Differential equations Questions and Answers

Find the measure of x 259 30 X Round your answer to the nearest hundredth

Which of the following represents z 6 3 6i in trigonometric form Oz 12 cos 210 i sin 210 O z 12 cos 150 i sin 150 O z 6 cos 210 i sin 210 O z 6 cos 150 i sin 150

If y 4 what is the value of y when x 5 O 125i 4 O 125i 4 O 125i 4 O5i 4 O 5i 4

Find for the following dy da y 1 lnx 4 O O O 2 ln x T 2 T 2 In x TV 1 ln z 3 In z T 1 In z

8 marks Classify the following differential equations as linear or non linear and give their order Give your reasons 2m each dy a t2d y t dt dt b c d 1 y d y dt dy dt d y dy t dt3 dt ty2 0 2y sint dt y et cos t y t

33 Solve d y dx dy 2 3y COS X dx

If f x has an inverse function f a could either the graph of for the graph of f 1 be symmetric with respect to the y axis Please explain your reasoning or use an example to illustrate your answer

by setting the x intercept and the asymptotes Provide your answer below 10 2 0 5 10 5 5 3 0 5 f x X 1 X 4 10

A distillation column processes two types of crude oil streams North Texas and Venezuelan Because of consumer demand the production of diesel fuel and home heating oil must be limited Based on the information in Table 1 estimate the optimum daily usage rate of each crude oil to maximize profit P The profit generated on processing North Texas A and Venezuelan crude B is 2 00 gal and 1 60 gal respectively Table 1 Crude oil usage and production information Diesel Fuel Home heating oil North Texas Venezuelan 8 30 10 50 Maximum Production rate gal day 2000 6000

dz z se the functions below to a express and as functions of u and v both by using the Chain Rule and by du v Expressing z directly in terms of u and v before differentiating Then b evaluate z 3 tan a Express z 1 dz 8 x ucos v y u sin v u v and dz dy as functions of x and y 2 4 on oxact answer Find dz dz du and dz v at the given point u v

Write the initial simplex table for the linear programming problem Do not solve A baker has 60 units of flour 132 units of sugar and 102 units of raisins A loaf of raisin bread requires 1 unit of flour 1 unit of sugar and 2 units of raisins A raisin cake requires 2 units of flour 4 units of sugar and 1 unit of raisin If raisin bread sells for 3 a loaf and a raisin cake for 4 how many of each should be baked so that the gross income is maximized What is the maximum gross income Let x be the number of raisin bread loaves and x2 be the number of raisin cakes O x1 x2 1 52 53 z 1 2 1 0 0 0 60 1 2 0 1 0 0 132 41 0 0 1 0 102 34000 1 0 O x1 x2 1 52 53 z 1 2 1 0 0 0 60 1 4 0 1 0 0 132 2100 0 102 34000 1 0 O x1 x2 1 52 53 z 1 2 1 0 0 0 60 1 4 0 1 0 0 102 2 1 0 0 1 0 132 34000 1 0 x1 x2 1 2 53 z 1 21000 60 1 40 100 132 1200 1 0 102 34000 1 0

Find equations for a the tangent plane and b the normal line at the point Po 2 2 1 on the surface 2x 4y 3z 27 a Which equation describes the tangent plane OA 4x 5y 3z 21 OB 4x 8y 3z 27 OC 7x 4y 5z 27 www

4 a b c d e What is the Estimation the error if P x x x 3 x 5 is used to estimate the value of sinzat z n 10 100 0 215 x 10 5 1 335 x 10 6 3 505 x 10 7 2 512 x 10 6 4 275 10 8

36 The graph of his given below Use the graph of h to determine the derivative of h at x 7 10 9 8 7 6 5 4 3 2 n 7 10 94 8 7 6 43 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10

Find the area of the figure pictured below 9 cm 4 cm Area 11 cm 3 cn a cm Question Help Worked Example 1 Message Calculator

y 4y 5y 2e t y 0 1 y 0 3 using the method of Laplace transforms Evaluate the particular solution at t i find the value Keep two decimal places in your answer Your Answer 1

a Define forward and backward differencing with examples Prove that f 2f 3f 6f 1 2 0 h 2 6h

The length of a rectangle is increasing at a rate of 5 cm s and its width is increasing at a rate of 7 cm s When the length is 9 cm and the width is increasing in cm s cm s

The slope of the tangent to the curve y x 1 O 15 11 Listen 0 18 Page 13 c

To be used in the design of the structure to be built in the ground environment the information of which is given below Calculate the soil bearing capacity according TBDY 2018 YASS 4 yd yd2 yd3 DP B 2 yw 10 kN m yd 20 7 kN m yd 21 9 kN m yd 3 19 7 kN m DR 1 71 m B 4 71 m L 8 17 m c 17 kN m 21

The slope of the tangent line to a curve is given by f x 7x 7x 3 If the point 0 7 is on the curve find an equation of the curve

Find the following indefinite integral 3 S 5e 0 5x dx X SC Type an exact answer in terms of a 3 X 5e 0 5x dx dx

P x 37 x 259 x 433 4 548922 20083 x 151 lao 24 6 24 1 what is the degree of the function 2 How many furns does it have 3 How many Zen s does it hake Do any of the 20 00f tell con you frire any multiplicity

Assume x and y are functions of t Evaluate dy dt y 2x 6 11 dx 5 x 1 y 2 dt dy dt for the following Round to two decimal places as needed

