Application of derivatives Questions and Answers

Rotate the axes to eliminate the xy term in the equation Then write the equation in standard form Use xp for x and yp for y in your answer Rotate the coordinate axes through an angle with 0 0 2 9x 24xy 16y 80x 60y 0 x 4y X Check which variable s should be in your answer Sketch the graph of the resulting equation showing both sets of axes y SP 2 4 2

A W30x99 hot rolled structural steel wide flange shape is used as a simply supported beam on a span of 32 ft as shown below The beam supports a superimposed uniformly distributed load of 4 0 kips ft in addition to its own weight Calculate the maximum bending stress A 4 kips ft 32 W30 x 99 fin 27 X I Sect A A X

Question 1 20 points 1 Find the areas bounded by the indicated curves y x y 2 x x 0 x 0

Find the area bounded by y x 27x 54 the x axis and the first coordinates of the relative maximum and minimu values of the function The area is 0 Simplify your answer GETTE

Suppose the temperature in degrees Celsius over a 9 hour period is given by T t 1 6t 32 a Find the average temperature b Find the minimum temperature c Find the maximum temperature a The average temperature is 32 degrees Type an integer or a decimal Round to one decimal place as needed b The minimum temperature is degrees Type an exact answer in simplified form

A company determines that its marginal revenue per day is given by R t where R t is the total accumulated rever in dollars on the fth day The company s marginal cost per day is given by C t where C t is the total accumulated cost in dollars on the fth day R t 130e R 0 0 C t 130 0 1t C 0 0 a Find the total profit P T from t 0 to t 10 the first 10 days P T R T C T S R t C t dt www The total profit is Round to the nearest cent as needed

36 A company is constructing an open top square based rectangular metal tank that will have a volume of 61 ft3 What dimensions yield the minimum surface area Round to the nearest tenth if necessary A 5 ft x 5 ft x 2 5 ft C 5 7 ft x 5 7 ft x 1 9 ft B 11 ft x 11 ft 0 5 ft D 3 9 ft x 3 9 ft 3 9 ft 36

Let y 8 sin 2x 6 What is the amplitude What is the period What is the phase shift NOTE If needed you can enter 7 as pi in your answers I Cal

Evaluate the limit lim 3 tan 4x Enter a simplified multiple of or or TU xxx X

olve the problem Round your answer if appropriate 31 Water is being drained from a container which has the shape of an inverted right circular cone The container has a radius of 4 00 inches at the top and a height of 8 00 inches At the instant when the water in the container is 6 00 inches deep the surface level is falling at a rate of 0 9 in sec Find the rate at which water is being drained from the container A 42 4 in 3 s B 36 8 in 3s C 24 3 in 3 s D 25 4 in 3 s

Suppose the angle is in the first quadrant 0 0 2 and cos 0 a csc 0 b cot 0 help fractions help fractions Find exact values as fractions not decimal approximations for the following

Evaluate the following expressions Note The answer must be given as a fraction NO DECIMALS If the answer involves a square root it should be entered as sqrt For instance the square root of 2 should be entered as sqrt 2 If cos 2 7 and tan 0 0 then sin 0 tan 0 cot 0 sec 0 csc 0 9

Evaluate the following expressions sin T cos tan 2 cot 2 sec csc 9 4

dA Find the amount A in an account with continuously compounded interest after t years if dt A 4 8 326 49 The differential equation is of the form Round to three decimal places as neer dy dt dy dt k M y dy dt with k dy ky ky M y ky rA A 0 8 000 and

From April through December 2000 the stock price of a company had a roller coaster ride The table below indicates the price of the stock in dollars at the beginning of each month during that period Find the average rate of change in price per month between June and September Round to the nearest cent as needed Month April May June July August September October November December Price 116 107 88 100 95 112 93 84 65 OA 12 00 per month OB 8 00 per month OC 8 00 per month OD 12 00 per month

function given in the graph below and that f is a piecewise function whose parts are either portions of lines or portions of circles as pictured 6 5 4 3 A 6 A 1 A 4 A 6 6 Let A be the function defined by A z f t dt X b Compute the followings A 6 1 a What does the Second FTC tell us about the relationship between A and f Write an equation to describe the relationship A 0 X 6 5 X 3 2 1 2 3 5 X c Compute the followings 4 X a

If 0 7 then 4 sin 0 equals cos 0 equals tan 0 equals sec 0 equals

f x 6 4 y 6 4 2 N 2 X 4 shows the graph of the function

Which of these functions are even OA f x x cos x B f 3 1 csc 3 cos 2x x OC f x OD f x 2 sin x cos x OE f a 1 sec a OF f t sec t

