has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require sured in number of infected cells per mL of blood plasma reaches a peak density at about t 12 days when a rash appears and then decreases fairly rapidly as a result of immune response The area under the graph of N t from t 0 tot 12 as shown in the figure is equal to the total amount of infection needed to develop symptoms measure in density of infected cells X time The function N has been modeled by the function f t t t21 t 1 Use this model with six subintervals and their midpoints to SECTION 5 1 b Show that R The Area and Distance Problems b f a Then draw a diagram to illustrate this equation ing that the n rectangles representing R Ln C reassembled to form a single rectangle whose a right hand side of the equation c Deduce that Ln b a R A n b a f b f a

3 a Estimate the area under the graph of f x 1 x from x 1 to x 2 using four approximating rectangles and right endpoints Sketch the graph and the rectangles Is your estimate an underestimate or an overestimate b Repeat part a using left endpoints

7 Polar Coordinates are just parametric equations Really Find the are lengths of the cardioid with polar equation r 1 cos 0 Also find also the surface area S generated by revolving the cardioid around the z axis

6 Consider the autonomous differential equation frac d y dt y 1 2 y 5 a Determine the equilibrium constant function solutions b Classify each of these equilibrium solutions as either stable unstable or semi stable c Sketch solution curves which correspond to y 0 0 y 0 3 and y 0 8

7 Find all solutions of these differential equations a b c b dy dx c dy dt dy dx u 1 t 8 Solve the initial value problem a u t y dy dx dy ye x y P 3t y x y 4 1 sinh x y 0 1

For each differential equation below state its order and determine whether it is linear or nonliear A y y xy cost 2y BI 2ry ysinx

Find all points where the function is discontinuous O x 1 O None O x 2 x 2 x 1