Application of derivatives Questions and Answers

Use part I of the Fundamental Theorem of Calculus to find the derivative of 9 y t si t sint dt g y NOTE Enter a function as your answer Make sure that your syntax is correct in romombor to put all the

Question 1 What is the value of 24cos x sin x dx

3 Using the first principals definition of the derivative determine the 1 derivative of the following y KU 2 A 2 2x 3 4 Let f 2 16 f 2 3 g 2 9 and g 2 1 Determine a value for h 2 for the following function h x f x g x A 2 T 2

1 point Find the derivative of the function h x h x BEE 1 1 r5 dr

8 So 1 point Suppose g x 8 t dt Calculate g x in two different ways Use Part 1 of the Fundamental Theorem of Calculus g x Now differentiate your answer B www Use Part 2 of the Fundamental Theorem of Calculus to evaluate the integral g x BEN

Use the given graph of the function f to answer the following questions Answer Formats Answer Formats Type your answers in interval notation Use INF and INF to denote o and o 5 3 4 2 1 Type your answers in interval notation Use INF and INF to denote 00 and oo Enter NONE if it is not positive negative If there is more than one interval type your answer as a comma separated list For example a b1 az b 2 3 19 Enter NONE if it is not positive negative If there is more than one interval type your answer as a comma separated list For example a b a2 b Part 1 The First Derivative Find the open interval s on which f z is positive negative Type your answers using interval notation 1 f z is positive 2 f z is negative Part 2 The Second Derivative Find the open interval s on which f z is positive negative Type your answers using interval notation 1 f z is positive 2 f 2 is negative

Distance Figure 19 shows a 75 foot flagpole Use the information in the diagram to work Problems 57 and 58 D 75 ft 60 A FIGURE 19 B A X B 57 a If angle ABC 60 and AB 25 feet find AC b Using part a find AD y 0 G 14 F FIGURE 20 D E

61 Distance a Bullet Travels A bullet is fired into the air at an angle of 45 How far does it travel before it is 1 000 ft above the ground Assume that the bullet travels in a straight line neglect the forces of gravity and give your answer to the nearest foot

se the revenue and cost graphs on the right to find the approximate cost of manufacturing 8 000 cell phones What is the approximate cost of manufacturing 28 000 phones Dollars in millions 5 4 3 2 1 0 8 Rovenub Cast 16 24 32 40 48

The number of fatalities in coal mines in a certain country for recent years is given in the table below Let x 10 correspond to the year 2010 Sketch the graph of the information in the table Year 2010 2011 2012 2013 2014 2015 2016 Number 43 23 21 21 15 12 7 Choose an appropriate window for the graph The first value on the x axis should be 10 and the last value on the x axis should be 16 Since the maximum y value in the table is 43 the will need to go as high as 45 Sketch the graph Choose the correct answer below OA Coal fatalities 45 04 9 9 Year 16 Q B 45 04 9 9 Year 16 Q O C 45 04 9 9 Year 16 O D 45 0 9 9 Year 16 Q

List the x intercepts and y intercepts of the graph List all the x intercepts Select the correct choice below and if necessary fill in the answer box es to complete your choice A The graph has two x intercepts The leftmost x intercept is 2 5 and the rightmost x intercept is 7 Type integers or decimals B The graph has one x intercept Type an integer or a decimal OC There are no x intercepts List all the y intercepts Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The graph has two y intercepts The bottommost y intercept is topmost y intercept is Type integers or decimals OB The graph has one y intercept Type an integer or a decimal OC There are no v intercepto and the Ay

Use the revenue and cost graphs on the right to find the approximate profit from manufacturing 24 000 cell phones What is the approximate profit from manufacturing 24 000 phones Dollars in millions 5 4 3 0 0 Revenue Cast 8 16 24 32 40 Cell phones in thousandel CO 48 G

Find the indefinite integral S 2 I 25 da C

The following table gives amount of annual spending of a country s defense budget in billions of dollars for military construction Let x 12 correspond to the year 2012 Sketch the graph of the information in the table Year 2012 2013 2014 2015 2016 Spending 11 8 7 8 8 3 5 4 6 9 Choose an appropriate window for the graph The first value on the x axis should be maximum y value in the table is the y axis will need to go as high as and the last value on the x axis should be Since the

