Application of derivatives Questions and Answers

Find the area of the region inside one leaf of the three leaved rose r 3 cos 30 The area of the region is Type an exact answer using as needed

Suppose the derivative of a function f is f x x 2 x 5 x 6 On what interval is f increasing Enter your answer interval notation

Graph the polar equation r 5 cos 50 O A 10 10 10 10 Q OB 10 10 10 TT 10 O C 10 AT 1 TTTTT 10 10 10 O D 10 10 10 10

Use the graph of the rational function shown to complete the statement As x o f x 10 8 6 4 10 8 6 4 2 3 Ay 10 2 4 6 8 10 O A 1 OB o OC 0 OD 0

Given that x and y are functions of time find the indicated rate of change dy dx Find when x 2 and dt dy dt 3 given that x4 y4 97 dt Round to two decimal places as needed

Which of the following functions makes the most sense as a model for the probability density function representing the time in minutes starting at t 0 that the next customer walks into a store Op t 3e 3t for t 0 Op t e 3t for t 0 O O p t f for 0 t 5 5 p t cost 0 t 2 e 2n t 2n

A delivery company requires that any box delivered must have a length plus girth distance around totaling no more than 120 inches Find the dimensions of the box with maximum volume that can be sent The dimensions of the box with maximum volume that can be sent are Simplify your answer Use a comma to separate answers as needed inches CE

The volume of a right circular cone with radius r and height h is V h 3 a Approximate the change in the volume of the cone when the radius changes from r 5 5 to r 5 9 and the height changes from h 4 20 to h 4 11 b Approximate the change in the volume of the cone when the radius changes from r 5 17 to r 5 16 and the height changes from h 14 0 to h 13 93 a The approximate change in volume is dV Type an integer or decimal rounded to two decimal places as needed b The approximate change in volume is dV Type an integer or decimal rounded to two decimal places as needed LO

Find the domain of the following function b 5 4 3 2 1 5 t 3 2 1 2 3 4 2 3 4 5 6 7

Write the equation of the vertical asymptote of the function Do not use space in your answer 5 4 3 2 1 y 3 321 2 1 2 3 4 5 6 7 8 type your answer 2 3 4 5 6 7 X

Find the Taylor series generated by fat x a 15 f x a 1 2 What is the Taylor series generated by fat x a 20 k 0

Find the length of the curve x 6t y 9t 0sts 8 The length of the curve x 6t y 9t on 0sts 8 is Type an integer or a fraction

se a graphing utility to graph each function in the interval 0 27 y 5 cos x y 5e x 5x 5 a Write an equation whose solutions are the points of intersection of the graphs b Use the intersect feature of the graphing utility to find the points of intersection to four decimal places Enter the point of intersection whose x coordinate is within the interval there is no solution enter NO SOLUTION x y x

Write the expression as the sine cosine or tangent of an angle 3T COS 57 COS 7 cos 9 8 sin sin

You and a friend are riding your bikes to a restaurant that you think is east and that your friend thinks is north You both leave from the same starting point with you riding 16 miles per hour east and your friend riding 12 miles per hour north After you travel 4 miles at what rate is the distance between you changing Do not include units in your answer Provide your answer below

You are tasked with constructing a rectangular box with a square base an open top and a volume of 184 in Determine what the dimensions of the box should be to minimize the surface area of the box What is the minimum surface area Keep your answer in radical form and omit units

Verify the identity cos 7x 7y cos 7x 7y 2 cos 7x cos 7y cos 7x 7y cos 7x 7y cos 7x cos 7y 63 13 sin 7x sin 7y

Find the exact value of the expression tan 25 tan 110 1 tan 25 tan 110

Find the exact values of the sine cosine and tangent of the angle 75 sin 75 cos 75 tan 75

A 10 foot ladder is leaning against a wall If the top of the ladder slides down the wall at a rate of 2 feet per second how fast is the bottom of the ladder moving along the ground when the bottom of the ladder is 5 feet from the wall Simplify your answer as much as possible and do not include units Give an exact answer Provide your answer below

