Differential equations Questions and Answers

Find the minimum value as well as the point at which the minimum occurs of L 2x 5x2 x3 subject to the constraints VI X1 2X2 X3 6 X2 2X3 6 2x2 x3 4 X1 X2 X3 0

Use the integral test to determine whether each of the following series converges or diverges For each fill in the integrand and the value of the integral Enter diverges if the integral diverges Then indicate the convergence of the sum A ZT n 1 Compare with f dx Evaluate your integral with bottom limit c 1 This sum O A converges O B diverges 8 B n 7 n 14n 2 8 Compare with fdx Evaluate your integral with bottom limit c 1 This sum O A converges B diverges

Suppose f x y tan x y and u is the unit vector in the direction of 3 2 Then a Vf x y b Vf 0 5 9 c fu 0 5 9 Du f 0 5 9

Instructions Use algebraic techniques to solve problems unless otherwise instructed Show the calcula to your answers except basic arithmetic when there is work to show Simplify and circle your final ans 1 8 pts Consider the equation 4x 4y 4x 24y 37 36 a What is the standard form of the equation of the circle Also find the coordinates of the cen radius and the diameter of the circle 2 2 4x 4y 4x 24y 37 36 4 x y x y 36 37 4 x y2 x 6y 1 2 x y x y XX 2 1 1 y by 9 x 1 y 3 9 2 center b1 3 2 padus 13 Diam 2xradius b 4 pts Sketch a graph of the circle 2x 3 E

Find the following Assume x can repre XXX A

3 Find the graph of a function that is defined and increasing for all x 20 has an inflection point at x 8 and is asymptotic to the line y x 15 Choose the correct graph O A Ay 15 N X 0 4 8 12 16 O B 15 X 0 4 8 12 16 O C 15 O X 4 8 12 16 D 15 a X 0 4 8 12 16

Let f x 7x 5x 3 Use interval notation to indicate the largest set where f is continuous BEEDRAA KEDAEN Largest set of continuity

12 Using the disk method determine the volume of the solid obtained by rotating the region bounded by y x 4x 5 x 1 x 4 and the x axis about the x axis

1 point Rebecca has 45 coins all nickels and dimes The total value of the coins is 3 60 How many of each type of coin does Rebecca have 13 dimes and 32 nickels 10 nickels and 35 dimes 18 nickels and 27 dimes 27 dimes and 18 nickels

The electric current i in A as a function of the time t in s for a certain circuit is given by i 5t t Find the average value of the current with respect to time for the first 5 0 s The average value of the current with respect to time is Type an integer or decimal rounded to one decimal place as needed

y Find the length of the curve x 4 1 2 8y The length of the curve is Type an integer or a simplified fraction from y 1 to y 3

On a standardized exam the scores are normally distributed with a mean of 300 and a standard deviation of 40 Find the z score of a person who scored 400 on the exam

What is the volume in cubic centimeters of a cylinder with a height of 12 centimeters and a base radius of 6 centimeters to the nearest tenths place Answer Attempt 1 out of 2

Find the length of the third side If necessary round to the nearest tenth 7 6

All exponential functions can be written in many forms Write the function f t 570 2 7 in the form f t ab Round all coefficients to four decimal places Answer Attempt 1 out of 2 t 15

Find the point on the graph of z 3x 3y at which vector n 12 18 1 is normal to the tangent plane P

X 9 x Let g x h g x 1 h x x3 3x 8 Calculate the function h g x

If dx 1 x xy x 0 and y 4 when x 1 when x 3 y OA 2 381 OB 4 711 OC 3 898 OD 3 898 OE 4 711

Given the values of f x shown in the chart below which of the following could be values for f x at x 11 212 and 3 X 1 2 3 f x 3 5 A B C D E X 2 1 2 31 12 f x 2 4 2 X X f x 2 5 X 1 X f x 2 f x 4 3 13 12 2 2 1 2 21 1 2 31 1 2 8 3 2 4 2 12 4 2 2 3 2

A particle moves along the x axis so that at any time t 0 its position is given by x t 3 12 2 36t For what values of t is the particle at rest A No values OB 2 only O C 3 only O D 6 only OE 2 and 6

If d dy x cos x and y 3 when x 0 when x y dx A 6 OB 1 647 OC 3 215 OD 2 OE 3T

From this table of data 1 9 1 99 1 999 2 001 2 01 2 1 g x 4 5 4 97 4 998 5 002 5 03 5 5 X it would appear that limx 2 g x O A 5 OB Nonexistent OC 2 OD 2 OE 5

If y x 1 y and y 1 then y OA y 3x C OB y Cel 2x 1 O C y sin x OD y C x 3 OE y C

The graph of y g x is shown below where the curve above the axis is a semi circle The value of fog x dx OA 4T 15 O B 5 2 OC 2 6 OD 6 2TT O E 5 2T 15 Click here for long description

Suppose the number of bacteria in a culture increases by 50 every hour if left on its own Assuming that biologists decide to remove approximately one thousand bacteria from the culture every ten minutes which of the following equations best models the population P P t of the bacteria culture where tis in hours O A 1 5P 1000 dt OB 1 5P 6000 O c 5P 1000 OD 5P 5P 1000 OE dP dt dP dt dP dt F 5P 6000

