Differential equations Questions and Answers

A factory is discharging pollution into a lake at the rate of r t tons per year given below where t is the number of years that the factory has been in operation Find the total amount of pollution discharged during the first 7 years of operation Round your answer to two decimal places r t t 1 tons

b x 24 2x3 5x2 1 x 2 4 4

2 Let y f x be the continuous function that satisfies the equation x 5x y 4y 0 and whose graph contains the points 2 1 and 2 2 Let be the line tangent to the graph of fat x 2 a Find an expression for y b Write an equation for line 1

1 Given the curve x xy 2y 6 a Find an expression for the slope of the curve at any point x y on the curve

1 Find the Taylor series for f x sin x centered at a

sec x y 6 sec ase AWY a The period is Type an exact answer in terms of b Select the correct choice below and fill in the answer box within your choice OA The phase shift is B The given function has no phase shift c Find the range of the function Choose the correct answer below O A 00 00 OC 6 6 units Type an exact answer in terms of OB 6 U 6 0 OD 6 U 6 00

5 Solve y 3x y y 0 1

Solve y y 1 Gal

18 f x 3x 15

Final Exam Practice Problems Consider the region bounded by y ln x over 1 e a Set up integrals using both methods if revolved about the z axis Repeat if revolved about the y axis Find each volume b Set up integrals if revolved about y 1 both methods c Set up an integral that computes the length of the curve d Set up an integral that computes the area if the curve is revolved about the z axi

Use absolute value on a graphing calculator to find the area bounded by the graphs of the equations over the given interval y e x y 0 1x 0 4 2 x 2 The area is

3 2 T 14 13 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 10 The curve above is the graph of a sinusoidal function It goes through the points 6 0 and 4 0 Find a sinusoidal function that matches the given graph If needed you can enter T 3 1416 as pi in your answer otherwise use at least 3 decimal digits f x

1 cosx icosx ifadesi a a dakilerden hangisine E

cal integration command on a graphing calculator Find the future value at 9 75 interest compounded continuously for 10 years of the continuous income stream with the rate of function f t 700t 11 000 GLOOB The future value of the income stream at 9 75 compounded continuously at the end of 10 years is Round to the nearest dollar as needed 0

3 The half life of a radioactive substance is 20 years If we start with 15 gram a how much is left after 5 years b how long will it take to have 5 grams left

Find the consumers surplus and the producers surplus at the equlibrium level for the given price demand and price supply equations Include a graph that identifies the consumers surplus and the producers surplus Round all values to the nearest integer p D x 36 0 06x p S x 11 0 04x The value of x at equilibrium is 250 The value of p at equilibrium is 21 The consumers surplus at equilibrium is

In 2012 34 1 of all glass containers were recycled A beverage company used 350 000 lb of glass containers per year After recycling the amount of glass in pounds still in use after t years is given by N t 350 000 0 373 a Find N 7 and explain its meaning b Find N 7 and explain its meaning c When will 6 of the original amount of glass still be in use a N 7 lb Round to two decimal places as needed

Determine which of the following functions is a solution to the differential equation Oy 4t Oy 3t Oy 14 Oy 4 Oy 1 dy dt 4y

Use a graphing calculator to approximate the area between the graphs of y In x and y 3xe on the interval 1 4 The area is ww

Determine whether or not the vector field is conservative If it is conservative find a function f such that F Vf F x y 6x 5y i 5x 4y j

Evaluate the line integral 4xyds where C is the line segment joining 4 5 to 5 4

A thin wire is bent into the shape of a semicircle x y 4 x 0 If the linear density is 4 find the exact mass of the wire

Find the work done by the force field F on a particle that moves along the curve C where F x y 5x 5y i 4xyj C r t 3t i t j 0 t 3

Convert this integral from rectangular coordinates to cylindrical coordinates Do not evaluate the integral S 4 2 16 2 So yzdzdydx

a Change the point 2 3 3 5 which is in rectangular coordinates to cylindrical coordinates Let r 0 and 0 0 2TT b Describe in words the surface whose equation is z 49

20 points Define g x y x x y 1 Compute the difference between the maxi and minimum of g over the region bounded by the y axis and the circle x y 4

8 5 points Use a calculator with definite integral capability to find decimal places A 2 4028 B 1040 0000 C 1037 7500 D 0 0039 E 0 7315 2 1 3 1 dr Give dr Give your answer t

2 Individuals with Pompe disease are unable to metabolize and are treated with a human enzyme produced by a glycogen transgenic hamster cells b lactose transgenic crops insulin cloned rabbits C d e cellulose transgenic hamster cells fats cloned hamster cells