Use a graphing utility to approximate the solutions to three decimal places of the equation in the given interval Enter your answers as a comma separated list 6 sec x tan x 9 0 7 7 2 X

Question Beginning 145 miles directly west of the city of Johnstown a boat travels due north If the boat is travelling at a speed of 47 miles per hour determine the rate of change of the distance between Johnstown and the boat when the boat has been travelling for 64 miles Do not include units in your answer and round to the nearest hundredth Provide your answer below

Question Beginning 145 miles directly south of the city of Hartville a car travels due west If the car is travelling at a speed of 42 m per hour determine the rate of change of the distance between Hartville and the car when the car has been travelling for

A boat starts off 23 miles directly east from the city of Uniontown It travels due south at a speed of 41 miles per hour Let 0 represent the angle measured from due east to the direction ray pointing from Uniontown to the boat After travelling 49 miles how fast in radians per hour is 0 changing Submit an exact answer in fractional form Provide your answer below I

A rectangular container with a square base an open top and a volume of 4 000 cm is to be made What is the minimum surface area for the container Enter only the minimum surface area and do not include units in your answer

Question A boat is travelling due south at a speed of 53 miles per hour If the boat started off 29 miles directly west of the city of Johnstown how fast in radians per hour is the angle opposite the southward path 0 changing when the boat has travelled 23 miles Submit an exact answer in fractional form Provide your answer below

A rock is being launched into the air 15 feet away from the launch site is a lamp If the rock is travelling vertically upwards at 6 feet per second what is the rate of change of the angle of elevation 0 from the lamp to the rock when the rock is 45 feet off the ground Submit an exact answer Provide your answer below

A lamp is placed on the ground at a distance of 17 feet from the point at which a rocket will be launched into the air If the rocket travels straight up at a rate of 7 feet per second determine the rate of change of the angle of elevation 0 from the lamp to the rocket when the rocket is at a height of 43 feet Submit an exact answer in fractional form Provide your answer below

A company plans to manufacture a rectangular box with a square base an open top and a 3 volume of 246 in The cost of the material for the base is 0 2 cents per square inch and the cost of the material for the sides is 0 3 cents per square inch Determine the dimensions of the box that will minimize the cost of manufacturing it What is the minimum cost Round any calculations and the final answer to the nearest hundredth if necessary

A window is composed of a semicircle placed on top of a rectangle If you have 40 feet of window framing materials for the frame all around the rectangle and all around the semicircle as pictured below what is the maximum size of the window you can create a The radius of the semicircle portion of the maximum size window is r b The dimensions of the rectangular portion of the maximum size window are feet help numbers feet enter a comma separated list help numbers

An airplane is flying at an altitude of 5 mi on a straight path that will take it over a radar tracking station If the distance s between the plane and the radar station is decreasing at a rate of 380 mph when s 8 what is the speed of the plane Keep your answer in rational form and omit units Provide your answer below

A company plans to manufacture a rectangular container with a square base an open top and a volume of 2 916 in Determine the dimensions of the container that will minimize the surface area What is the minimum surface area Enter only the minimum surface area and do not include units in your answer Provide your answer below

The cost for producing a widgets is given by C z 0 032 0 62 150z with 0 x 100 Determine the production level with the lowest average cost per widget and find the minimum average cost per widget The production level with the lowest average cost is widgets The minimum average cost per widget is

Two poles are connected by a wire that is also connected to the ground The first pole is 16 ft tall and the second pole is 22 ft tall There is a distance of 38 ft between the two poles Where should the wire be anchored to the ground to minimize the amount of wire needed The wire should be anchored to the ground at a distance of wire feet from the pole labelled A in the diagram above in order to minimize the length of the

1 point The figure below gives the behavior of the derivative of a fu graph G Click on the gr a At what x values does the graph of g x have inflection points b What x values give the global maxima and minima of g on 2 2 minimum at x 2 Enter your answer as a comma s E

