Application of derivatives Questions and Answers

Which pair of angles is a linear pair 8 Show your work here B and and G 6 Fland C 4 2 8 6 4 B 2G 4 6 8 F 4 A OB and F 6 None of the above 8

Solve for the unknown angle in degrees 8421 and 8431 84318618 find are complementary angles If 8431

Solve for y Y 52

5 12 pts How many years will it take for an initial investment of 20 000 to grow to 50 000 a Assume an annual interest rate of 4 3 compounded annually b Assume an annual interest rate of 4 3 compounded continuously Give exact and approximate solutions

A car rental agency rents 200 cars per day at a rate of 30 per day For each 1 increase in rate 5 fewer cars are rented At what rate should the cars be rented to produce the maximum income What is the maximum income The rental agency will earn a maximum income of when it charges per day

Let F x y f x y g x y be defined on R Find and graph the streamlines for the vector field F y 5 Oy 10x C Oy 10 OA C Choose the correct graph below Q 5 OB CHICCO OC 5 OD Q G KKERRE 22222 X

Let F x y f x y g x y be defined on R Find and graph the streamlines for the vector field F 4 x What is the equation that gives the streamlines of the vector field Oy 8x C Oy 8x C x O y O y C OA x 8 C Choose the correct graph below OB OC O D G

20 5 kg mass moves along an x axis Its acceleration as a function of its position is shown in the figure a m s 20 15 10 5 1 2 3 4 x m 5 6 7 8

A publishing company sells 625 000 copies of certain books each year It costs the company 1 to store each book a year Each time it must print additional copies it costs the company 500 to set up the presses How many books should the company produce during each printing in order to minimize its total storage and setup costs The company should produce books each printing in order to minimize costs

Starting with a 160 foot long stone wall a farmer would like to construct a rectangular enclosure by adding 960 feet of fencing as shown in the figure to the right Find the values of x and w that result in the greatest possible area X W ft ft CCES 160 feet

Let R be the region bounded by y x x 1 and y 0 Use the shell method to find the volume of the solid generated when R is revolved about the line y 3 Set up the integral that gives the volume of the solid using the shell method Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers 1 A OB 0 2x y 3 y dy Jo GEXER dx

Find the number of terms for this polynomial 8 18x840x 18 161

K Why do the values of cos x lie in the interval 0 1 Choose the correct answer below OA Cosine is undefined for negative values of x Cosine is only defined for values in the interval 0 B OC O D Cosine is only invertible for values outside the interval 0 Cosine is not one to one if any of the values outside the interval 0 are also included

Simplify the following expression 8 1 57 3 183 1 85k 17k1 13 1 8 Show your work here 8 Hint To add an exponent x type exponent or press A

A boat is towed toward a dock by a cable attached to a winch that stands 10 ft above the water level see figure Let 8 be the angle of elevation of the winch and let L be the length of the cable as the boat is towed toward the dock Complete parts a to c below a What is the rate of change of 0 with respect to L dL de 10 E OA dL L 10 de O C dL de 10L 100 de 10L OB dL 2 10 de 10 OD dL LL 100 10 ft

Points 0 of 1 A boat sails directly toward a 150 m skyscraper that stands on the edge of a harbor The angular size of the building is the angle formed by lines from the top and bottom of the building to the observer see figure Complete parts a and b below a Use a trigonometric equation to find an expression for the angle 0 in terms of x Type an equation 150 m Save

Find the absolute maximum value on 0 co for f x 10x 5x In x Select the correct choice below and if necessary fill in the answer boxes to complete your choice OA The absolute maximum is at x Round to two decimal places as needed OB There is no absolute maximum

Consider the differential equation dy dt a Complete the phase portrait below by placing the appropriate symbols on the line To place a symbol on the line Click a symbol button then click a point on the line To remove a symbol from the line Click the delete X button then click the symbol on the line decreasing increasing steady state slope undefined 4 y delete X b What happens to solutions with initial conditions y 0 4 as t increases If you entered they tend to a steady state what is the value of the steady state Enter NA if you chose another answer The steady state value is c Describe the behavior of solutions with initial conditions y 0 4 as t increases The steady state value is If you entered they tend to a steady state what is the value of the steady state Enter NA if you chose another answer

Use a right triangle to write the following expression as an algebraic expression Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x sin x COS Identify the right triangle that can be used to simplify the given expression Choose the correct answer below OB O C O A 8 a 1 0 X A O D A

The accompanying table shows the time of useful consciousness at various altitudes in the situation where a pressurized airplane suddenly loses pressure The change in pressure 0 274a where T measures time in minutes and a is the drastically reduces available oxygen and hypoxia sets in The upper value of each time interval is roughly measured by T 10 2 altitude over 22 000 in thousands of feet a 0 corresponds to 22 000 ft Complete parts a through c below Click the icon to view the time of useful consciousness table a A jet flying at 42 000 ft a 20 suddenly loses pressure when the seal on a window fails According to this model how long do the pilot and passengers have to deploy oxygen masks before they become incapacitated T S Type an integer or decimal rounded to the nearest tenth as needed

