Differential equations Questions and Answers
Calculus
Differential equationsFind 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
Calculus
Differential equationsUse 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
Calculus
Differential equationsSuppose 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
Calculus
Differential equationsInstructions 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
Calculus
Differential equations3 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
Calculus
Differential equationsLet f x 7x 5x 3 Use interval notation to indicate the largest set where f is continuous BEEDRAA KEDAEN Largest set of continuity
Calculus
Differential equations12 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
Calculus
Differential equations1 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
Calculus
Differential equationsThe 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
Calculus
Differential equationsy 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
Calculus
Differential equationsOn 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
Calculus
Differential equationsWhat 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
Calculus
Differential equationsFind the length of the third side If necessary round to the nearest tenth 7 6
Calculus
Differential equationsAll 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
Calculus
Differential equationsFind the point on the graph of z 3x 3y at which vector n 12 18 1 is normal to the tangent plane P
Calculus
Differential equationsIf 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
Calculus
Differential equationsGiven 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
Calculus
Differential equationsA 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
Calculus
Differential equationsIf 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
Calculus
Differential equationsFrom 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
Calculus
Differential equationsIf 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
Calculus
Differential equationsThe 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
Calculus
Differential equationsSuppose 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
Calculus
Differential equationsUsing 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
Calculus
Differential equationsSuppose 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
Calculus
Differential equationsyy 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
Calculus
Differential equationsConsider 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
Calculus
Differential equationsThe 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
Calculus
Differential equationsWhich 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
Calculus
Differential equationsA 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
Calculus
Differential equationsOA 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
Calculus
Differential equationsWhich 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
Calculus
Differential equationsFor 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
Calculus
Differential equationsFor 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
Calculus
Differential equations2 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
Calculus
Differential equations5 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
Calculus
Differential equations4 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
Calculus
Differential equations12200 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
Calculus
Differential equations13 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
Calculus
Differential equations17 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
Calculus
Differential equationsFind 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
Calculus
Differential equationsFor 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
Calculus
Differential equationsa 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
Calculus
Differential equationsor 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