Calculus Questions

Let and w represent the length and the width of the rectangle measured in m Let A represent the area of the rectangle measured in m Writing an equation for A in terms of I and w gives us the following A m Let P represent the perimeter of the rectangle measured in m Writing an equation for P in terms of and w gives us the following P m

oh Answer each of the questions throughout the problem below NOTE You will have to submit your scratchwork and also include a graph for parts A B for this problem in the next question for full credit Suppose z 4 2i and w 3 2i Calculate each of the following A 3z4w Select B w Select C 2w Select

If two numbers have a difference of 56 and we let x represent the smaller of these two numbers then the larger of these two numbers is given by the following

Find two numbers whose difference is 46 and whose product is a minimum smaller number larger number

DeMoivre s Theorem For a complex number in polar form r cos i sin r cos no i sinne Question Calculate 3 cos 60 i sin 60 NOTE You will have to submit your scratchwork for these problems in the next question for full credit For more information on this topic look in your textbook at section 6 5 You might also want to look at the Complex Numbers presentation 81 cos 240 i sin 240 12 cos 240 i sin 240 81 cos 60 i sin 60

Find all missing measurements for the triangle with b 22 7 c 14 2 and A 22 Using the Law of Cosines a 22 7 Select 2 22 7 Select Select a Select O cos

5x 2 sin cos 0 COS 2 Since it s something times something we can break our equation into smaller simpler equations 2 sin 2 0 OR cos sin 0 OR cos 2 First we ll solve sin sine of O That tells us that X 2 Select 2 sin 0 0 for 0 Select Select Select Select 5x 52 5x 2 Now we just multiply both sides of the equation by 2 to get x by itself and then pick out the solutions with 0 x T Choose the relevant solutions from the list below 0 Which angles make have a

Now we just multiply both sides of the equation by 2 to get x by itself and then pick out the solutions with 0 x T Choose the relevant solutions from the list below Select Select Select Select Select We still have to solve the other of our two smaller simpler equations COS 5x Which angles make have a cosine of 0 2 0 cos 0 0 for 0 Select That tells us that 5x and Select

Example Find all solutions to sin 3x sin 2x with 0 x 2 First we bring everything to one side of the equation sin 3x sin 2x 0 Next we use a sum to product formula from section 5 4 of our textbook to rewrite the left side of our equation as a product The relevant formula is sina sin 2 sin In the equation we re solving a Select a which means that B 2 a 2 Select x Select Select a x cos a 2 2 and 3 Our rewritten equation now looks like this 5x and

he sum and difference formulas are in section 5 2 of your textbook Example Find sin 105 According to section 5 2 of our textbook sin a sin a cos 6 cos a sin We know 150 45 105 so we can use a 150 and 3 45 We chose those two because we already know the sines and cosines of those two angles sin 105 sin 150 45 sin 150 cos 45 cos 1 3 1 2 2 2 2 2 6 2 4 6 4

Suppose oil spills from a ruptured tanker and spreads in a circular pattern If the radius of the oil increases at a constant rate of 2cm sec how fast is the area of the spill increasing when the radius is 30cm 607T 120 T 900TT

Using the accompanying Colleges and Universities data to find 95 confidence intervals for the median SAT for each of the two groups liberal arts colleges and research universities Based on these confidence intervals does there appear to be a difference in the median SAT scores between the two groups Click here to view the Colleges and Universities data For the liberal art colleges the 95 confidence interval is 1240 8752 1275 7648 Use ascending order Round to four decimal places as needed For the research universities the 95 confidence interval is 1231 1310 Use ascending order Round to four decimal places as needed A 1 Liberal Arts Colleges Median SAT score 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1166 1205 1207 1207 1222 1220 1222 1239 1250 1241 1244 1254 1258 1251 1261 1268 1294 1289 1300 1305 1297 1300 1311 1310 B Research Universities Median SAT score 1112 1139 1147 1171 1202 1225 1233 1215 1231 1277 1279 1275 1272 1278 1286 1310 1309 1328 1341 1342 1348 1360 1380 1399