Using separation of variables technique solve the following differential equation with initial condition 1 and y 3 5 The solution is dy yx 5x dz O A In y 5 ln x 1 O B ln y 5 arctan3 n10 arctan 3 C y ln x 1 50 In 10 O D y In x 1 25 In 10 O E In y 5 ln x 1 ln 10

Suppose you want to find an equation for a curve whose slope at any point x y is x 6y2 You d need to solve the differential equation x 6y Using separation of variables to solve this equation you get where C is an arbitrary constant of integration O A y x C B 3y 2x C C 2y x C O D 2y x C E 6y x C

yy ex 0 and y 4 when x 0 which means that O A In x 4 6 OB 2 4x 3 O c y 2x x OD y x In x 4 O E 2 2ex 14

Graph I Graph II Graph III Click here for long description Graph IV

Consider the position function s t 4 9t 25t 25 Complete the following table with the appropriate average velocities and then make a conjecture about the value of the instantaneous velocity at t 3 Complete the following table Type exact answers Type integers or decimals Time Interval Average Velocity 3 4 3 3 1 3 3 01 3 3 001 3 3 0001 GOLA

The differential equation dy dx O A y Cekt OB y C x 4 O C y Ce x O D y e C E x y C af has a solution given by

Which of the following is not a differential equation A y 0 dx OB 4 y 2x dy c x x dx 4 y Y O D t 1 2 4 BOULL d y dt2 2 E t s ds

A separable differential equation is a first order differential equation that can be algebraically manipulated to look like A Both f y dy g x dx and f x dx f y dy B f x dx f y dy O c f y dy g x dr O D f x dx f y dy O E g y dx f x dx

OA ye dy xdx O B ydy xe d C None of these The equation is not separable OD ydy xe dx E udy re 2 dr

Which of the following is a separable first order differential equation O A y 2 In xy x y OB d x Y O c dy da xy x y dy O D sin x O E y x y 2x

3 Write the equation of a vertical line passing through 8 5

For what value does the function fail to exist f x A x 4 C x 4 3 X 4 B x 3 D x 0

For what values does the function fail to exist A x 5 and x 0 f x C x 1 5 and x 0 X x2 5x B x 0 and x 5 D x 1 5 and x 0

Find the limit es lim t 0 t 1 t

2 Camille launches a model rocket in an open field near her house The rocket has a bit of a problem being slightly off balance Its trajectory is described by the function y 60 In x 1 6x for 0 x 36 15 where y is the rocket s height in feet above the ground and x is the horizontal distance in feet between the launch site and a point directly below the rocket Below is a graph of the trajectory of the rocket 801 60 40 201 0 5 10 15 20 25 30 35 A What is the maximum height of the rocket during its flight

5 Consider the differential equation y 1 x A Use the axes provided to sketch a slope field for the given differential equation at the eleven points indicated 2 points 1 1 1 B Use the slope field for the given differential equation to explain whether a solution could have the graph shown at right 2 points H C On your slope field from part a sketch the graph of the particular solution y f x such that f 0 0 5 1 point

4 At high temperatures nitrogen dioxide NO2 decomposes into NO and O2 If y t is the concentration of NO2 in moles per liter then at 600 K y t changes according to the reaction law 05y for time t in seconds A Express y in terms of t and the initial concentration y B Assuming that the concentration of NO is twice as high at t 20 seconds as it is at 100 seconds find the exact initial concentration of the NO Reminder Exact means no

12200 4 Consider a lake of constant volume 12200 km3 which at time contains an amount y t tons of y 1 pollutant evenly distributed throughout the lake with a concentration tons km Assume that fresh water enters the lake at a rate of 67 1 km yr and that water leaves the lake at the same rate Suppose that pollutants are added directly to the lake at a constant rate of 550 tons yr Among the many simplifying assumptions that must be made to model such a complicated real world process is that the pollutants coming into the lake are instantaneously evenly distributed throughout the lake A Write a differential equation for y t B Solve the differential equation for initial condition y 0 200000 to get an expression for y t Use your solution y t to describe in practical terms what happens to the amount of pollutants in the lake as goes from 0 to infinity

13 14 Use the graph of the function f to state the value of eac limit if it exists If it does not exist explain why a lim f x x10 b lim f x x40 13 f x x 1 x c lim f x x 0 14 f x 2 el x e x 1

17 Consider the differential equation given by A On the axes provided below sketch a slope field for the given differential equation at the nine points indicated B Let y f x be the particular solution to the given differential equation with the initial condition Based on your slope field how does the value of f 0 2 compare to f 0 Justify your answer C Find the particular solution y f x to the given differential equation with the initial condition f 0 3 Use your solution to find 0 2 y 1 3 2 dy xy dx 2 1 X

Find an equation of the line below Slope is 3 3 6 on line What is an equation for a line that has a slope of 3 and passes through the point Type an equation Type your answer in point slope form

Factor the given polynomial x 16x 55 x 16x 55

For a function y g x the differential dy is given by O A dy g x dx OB dy g u dx O c dy g x dx O D dy dda O E dy da

a Set up an integral for finding the Laplace transform of f t 1 6 B F s f t A where A and B help formulas b Find the antiderivative with constant term 0 corresponding to the previous part c Evaluate appropriate limits to compute the Laplace transform of f t F s L f t d Where does the Laplace transform you found exist In other words what is the domain of F s

or a certain electric circuit the current in A is given by i 4 31e 52 4t di sin 219t where t is the time in s Find the expression for dt HILF E