Calculus Questions

A tank contains 2 000 L of brine with 16 kg of dissolved salt Pure water enters the tank at a rate of 20 L min The solution is kept thoroughly mixed and drains from the tank at the same rate a How much salt is in the tank after t minutes kg yo b How much salt is in the tarik after 30 minutes Round your answer to one decimal place y kg

Points Let S be the solid obtained by rotating the region shown in the figure about the y axis Assume a 3 and b 2 y axx b Sketch a typical approximating shell y 8 6 4 x Fi 3 N y

Solve the differential equation de t sec 0 dt Beta

Find the solution of the differential equation that satisfies the given initial condition dA Ab2 cos br A 0 b dr

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y axis y y 0 x 2 V Sketch the region and a typical shell y 10 8 6 4 X 3 2 4 X 2 10 2 y 8 6 2 2 4 X

Solve the differential equation dy dx 7x y for y 0

Write a differential equation that models the given situation The stated rate of change is with respect to time t Use k for the proportionality constant For a car with maximum velocity M the rate of change of the velocity v of the car is proportional to the difference between M and v dv dt Need Help Read

Suppose that x 2y t and y tsin 8 Find 3 0 dy at and at

Suppose that x 2y and y tsin 9 Find 8 oy dy and

Select the correct answer Which equation has infinitely many solutions A 5 x 2 4x 12 x 2 B 3x 4 2x 5 x 4 C 5 x 7 5 x 7 D 6x 2 x 2x 2 7x

Along the curve r t t i j tk 0 st 1 evaluate each of the following integrals a x y z dx b x y z dy x y z dy c f x y z a b C C C K y z dx 13 6 CA C f x y z dy z dy Type an integer or a simplified fraction C Type an integer or a simplified fraction dm X

Evaluate f x y x y ds where C is the straight line segment x 2t y 8 2t z 0 from 0 8 0 to 8 0 0 C x y ds C Type an exact answer

Integrate f x y z The integral of f is x y z over the path r t 2t i 3t j tk 0 astsb 2 x y z

3 Solve the following system of non linear equations by using the fi x y x x 1 2y 18 0 Newton Raphson method Start with xo 2 and yo 11 and carry out the first two iterations Firstly write the general formula of this method f2 x y x 1 y 6 25 C for solving system of non linear equations and analyse the convergence criteria of this method olution 3 Solution by the Newton Raphson method

A plane flying horizontally at an altitude of 2 mi and a speed of 600 mi h passes directly over a radar station Find the rate at which the distance from the plane to the station is increasing when it is 3 mi away from the station

Integrate f x y z The integral of f is x y z 2 2 x y z over the path r t ti tj 2t k 0 ast b

Find the line integrals of F 4yi 2xj zk from 0 0 0 to 1 1 1 over each of the following paths a The straight line path C r t ti tj tk 0 st 1 b The curved path C r t ti t j t k 0st 1 c The path C3UC4 consisting of the line segment from 0 0 0 to 1 1 0 followed by the segment from 1 1 0 to 1 1 1 0 0 0 C 1 1 CS CA 1 1 0 a The line integral of F over the straight line pa C is Type an integer or a simplified fraction

Find the gradient field of the function f x y z 3x 3y 4z 1 2 The gradient field is Vf i j k

Evaluate fc xy x xy x z ds along the curve r t 2ti tj 5 2t k 0 st 1 c xy c xy x z ds Simplify your answer ww

Find the line integral of f x y ye along the curve r t 4t i 3t j 1sts 1 he integral of f is Type an exact answer

Find the line integral along the given path C fix x C y dx where C x t y 7t 2 for 0 st 4 x y dx

Evaluate x y ds where C is the straight line segment x 2t y 8 2t z 0 from 0 8 0 to 8 0 0 f x y ds 0

Find the area of the region bounded by y 1 1 x2 y 2 x 1 and the y axis Find the area of the region bounded by x 6y2 4y x 3y 2

Given that the directional derivative of f at the point 1 5 5 in th direction of a 9i 6j 2k is 4 and that Vf 1 5 5 4 find Vf 1 5 5 f 1

Use Euler s method with step size 0 2 to estimate y 1 where y a is the solution of the initial value problem y 5x y y 0 0 y 1 11 95005255 x

1 point Suppose the clean water of a stream flows into Lake Alpha then into Lake Beta and then further downstream The in and out flow for each lake is 400 liters per hour Lake Alpha contains 500 thousand liters of water and Lake Beta contains 200 thousand liters of water A truck with 400 kilograms of Kool Aid drink mix crashes into Lake Alpha Assume that the water is being continually mixed perfectly by the stream a Let x be the amount of Kool Aid in kilograms in Lake Alpha hours after the crash Find a formula for the incremental change in the amount of Kool Aid Ax in terms of the amount of Kool Aid in the lake x and the incremental change in time Ar Enter At as Deltat Ax kg b Find a formula for the amount of Kool Aid in kilograms in Lake Alpha t hours after the crash kg x 1 c Let y be the amount of Kool Aid in kilograms in Lake Beta 1 hours after the crash Find a formula for the incremental change in the amount of Kool Aid Ay in terms of the amounts x y and the incremental change in time Ar Enter At as Deltat kg Ay d Find a formula for the amount of Kool Aid in Lake Beta 1 hours after the crash kg y t