Determine the period for the function y 10 sec 1 x 1 2 10 Type pi for 7 in your answer s if needed Period

Find the length and direction when defined of uxv and vxu u 4i 4j 2k v 2i 2j 4k uxv Type an exact answer using radicals as needed Select the correct choice and fill in any answer boxes in your choice below O A The direction of uxvis i j k Type exact answers using radicals as needed OB The direction of u xv is undefined vxu Type an exact answer using radicals as needed Select the correct choice and fill in any answer boxes in your choice below OA The direction of vxu is i j k Type exact answers using radicals as needed B The direction of vxu is undefined

For the curve determine each of the following Note Use a negative for a shift to the left y 17 tan 7x 2 Period Phase shift

The weekly marginal cost of producing x pairs of tennis shoes is given by the following function where C x is cost in dollars 200 Xx 1 C x 11 If the fixed costs are 2 000 per week find the cost function What is the average cost per pair of shoes if 1 000 pairs of shoes are produced each week The cost function is C x

The weekly marginal revenue from the sale of x pairs of tennis shoes is given by the equation below where R x is revenue in dollars 200 x 1 Find the revenue function Find the revenue from the sale of 1 000 pairs of shoes R x 41 0 03x R 0 0 Find the revenue function R x R x Use integers or decimals for any numbers in the expression

Find the derivative of y with respect to x 12x 3 29 y cos 5 A 12 1 12x 3 2 60 1 12x 3 B 12 25 12x 3 12 25 12x 3 D

E Question 2 7 5 points Ov NEXT 9 2 Domain for f x Range for f x Asymptote for f x End Behavior for f x What function is f x BOOKMARK What function is g x Domain for g 2 Range for g x Asymptote for g a End behavior for g x f z Given that f x and g x are inverses provide the following characteristics 183

Find where the function f x 32 8x 144x 9 is increasing and where it is decreasing 4 2000 4000 6000 8000 4 T 0 6 f x 12x 24x 288x 12x x smallest 0 and largest and arrange To use the I D Test we have to know where f x of the three factors of f x namely 12x x intervals whose endpoints are the critical numbers our work in a chart A plus sign indicates that the given expression is positive and a negative sign indicates that it is negative The last column of the chart gives the conclusion based on the I D Test For instance f x 0 for 0 x 6 so f is Select an answer on 0 6 It would also be true to say that f is decreasing on the closed interval 0 6 Interval HA 4 12x x 6 x 4 f x E x 0 and where f x and a This depends on the signs We divide the real line into f decreasing on 4 Select an answer on 4 0

Consider the function below Use U to denote the union of two intervals f x 4 2x x a Find the interval s of increase Enter your answer using interval notation Find the interval s of decrease Enter your answer using interval notation b Find the local minimum value s Enter your answers as a comma separated list If an answer does not exist enter DNE Find the local maximum value s Enter your answers as a comma separated list If an answer does not exist enter DNE c Find the inflection points x y smaller a value x y larger x value Find the interval s where the graph is concave upward Enter your answer using interval notation Find the interval s where the graph is concave downward Enter un

Find the inflection point of the function Hint g 0 does not exist g x 9x x x g x

Find the points of inflection of the curve y x X X 10 smallest value largest value 1 x 1 x Hint All three lie on one straight line

Suppose the derivative of is x x 2 x 3 x 7 On what interval is f increasing Enter your answer in interval notation

This exercise uses Newton s Law of Cooling A roasted turkey is taken from an oven when its temperature has reached 185 F and is placed on a table in a room where the temperature is 76 F a If the temperature of the turkey is 150 F after half an hour what is its temperature after 45 min Round your answer to the nearest whole number OF b After how many hours will the turkey cool to 100 F Round your answer to one decimal place hr

Find the area under the graph of f over the interval 2 3 x 1 x 1 2x x 1 f x The area is Simplify your answer

This exercise uses the population growth model Some bullfrogs were introduced into a small pond The graph shows the bullfrog population for the next few years Assume that the population grows exponentially 12 A 700 600 500 Frog population 400 300 200 100 0 2 225 1 2 3 a What was the initial bullfrog population bullfrogs 4 5 b Find a function that models the bullfrog population t years since the bullfrogs were put into the pond Round your r value to four decimal places n t c What is the projected bullfrog population after 15 years bullfrogs d Estimate how long it takes the population to reach 124 000 Round your answer to the nearest whole number

Find the area of the region bounded by the graphs of the given equations y x 2 y x The area is Type an integer or a simplified fraction