9 points Find all local absloute maxima minima of the function f x 2x 1 1 x 5 on the real line o

9 points Let S t be the function which specifies the distance in km from a runner to the start line at time t in hours of a race The graph of S t is given below 4 3 2 1 y km 2 3 b What happened between times 3 and 4 S t Use the graph to give approximate answers to the following problems a When was the runner running the fastest c What is the meaning of S 1 t h

Which of the following functions satisfy the given differential equation y y a ex b xe x c 3e x d ex e xex

A worker has developed a process for assembling a new product Initially assembly will be slow because the process is new but eventually the process will allow for close to 90 items to be produced a day This can be modeled by the equation f x 90 110 x 3 where f x represents the rate of production in items per day How many items will be produced from day one to day two Round your answer to the nearest whole item Provide your answer below

3 a Fill in the table with the exact values and their approximations where necessary 27 14 PK 2n 7n 4 S R T 37 KA 5 4 31 JT 4 2 X 1 sinx COSX b Sketch y tan x for 27 x 2 tan x m sin COSX 10 0 I 5 1 4 0 2 1 2 0 2 1 4 0 und it 2 31 TC 5 4 4 42 1 3 4 0 0 I E It 5 T L und 77 27 4 2 1 0 3 7 2n 2

Suppose the rate of growth of bacteria in a Petri dish is given by q t 6 where t is given in hours and q t is given in thousands of bacteria per hour If a culture starts with 7000 bacteria find a function Q t that gives the number of bacteria in the Petri dish at any time t How many bacteria are in the dish after 4 hours You should round your answer to the nearest whole number Do not include any commas in your final answer if applicable Provide your answer below bacteria

Simplify the expression cos t 1 sin t sin t cos t 1 as much as possible help formulas

6 15 Use the method of variation of parameters to find the general solution to the differential equation y 2y y x 1 ex No credit for any other method

Find f Use C for the constant of the first antiderivative and D for the constant of the second antiderivative f x 10x sin z f x

6 8C6

3 4 The graph of a twice differentiable function is shown in the figure above Which of the following is true A 1 1 0 B 1 0 S 1 C 0 1 S 1 D 1 f 1 f 0 E I 0 1

6 If x4 x2 x 1 is divided by x2 1 the remainder is

From the graph determine the x and y intercepts and the vertical and horizontal asymptotes If an answer does not exist ONE Enter your asymptotes as a comma separated list of equations if necessary x intercept y intercept vertical asymptote horizontal asymptote 10 5 y 10 S 5 S 10 M

rms of Linear Equations Show all calculations in the space provided Please remember to indicate the formula s you use Describe the equations below List and identify the information provided by the equations 1 y 8 x 3 2 y x 3 Place the equation in slope intercept form 3 3x 2y 18 0 3 y 4x 4 y 2 x 5

3 The rate of growth of a fish population was modeled by the equation 0 61 G t 60 000e 1 5e06 where t is measured in years since 2000 and G in kilograms per year If the biomass was 25 000 kg in the year 2000 what is the predicted biomass for the year 20303

Suppose the temperature in degrees Celsius over a 9 hour period is given by T t 1 7t 39 a Find the average temperature b Find the minimum temperature c Find the maximum temperature a The average temperature is degrees Type an integer or a decimal Round to one decimal place as needed

Asso W Sketch the region enclosed by the given curves x 1 x3 X 1 x2 0 5 0 5 y 1 0 0 5 0 5 1 0 y 1 0 0 5 0 5 y 0 5 0 5 1 0 1 0 X M 0 5 0 5 1 0 0 5 0 5 1 0 0 5 0 5 y 1 0 0 5 webassign net 0 5 1 0 1 5 1 0 2 0 X X

89 M Sketch the region enclosed by the given curves Decide whether to integrate with respect to x or y Draw a typical approximating rectangle 2x y 8 x y 4 2 1 1 2 4 4 2 y 1 C JS Sec

2 12 pts The half life of cesium 137 is 30 years Suppose we have a 10 gram sample a Find a function m t moet that models the mass remaining after t years b Calculate the mass remaining after 50 years Give exact and approximate answers 1 5 grams of the sample remain Give exact and approximate solutions

Find the solution of the simultaneous linear differential equations dx dt dy 2x2y 4e 2t dt that satisfies the initial conditions 0 7 y 0 4 5x 3y 17e 2t

s x sin x cos x v x sin x X t x w x X cos x sin x u x x 4x 5x x 3x 2 y x 2x 5x 1

Calculate the second and the third derivatives of the following functions f x 4x 6x 2x 1 k x sin 2x x 1 g x x 1 h x cos 2x x 2x 1 m x 2 n x x 4x x 1