A rectangular box is going to be made with a volume of 274 cm The base of the box will be a square and the top will be open The cost of the material for the base is 0 3 cents per square centimeter and the cost of the material for the sides is 0 1 cents per square centimeter Determine the dimensions of the box that will minimize the cost of manufacturing it What is the minimum cost Round any calculations and the final answer to the nearest hundredth if necessary

To carry a suitcase on an airplane with a certain airline the length width height must not exceed 61 inches Part 1 Assuming the base is a square find a formula for the maximum volume that depends only on the height h in Part 2 What height allows you to have the largest volume

0 5 Shown above is the graph of two functions 5 Both are defined for x 0 a The red line shows the function f x or 10 b Find the three values of at which f x g x Give your answers in order from smallest to larges f x log x 4 or 3 or and g x log x 5 0 c Find the two values of x at which f x and g x have parallel tangent lines Give your answers in order from smallest to largest 5 5 10

Note If you are asked for value s give your answer as a single value e g 17 or a comma separated list e g 1 2 3 or NONE If you are asked for an interval give your answer in interval notation Type INF for oo and INF for 00 Suppose that f x x 3x a List the x values of all critical points of f x 0 9 4 B Use interval notation to indicate where f x is increasing A function is increasing if larger x values give larger y values Don t include points where the function changes from increasing to decreasing or from decreasing to increasing Increasing for a in 9 4 Inf C Use interval notation to indicate where f x is decreasing A function is decreasing if larger x values give smaller y values Don t include points where the function changes from increasing to decreasing or from decreasing to increasing Decreasing for a in D List the x values of all local maxima of f x values of local maxima none E List the values of all local minima of f x

WW7 Problem 17 1 point Consider the following graph of a derivative f a At what x values do the points of inflection of f occur 3 b On which interval s is f concave down f is concave down for in Click on the graph to enlarge Enter your answer as a value or a comma separated list 1 0 Enter your answer using interval notation 1 1

Find two positive integers such that their sum is 64 and a maximize and b minimize the sum of their squares The two positive integers that maximize the sum of their squares are help numbers The two positive integers that minimize the sum of their squares are help numbers

You are building five identical pens adjacent to each other with a total area of 1400m as shown in the figure below What dimensions should you use to minimize the amount of fencing

You are constructing an open top box for your cat to sleep in The plush material for the square bottom of the box costs 4 ft and the material for the sides costs 4 ft You need a box with volume 4 ft Find the dimensions of the box that will minimize the cost The dimensions of the box that will minimize the cost are feet enter a comma separated list help numbers

You have 790 ft of fencing to make a pen for hogs You have a river on one side of your property and will use the river as one side of the pen You wa to maximize the area of the pen using all of the available fencing Use the picture below for this problem W The area of the pen as a function of its length 1 is A 1 ft For what value of I will A l 0 The area of the pen as a function of its width w is A w ft For what value of w will A w 0 W

You need to construct fence around an area of 2601 ft What are the dimensions of the rectangular pen that minimize the amount of material needed ft by

A company plans to manufacture a rectangular box with a square base an open top and a volume of 108 cm Determine the dimensions of the box that will minimize the surface area What is the minimum surface area Enter only the minimum surface area and do not include units in your answer

A box with a square base and open top must have a volume of 119164 cm We wish to find the dimensions of the box that minimize the amount of material used First find a formula for the surface area of the box in terms of only x the length of one side of the square base Hint use the volume formula to express the height of the box in terms of x Simplify your formula as much as possible A x Next find the derivative A x A x Now calculate when the derivative equals zero that is when A x 0 Hint multiply both sides by x A x 0 when

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle Find the area of the largest possible Norman window with a perimeter of 20 feet Round your answer to one place after the decimal square feet

A 22 g bullet traveling 255 m s penetrates a 2 0 kg block of wood and emerges going 150 m s Part A If the block is stationary on a frictionless surface when hit how fast does it move after the bullet emerges Express your answer to two significant figures and include the appropriate units

An object at rest is suddenly broken apart into fragments A and B by an explosion The fragment A acquires six times the kinetic energy of the fragment B Part A What is the ratio of their masses Express your answer using two significant figures VAX