Divide 9x 21x 8x 30x 5 by 3x 1

Given that f x x find f of x f of x Simplify your answer

Given that f x 2x 1 find f of 3 fof 3 Simplify your answer

Given that f x 6x 1 and g x x find fog 2 fog 2

Find a particular solution yp to the nonhomogeneous equation y y 2y 2x 5 by guessing yp Yp Hint Ax B

Step 2 Next find the horizontal asymptote if any This is done by determining the ratio of the degree of the polynomial in the numerator to the degree of the denominator polynomial If that ratio is less than one then y 0 is a horizontal asymptote If the ratio is equal to one then the horizontal asymptote is determined by the quotient of the leading coefficients of the numerator and denominator polynomials If the ratio is greater than one the graph of g has no horizontal asymptote Enter NONE in any unused answer blanks Exercise d Plot additional solution points as needed to sketch the graph of the rational function Step 1 To graph the function you begin by plotting the intercepts and sketching the asymptotes Then to clarify the curve you plot at least one point between and one point beyond each x intercept and vertical asymptote This shows the behavior near the asymptotes and shows the shape of the curve elsewhere Complete the table of values below to determine points on the graph 6 4 2 0 2 4 6 Y Y

back to the skipped part Tutorial Exercise A population of a particular yeast cell develops with a constant relative growth rate of 0 4775 per hour The initial population consists of 3 4 million cells Find the population size in millions of cells after 6 hours Step 1 Since the relative growth rate is 0 4775 per hour then the differential equation that models this growth with P in millions of cell and t in hours is dP dt toome back 59 67 A

4 Draw any four level curves contours of the surface given by the formula 3z 4x 2y 0 Describe the shape of the surface in words

3 Find the slope if possible Write an equation for each line V a b slope equation 2 Graphing by using the y intercept and 2 3 X 3 Graphing a linear equation 4 3 2 1 slope equation 3 Example y x 1 1 Plot the y intercept 0 1 3 4 1 2 11 2 3 X c 4 3 2 HT slope equation 2 Locate a second point 772 move up 3 units 12 21 11 2 3 Graph the line 3

ASSUME THE INITIAL POPULATION Y 0 yo 7 ay 1 4 2 dy dt 2 WHAT DOES a AND represent a 0 75

Find the area of the trapezoid shown below 19 9 Question Help Video Message instructor Calculator

The amount of a radioactive substance y decays exponentially such that its half life is 8 days In other words the amount of any sample decays to exactly half in that time period Write a differential equation in terms of y and t in days to represent the rate of decrease in the amount of the substance O O O O di 316 316 318 dy dt dt dt 0 0866y 0 1732y 0 0433y 0 1732y 90 91 92 93 94 95 96 97 98 99 Finish

1 2 a Let f x y arctan direction cos i sinej for which Daf 0 1 2 15p b Does the function f x y ln x y has a saddle point Explain 10p

Use the geometric series formula 2 X 9 1 2 X 3 W n 0 1 1 y 8 y to express the function as a series n 0 0

y tan x sec x y T 0 Which of the following statement s are true regarding this initial value problem 1 Solution of the initial value problem is y x II Solution is valid on lim y x O A Only I B Only II OC Only III ODII and III 3T 2 2 O EXI II and III 856 105 sec x 7 2 sec x 5 sec x 3

IF a parabolic mirror passes through the points 0 0 and 10 4 find the focal point

1 point Find a function f x so that h x g f x given that g x 5x 2x 2x 3 and h x 5 2x 2 2 2x 2 2 2x 2 x

0 5 88 Points DETAILS PREVIOUS ANSWERS dx LARCALCET7 8 5 015 Use partial fractions to find the indefinite integral Remember to use absolute values where appropriate Use C for the constant of integration 7 x

For the given function f evaluate f 1 f 0 f 3 and f 7 f x f 1 f 0 f 3 7 2x 3 if 2x 6 if x 0 x 0

In each part list the points A F on the graph of f whose x coordinates satisfy the given conditions a f x 0 and f x 0 b f x 0 and f x 0 c f x 0 and f x 0 d f x 0 and f x 0 e f x 0 and f x 0 A B C D E F D 2 E Q Drag each of the points given above into the appropriate category below depending on whether or not they satisfy the conditions in parts a thro

Find the largest open interval s where the function is concave upward f x x 3x 9x 5 OA 1 00 OB 1 OC 1 00 1 O D None

Soru 10 Puan x 1 By using the partial fractions method calculate the integral f dx x x 1 A 2 In x 2 ln x 1 C x B In x 1 2ln x C C Inx 2 lnx 1 C 1 D 2 In x In x 1 C E In x ln x 1 C x

Evaluate the following 9 9 x4 dx

Let S be the surface given by 2x 3y 6z 9 inside the cylinder r y 7 Find the surface area of S Give a parametrization of the boundary curve of S Is S orientable If yes verify Stoke s theorem for the vector field F xi y zk

5 4 3 2 1 3 2 1 2 3 ZA 2 3 4 5 Le At the point shown on the function above which of the following is tru Of 0 f 0 of 0 f 0 Of 0 f 0 f 0 f 0

Let g x fre f t dt where f is the function whose graph is shown y 6 g 6 9 g 9 0 3 6 a Evaluate g x for x 0 3 6 9 12 15 and 18 9 g 0 g 3 0 g 12 9 g 15 18 g 18 21 X X x X X X X 12 15 18 21 e b Estimate g 21 Use the midpoint to get the most precise estimate g 21 13 5 X d Sketch a rough graph of g c Where does g have a maximum value Where does it have a minimum value minimum x x 18 maximum X 6 x