Evaluate the following integral x 11 3 x 11 dx fx 11 dx 3 11x 1 4 4x 8

The figure to the right shows the graphs of y 2 y 2y 3 and y 3 Match each curve with the correct function The function y 2 is shown by the graph graph The function y 2 is shown by the graph The function y 3 is shown by the graph P The function y 3 is shown by the

Evaluate the derivative of the following function 6x H x 2x 5 x 6x 2x x 5 H x 6 2x 6 log 2x 6

Suppose you have just poured a cup of freshly brewed coffee with temperature 90 C in a room where the temperature is 25 C Newton s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings Therefore the temperature of the coffee T t satisfies the differential equation A What is the limiting value of the temperature of the coffee where Troom 25 is the room temperature and k is some constant Suppose it is known that the coffee cools at a rate of 1 C per minute when its temperature is 75 C lim T t tx B What is the limiting value of the rate of cooling dT lim t 00 dt C Find the constant k in the differential equation k EB dT dt CER k T Troom D Use Euler s method with step size Ax 3 minutes to estimate the temperature of the coffee after 15 minutes

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

Match each differential equation to a function which is a solution FUNCTIONS A y 3x m B y e C y sin x D y t E y 7 exp 8m DIFFERENTIAL EQUATIONS Enter the capital letter that corresponds to one of the functions above eg A 1 On 1 y y 0 2 2m y 3my y 3 y 8y

Consider the slope field shown a For the solution that satisfies y 0 0 sketch the solution curve and estimate the following and y 1 y 1 b For the solution that satisfies y 0 1 sketch the solution curve and estimate the following E and y 1 y 1 c For the solution that satisfies y 0 1 sketch the solution curve and estimate the following y 1 and y 1 1 2 7 7 7 11 12 12 1 xxx 111 7 11 AVAI 2 4 7 7 7 71 SAVAI p IV X VA 11121 AIZLI W l l l l l l L 7 7 7

In the figure below which of the following is a solution to the differential equation dy dt with initial value y 0 3 4 Note pay attention to axis labels 1 0 8 25 0 5y IV 1 I I 1 I 1 1 I I 1 1 1 0 I 1 I 0 25 II IV

Verify that every member of the family of functions y following questions Answer y log x C X 1 Find a solution of the differential equation that satisfies the initial condition y 3 10 is a solution of the differential equation x2y xy 1 Answer the Answer 2 Find a solution of the differential equation that satisfies the initial condition y 10 3

This problem is an example of a differential equation an equation that relates a function to one or more of its derivatives You can solve this problem by doing some educated guessing Educated means remember what we did in the past Suppose f is the function that satisfies for all a in its domain and Then f x f x f 1 1 f x Hint try some of your familiar functions parabolas hyperbolas exponential trig nowor function

8 Using the second derivative test find the maximum and minimum points for the function f x x 8x 5 What conclusion can be made if a A function changes from a decreasing interval to an increasing interval b lim f x and lim f x 0 x 0 x 0 If the minimum of the function y x 4x 21 occurs at x 2 what are the coordinates of the minimum point

Helium is pumped into a spherical balloon at a constant rate of 3 cubic feet per second How fast is the radius increasing after 2 minutes At what time if any is the radius increasing at a rate of 140 feet per second The radius is increasing at a rate of ft sec Type an integer or a decimal Do not round until the final answer Then round to four decimal places as needed

Find an equation for the parabola that has its vertex at the origin and has its focus at the point 8 75 0

A drug that stimulates reproduction is introduced into a colony of bacteria After t minutes the number of bacteria is given approximately by the following equation Use the equation to answer parts A through D N t 1000 271 1 Ost 18 C Sketcn the grapns or N and on the same coordinate system Choose the correct grapn below O C O D A Ay 4 000 0 0 18 O 5 B 4 000 Ay 0 D What is the maximum rate of growth The maximum rate of growth at minutes is 18 Q Q 4 000 0 bacteria per minute 18 Q Ay 4 000 0 0 18 O O 5

A T shirt manufacturer is planning to expand its workforce It estimates that the number of T shirts produced by new workers is given by T x 0 5x 18x 0 x 27 When is the rate of change of T shirt production increa and when is it decreasing What is the point of diminishing returns and the maximum rate of change for T shirt production Graph T and T on the same coordinate system The rate of change of T shirt production is increasing on Type your answer in interval notation

The marketing research department of a computer company used a large city to test market the firm s new laptop The department found the relationship between price p dollars per unit and the demand x units per week was given approximately by the following equation p 1050 0 14x 2 0 x 75 So weekly revenue can be approximated by the following equation R x xp 1050x 0 14x 0 x 75 Use this information to answer parts A and B B There is no local maximum What is are the local minimum a Select the correct choice below and if necessary fill in the answer box to complete your choice OA The local minimum a is are at x Simplify your answer Use a comma to separate answers as needed