1 point Initially 10 grams of salt are dissolved into 40 liters of water Brine with concentration of salt 4 grams per liter is added at a rate of 2 liters per minute The tank is well mixed and drained at 2 liters per minute a Let x be the amount of salt in grams in the solution after 1 minutes have elapsed Find a formula for the incremental change in the amount of salt Ax in terms of the amount of salt in the solution x and the incremental change in time At Enter At as Deltat Ax grams b Find a formula for the amount of salt in grams after t minutes have elapsed x t grams c How long must the process continue until there are exactly 20 grams of salt in the tank minutes

c1 e 1 16 c2 e 4 t 3 c1 e 1 12 c2 e 4 t 4 c1 e 1 c2 e 4 t 2 5 3 e 1 7 e 4 t 3c1e 1 12c2e 4t b Find x 1 if x 0 4 and x 0 2 x t 2 5 3e t 7e 4t c e 16c e4 3c e 12c e4 help formulas 4 c e c e 4 c1e t c2e 4t 7e4 correct least one of the answers above is NOT correct point Suppose x c e c e t a Verify that x c e c e is a solution to x 3x 4x 0 by substituting it into the differential equation Enter the terms in the orde given Enter c as c1 and c as c2 c1e t 16c2e 4t 0 help formulas correct correct incorrect

1 point Find a non constant solution to x x 4 using your knowledge of derivatives from basic calculus x t help formulas

1 point Suppose x c e a Verify that x cje c e4 is a solution to x 3x 4x 0 by substituting it into the differential equation Enter the terms in the order given Enter c as c1 and c as c2 0 help formulas b Find x 1 if x 0 4 and x 0 2 help formulas

2 dx 1 Enter your answers below in terms of the independent variable I in the order in which 1 point Show that y cos t is a solution to dt the terms were given Be sure you can justify your answer help formulas

1 point Suppose y c cos kx c sin kx where k 0 is a constant and c and c are arbitrary constants Find the following Enter c as c1 and C as c2 a b dy dx d y dx c Rewrite your answer to the previous part in terms of y d y dx d Find the general solution to help formulas d y dx help formulas help formulas 9y Enter your answer as an equation y help equations

e 1 point Suppose x a Find the value of the expression x 6x 12x 8x in terms of the variable r Enter the terms in the order given help formulas b Simplify your answer to the previous part to obtain a differential equation in terms of the dependent variable x satisfied by x e x 6x 12x 8x help formulas c Is x e a solution to your differential equation in the previous part Choose Be sure you can justify your answer

Simplify the trigonometric expression below considering only the values of for which the expression is defined cos B tan B esc B cot B Remember to put parentheses around the argument of any trigonometric function

Considering only the values of 0 for which the expression is defined which of the following is equivalent to the expression below Select the correct answer below sin 0 01 sin 0 O 1 cos 0 tan 0 csc 0

Considering only the values of for which the expression is defined which of the following is equivalent to the expression below Select the correct answer below O cot cot 0 1 cos tan sec cot

Considering only the values of a for which cos a csc a sec a is defined which of the following expressions is cos a csc a sec a Select the correct answer below Ocot a tan a tan a

Provide your answer below TT TT 2 2 34 1 0 1 2 TT 2 3 4 0 TT VI X J

A surveying team set up some equipment 133 ft from the base of a tree in order to sight the top of the tree From ground level they measure the angle of elevation to be 34 If the calculated height needs to be accurate to within 1 what is the allowed error in the angle measurement Assume the 133 ft measurement is 100 accurate The allowed relative error is The allowed percent error is Notor ontor on on OF DOOUroto to 4 decimal plagos

1 Evaluate a sin sin 1 sin cos d b sin tan 2 3 e sin cos x c f cos tan tan sin 5 P Vp 9

On the graph of f x cot x and the interval 2 0 for what value of x does f x 1 Choose the points in the figure Provide your answer below 2TT 3 2 0 3TT 8 TT 3 TT 2 2 1 2 0 0 1 2 3

4 U U 3 2 5 2 1 TI 2TT 3TT 4TV 5T 1 100 n 2 Note Enter your answer in cosecant form Provide your answer below k 2TT TT 0

Find a function x y that is harmonic in the region of the first quadrant between the curves xy 2 and xy 4 and takes on the values 3 on the lower edge and the value 5 on the upper edge Hint Begin by considering z

Given the side length a 13 and the angle A 41 on the triangle below find the lengths of b and c and the measure of angle B Do not round during calculations Round your answers for lengths of b and c to one decimal place Round your answer for angle B to the nearest whole number B Provide your answer below b c B 13 C b C 41 A

Given the side length b 15 and the angle B 26 on the triangle below find the lengths of a and c and the measure o angle A Do not round during your calculations but round your final answers to one decimal place 15 C Provide your answer below A 26 a B

What is the amplitude and period of the function f x 2 sin 7x Provide your answer below

Given the triangle below find the angle 0 0 Provide your answer below 10 9 Give your answer in degrees rounded to the nearest tenth Do not include the degree symbol in your ans

Sorry that s incorr 0 to Tou 7 cimal places LO 5

y 2 cot 2x Drag the movable red point to set the asymptotes for one period of the given function Also place the blue point at the correct set of coordinates You may click on a point to verify its coordinates Provide your answer below TT 2 TT 4 6 5 4 3 2 16 0 1 2 3 4 TT 4 TT 2 3TT 4 3 5TT 4 3TT 2 7TT 4 2TT

Find a basis of solutions by the Frobenius method y x 1 y 0