Let g be the function given by g x x At what value of x on the closed interval 1 1 does g have an absolute maximum O 1 0 1 04 3

How many points of inflection are on the graph of the function f x 12x 14x 7x 9 Select the correct answer O O O O 0 1 2 3 4

10pts Section 16 2 Evaluate J Laye x y 2 Include a sketch of the region of integration in your solution xy dA where R is the region bounded by the lines y x y 2x and

Let f be a function such that f 1 2 At each point x y on the graph of f the slope is 5xy x y 5 Which of the following statements is given by da true f has a relative maximum at x 1 f has a relative minimum at x 1 f has neither a relative minimum nor a relative maximum at x 1 There is insufficient information to determine whether f has a relative minimum a relative maximum or neither at x 1

Find the values of a for which y ea satisfies the equation y 4y 8 a 1 and a 1 a 3 and a 3 a 1 and a 3 a 1 and a 3

Differentiate the function Y O e e e e O2e 2e 2 ez e O 4 e e 0 4 O ez e z x ete ex e

Calculate the derivatives of the following function sin 1 x x Ocos x x 1 sin O 1 2 3 ONA 2 1 x 3 x 1 1 x 2

What is the equation of the tangent line to the curve y arctan 2x at the point which x 1 Write the equation in point slope form Oy x arctan 2 Oy x arctan 2 O y x arctan 2 Oy x arctan 2

Find two positive numbers whose product is 144 and whose sum is a minimum Select the correct answer 4 36 2 72 12 12 6 24 3 48

50 100 Select the correct answers 8 6 4 y f x 3 MAMM

f x 13 0 1 02 3 5 17 28 41 Selected values of a differentiable function f are given in the table above What is the fewest possible number of values of c in the interval 0 11 for which the Mean Value Theorem guarantees that f c 4

If y 2x 6x and 6 find dos s 0 948 836 946 928 none of these when x 5 Select the correct answer

Find the absolute maximum value of y 36 x on the interval 6 6 Select the correct answer 0 1 05 6 7

4 5 8 5 0 5 1 5 10 Select the correct answer O 1 1 8 9 2 6 O 26 u 8 9 12 x

Find the critical numbers of f x x x 3 Select the correct answer 27 7 20 O 0 2 O 0 3 0 3 22 1 7 2 2 2 O 0 2 0 2 1 1 12 2 7 0 2 12

Find any absolute or local maximum and minimum values of f x 8 2x if x 6 Select the correct answer O 4 is a local maximum 4 is an absolute minimum 6 is an absolute minimum O 4 is an absolute maximum 6 is an absolute maximum 6 is a local minimum

Suppose that F x f g x and g 14 2 g 14 5 f 14 15 and f 2 12 Find F 14 Select the correct answer 24 60 15 12 017

Find the inflection points for the function given f x 8x 2 sinx 0 x 3x Select the correct answer 8x 2x 16x 2 x 2 2 16x 2 x 8x 2 2x 16x 2 x 8x 2x 16x x 8x 2 2 16x

Estimate the extreme values of the function y x x 9x 72x 2 Select the correct answer 1 226 223 146 182 150 223 383 18 200 95 O

Calculate y x4 x y x 3y Select the correct answer Oy y4 2xy 4x x 2 Oy 1 y 2xy 4x x 3 2 x 2x 3 x 4x 1 Oy 1 34 2x 4x x 3 3 Oy none of these

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval f x 2 x 0 9 Select the correct answer Oc 914 c 0 Oc 1 4 Oc s none of these

If a varies jointly as b and c and a 27 when b 18 and c 3 find a when b 6 and c 1 3

Two children are playing with a ball The girl throws the ball to the boy The ball travels in the air curves 12 ft to the right and falls 37 ft away from the girl see the following figure If the plane th contains the trajectory of the ball is perpendicular to the ground find the plane s equation 37 ft X 12 ft