Instructions Use algebraic techniques to solve problems unless otherwise instructed Show the calculations leading to your answers except basic arithmetic Simplify and circle your final answers 1 12 pts The number of bacteria in a culture was initially 100 and it was 650 after 3 hours a Find a function of the form n t n et that models the number of bacteria after t hours 3r ICD ye 650 ean 13 R IW 3v en 65 6 5 P 3 un 4 5 0 623934059 Ronden 624 b What is the relative growth rate of the population 0 2 62 4 c Calculate the number of bacteria after 12 hours Give exact and approximate answers 100 xe0424x12 178647 5565 178448 d When will the number of bacteria reach 8 000 Give exact and approximate solutions

This exercise uses the radioactive decay model If 250 mg of a radioactive element decays to 240 mg in 48 hours find the half life of the element Round your answer to the nearest whole number hr

Instructions Use algebraic techniques to solve problems unless otherwise instructed Show the calculations leading to your answers except basic arithmetic Simplify and circle your final answers 1 12 pts The number of bacteria in a culture was initially 100 and it was 650 after 3 hours a Find a function of the form n t noe that models the number of bacteria after t hours

Completely factor the following polynomial functions 6 x 16 7 x 2x 24 8 3x 9x 18x

Problem 6 Find the arc length of the following function on the given interval f x 4 24 1 4x n 1 x 2 Enter your answer as an exact fraction or as a number correct to 4 decimals

Problem 7 Find an expression for the arc length of the upper half of a circle of radius 22 on the interval 0 t where 0 t 22 Problem 7 Enter your answer as a symbolic function of t as in these

Problem 6 Consider the partial fraction decomposition of the following Problem 6 1 x 1 x 8 One of the terms in the partial fraction decomposition is of the form Enter your answer symbolically as in these examples B x 8 Find the value of B

A rectangle is inscribed with its base on the x axis and its upper corners on the parabola y 5 x What are the dimensions of such a rectangle with the greatest possible area Width Height 15 3 10 3

K 5 9 A strain of bacteria is placed into a petri dish at 30 C and allowed to grow The following data are collected Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth The population is measured using an optical device in which the amount of light that passes through the petri dish is measured Complete parts a e Time hours x 0 25 3 5 4 5 6 HO Population y 0 27 0 53 0 76 1 03 1 46 A strain of bacteria is placed into a petri dish at 30 C and allowed to grow The following data are collected Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth The population is measured using an optical device in which the amount of light that passes through the petri dish is measured Complete parts a e Time hours x 0 2 5 35 45 6 Population y 0 27 0 53 0 76 1 03 1 46 CCC a Treating time x as the predictor variable use a graphing utility to fit an exponential function to the data yab 0 2698 1 3328 Round to four decimal places as needed b Express the function found in part a in the form N t Noe where x t N t 0 26980 28731 Round to four decimal places as needed c Graph the data together with the exponential function found in part a or part b on a scatter diagram Which graph correctly illustrates the exponential model found for this data Question 1 5 9 1 GC Part 4 of 5 Q NEUV ZUBOE Round to four decimal places as needed c Graph the data together with the exponential function found in part a or part b on a scatter diagram Which graph correctly illustrates the exponential model found for this data G 200 G i C E HW Score 8 33 1 of 12 points Points 1 of 5 d Use the exponential function from part a or b to predict the population at x 7 hours 0

If AB i 5j 4k and B is the point 9 4 1 find A

The financial director of a hotel finds that the income stream measured in thousands of dollars varies depending on the season She estimates that the total income can be modeled using the continuous distribution f t t 361 288t where t is measured in months What is the total income in the first year Give your answer in thousands of dollars to the nearest thousand dollars Provide your answer below

A strain of bacteria is placed into a petri dish at 30 C and allowed to grow The following data are collected Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth The population is measured using an optical device in which the amount of light that passes through the petri dish is measured Complete parts a e Time hours x 0 2 5 3 5 4 5 6 Population y 0 27 0 53 0 76 1 03 1 46 a fialny w a as the pituition and the a graping unity to in an OAUTOT CE to o to y ab 0 2698 1 3328 Round to four decimal places as needed b Express the function found in part a in the form N t Noekt where x t N t 0 2698e0 2873t Round to four decimal places as needed c Graph the data together with the exponential function found in part a or part b on a scatter diagram Which graph correctly illustrates the exponential model found for this data A OB O C 1 2 0 2 0 2 Q Q Q Q Q

If q r S d r cost t2 cos 5 0 2r dt then syntax error Question Help Video Add Work X

Let f x 5e 2 3 3 1 1 6 de t dt 0 1 List all values of x for which f x has a local maximum X

If q r d r cost9 t2 cos r 2r F Quoction Hair mucu dt then X