A drug that stimulates reproduction is introduced into a colony of bacteria After t minutes the number of bacteria is given approximately by the following equation Use the equation to answer parts A through D N t 1000 27t t 0 t 18 A When is the rate of growth N t increasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The rate of growth is increasing on Type your answer in interval notation Use a comma to separate answer as needed OB The rate of growth is never increasing

X log 8 3 on for x Type an integer or a simplified fraction wwwwww

K Enhanced Homework a Find the critical points of f x 6x e b Use the First Derivative Test to locate the local maximum and minimum values c Determine the intervals on which f is concave up or concave down Identify any inflection points A The function has critical points at x 0 4 Use a comma to separate answers as needed Question 24 7 2 98 Part 4 of 6 OB The function has no critical points b Identify the location of all the local maxima Select the correct choice below and fill in any answer boxes within your choice HW Score 80 56 198 Points 0 33 of 1 A x 4 Use a comma to separate answers as needed B There are no local maxima Identify the location of all the local minima Select the correct choice below and fill in any answer boxes within your choice Ax 0 Use a comma to separate answers as needed OB There are no local minima Identify all the intervals where f is concave up Select the correct choice below and fill in any answer boxes within your choice A fis concave up on the open interval s Use interval notation Use a comma to separate answers as needed OB I is not concave up on any intervals

a Find the inverse of the function below on the given interval and write it in the form y f x b Verify the relationships f f x x and f f x x f x x 36 for x 0 B a f x x 36 b Verify the relationship f f x x Select the correct choice below and fill in the answer box es to complete your choice f F x x 36 Type integers or simplified fractions OA f f x x 36 Verify the relationship f f x x Select the correct choice below and fill in the answer box es to complete your choice 36 x f f x 36 36 Type integers or simplified fractions OB x 36 Question 8 7 1 32 Part 3 of 3 X X HW Score 80 56 Points 0 4 of

a Find the critical points of f x 6x e b Use the First Derivative Test to locate the local maximum and minimum values c Determine the intervals on which f is concave up or concave down Identify any inflection points Part 3 of 6 a Identify all the critical points Select the correct choice below and fill in any answer boxes within your choice A The function has critical points at x 0 4 Use a comma to separate answers as needed x 4 Use a comma to separate answers as needed OB The function has no critical points b Identify the location of all the local maxima Select the correct choice below and fill in any answer boxes within your choice HW Score 79 86 19 17 om Points 0 17 of 1 OA x Use a comma to separate answers as needed OB There are no local minima OB There are no local maxima Identify the location of all the local minima Select the correct choice below and fill in any answer boxes within your choice

Part 1 of 3 A glass has circular cross sections that taper linearly from a radius of 6 cm at the top of the glass to a radius of 5 cm at the bottom The glass is 18 cm high and full of onenge jus How much work is required to drink all the juice through a straw if your mouth is 4 cm above the top of the glass Use 1000 kg m for the density of orange juice and 8 0 m s acceleration due to gravity Find the cross sectional area A y of a horizontal slice at y Use the center of the bottom of the glass as the origin in your coordinate system Choose the corect answer be OA A y y 0 9 OB A y x y 0 9 18 OC A y y 0 91 18 OD A y x y 0 9 HW Score 48 8 11 71 of 24 points O Points 0 of 1 2 Save

25 The lower edge of a dam is defined by the parabola y see figure Use a coordinate system with y 0 at the bottom of the dam to determine the total force on the dam Lengths are measured in meters Assume the water level is at the top of the dam CHOCO Find the width function w y for each value of y on the face of the dam Choose the correct answer below OA w y 10 y OB w y 10y Oc wy 5 y OD w y 5y Points 0 of 1

Part 1 of 6 a Find the critical points of f x 6x e b Use the First Derivative Test to locate the local maximum and minimum values c Determine the intervals on which fis concave up or concave down Identify any inflection points a Identify all the critical points Select the correct choice below and fill in any answer boxes within your choice OA The function has critical points at x Use a comma to separate answers as needed OB The function has no critical points

Compute the derivative of the following function 12 x 1 y 0 Type an exact answer Qu

W y 10 8 6 4 2 9 x 2 4 6 8 10 S Write the standard equation for the hyperbola graphed above

Compute the following derivative d dx cos cos 4e d dx Type an exact answer cos cos 4e

a Find the inverse of the function below on the given interval and write it in the form y f x b Verify the relationships f f x x and f f x x f x x 36 for x 0 1 4 a f x x 36 b Verify the relationship f f x O A f f x x 36 D Type integers or simplified fractions OB f f 1 x 3 x Select the correct choice below and fill in the answer box es to complete your choice 36 x BETER EX

Evaluate the following integral Include absolute values only when needed dx x 7 9x 7 dx 9x 7 www

Find an equation of the line tangent to the following curve at the point a f a f x 18e 2x a 0 y BA