1 point Consider the initial value problem y 3 2y y 2y 0 y 0 1 y 0 2 y 0 0 Suppose we know that y t e y2 t y3 t e21 are three linearly independentt solutions Find a particular solution satisfying the given initial conditions y t help formulas

1 point Find the general solution to y y 7y 7y 0 In your answer use c c and c3 to denote arbitrary constants and x the independent variable Enter c as c1 c as c2 and c3 as c3 help equations

1 point Find the general solution to 3y 108y 0 Give your answer as y In your answer use c and c to denote arbitrary constants and x the independent variable Enter c as c1 and c as c2 help equations

y c1 e 0 x c2 e 3 x y 1 33 e 0 x 0 33 e 3x y c e x c e 3x y 1 33e x 0 33e 3x Find the particular solution that satisfies y 0 1 and y 0 1 y 1 33e 0x 0 33e 3x help equations incorrect incorrect least one of the answers above is NOT correct point a Find the general solution to y 3y 0 Give your answer as y In your answer use c and c to denote arbitrary constants and x me independent variable Enter c as c1 and as c2 y c1e 0x c2e 3x help equations

point Find the general solution to y 4y 29y 0 Give your answer as y In your answer use c and to denote arbitrary constants nd x the independent variable Enter c as c1 and c as c2 help equations

1 point Are the functions f g and h given below linearly independent f x e x g x xex h x x e x If they are independent enter all zeroes If they are not linearly independent find a nontrivial solution to the equation below Be sure you can justify your answer e4x xex x e4x 0

1 point a Find the general solution to y 3y 0 Give your answer as y In your answer use c and c to denote arbitrary constants and x the independent variable Enter c as c1 and c as c2 help equations b Find the particular solution that satisfies y 0 1 and y 0 1 help equations

1 point a Find the general solution to y 10y 25y 0 Enter your answer as y In your answer use c and c to denote arbitrary constants and x the independent variable Enter c as c1 and c as c2 help equations b Find the solution that satisfies the initial conditions y 0 4 and y 0 0 help equations

1 point Consider the initial value problem y y 31 y 0 2 y 0 2 Suppose we know that a complementary solution is y t c cos t sin t and that a particular solution is y 3t Find a solution satisfying the given initial conditions y t help formulas

1 point Solve the initial value problem y 9y 20y 0 y t y 0 1 y 0 0

1 point Find the two values of r for which is a solution of the differential equation smaller value larger val y x ex y 18y 77y 0

1 point Use the Wronskian to show that the functions y e x and y ex are linearly independent Wronskian det These functions are linearly independent because the Wronskian Choose zero for all x

Differentiate g x log 37 3x 2 x 0

Differentiate y log 10X d log 10X

Using the accompanying Accounting Professionals data to answer the following questions a Find and interpret a 99 confidence interval for the mean years of service b Find and interpret a 99 confidence interval for the proportion of employees who have a graduate degree Click the icon to view the Accounting Professionals data a A 99 confidence interval for the mean years of service is 10 66 20 64 Use ascending order Round to two decimal places as needed Interpret the confidence interval Choose the correct answer below Save There is a 99 chance that the population mean years of service will be between the confidence interval A limits OB We are 99 confident that the sample mean years of service will be between the confidence interval limits Oc Of all the data values 99 of the population will fall between the confidence interval limits We are 99 confident that the population mean years of service will be between the confidence interval limits D b A 99 confidence interval for the proportion of employees who have a graduate degree is Use ascending order Round to two decimal places as needed 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 A Employee 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 B Years of Service Graduate Degree 19 Y 9 Y 9 N 14 Y 8 Y 26 Y 8 Y 20 N 10 N 10 N 6 N 8 N 16 Y 21 N 26 N 19 Y 28 N 17 N 8 Y 31 N C D