Use the revenue and cost graphs on the right to find the approximate profit from manufacturing 24 000 cell phones What is the approximate profit from manufacturing 24 000 phones Dollars in millions 5 4 3 1 0 0 Revenue IN Cast 8 16 24 32 40 Cell phones in thousands 48 Q

d expressions for the Revenue Cost and Profit from selling x hundred items tem Price Fixed Costs Variable Costs 5 00 18 000 344x Which expression represents revenue A R 5x 18 000 O C R 5x OB OD F

Without sketching the graph find the x intercepts and y intercepts of the graph of the equation 7x 3y 21 Select the correct choice below and if necessary fill in the answer box to complete yo choice OA The x intercept s is are Type an integer or a simplified fraction Use a comma to separate answers as needed OB There are no x intercepts

1 point Evaluate the integral by interpreting it in terms of areas 8 6 8 64 x dx

49 x 2 2 dx If applicable enter your a r Use the formula for the area of a circle to calculate answer in terms of it

Problems 15 and 16 refer to Figure 18 D FIGURE 18 15 Find AB if BC 4 BD 5 and AD 2 A B C

Find and sketch an angle that is complementary to the angle O 11 79 179 101 79

There are 320 available to fence in a rectangular garden One side of this garden will be next to a road The fencing for the side of the garden next the road costs 6 per foot and the fencing for the other three sides costs 2 per foot Find the dimensions of the largest possible garden Dimensions x includo hoin funite in

A rectangular garden of area 75 square feet is to be surrounded on three sides by a brick wall costing 10 per foot and on one side by a fence costing 5 per foot Find the dimensions of the garden that minimize the cost of materials Dimensions x include help units in your answers

Find the maximum value of Q xy provided x y 18 Maximum Value Also list the x and y values where this maximum occurs 1 x 0 0 2 y

Consider the following function f x on the interval 6 3 f x x 16x 62 Absolute Maximum 1 Find the value where the Absolute Maximum occurs on the given interval 2 Find the Absolute Maximum Value of f x on the given interval x Maximum Value Absolute Minimum 1 Find the value where the Absolute Minimum occurs on the given interval 2 Find the Absolute Minimum Value of f x on the given interval x Minimum Value

Find all critical numbers of the following function Answer Formats If the function does not have any enter NONE f x 3 4x 6x x

A farmer has 1500 available to build an E shaped fence along a straight river so as to create two identical pastures The materials for the side parallel to the river cost 6 per foot and the materials for the three sections perpendicular to the river cost 5 per foot Find the dimensions for which the total area is as large as possible Dimensions x include help units in your answers

Find all critical numbers of the following function Answer Formats If the function does not have any enter NONE f x x 14x 54 0 x

Complete the table f s S f s Is 21 S 2 0 1 3 2 N G 4

valuate the function at each specified value of the independent variable and simplify If an answer is undefined enter NDEFINED g y 5 3y a g 0 b g c g s 3

valuate the function at each specified value of the independent variable and simplify If an answer is undefined enter NDEFINED g y 5 3y a g 0 b g c g s 3

Consider the following f x 7x 3 x a Use a graphing utility to graph the function 10 increasing 10 5 decreasing 5 20 10 10 20 10 10 20 5 5 10 10 X X 10 10 5 5 20 10 20F 20 10 0 20 5 b Determine the open intervals on which the function is increasing decreasing or constant Enter your answers using interval notation If an answer does not exist enter DNE 5

ketch the graph of the piecewise defined function 3 x x 1 h x y 10 10 5 5 J J 5 5 10 5 5 10 10 10 if x 0 if x 0 10 5 y 10 5 5 5 X 10 10 X 10 y 10 5 5 5 5

Use a graphing utility to graph the function and estimate its domain and range Then find the domain and range algebraically Enter your answers using interval notation X 41 domain range f x

Use the graph of the function to find the domain and range of f Enter your answer using interval notation domain range b f 1 c d LLLLLLL f 3 y 16 15 f 1 652229LSSOHNWA51 14 13 12 11 5 4 3 2 1 1 10 8 Use the graph to find the indicated function values a f 2 2 3 4 5 y f x 1 2 3 4 HA 5 6 X

Compute the following limit 9x 7 9x 2 lim 818 Select the correct choice below and fill in any answer boxes in your ch A lim X 8 9x 7 9x 2 Type an integer or a simplified fraction B The limit does not exist

Use limits to compute the derivative f 5 if f x 4x3 f 5 Simplify your answer

36 T limsin 3x 0 1 0 01 0 001 0 0001 x 0 X 0 1 0 01 0 001 0 0001 sin 3x X a b C d X 0 1 0 01 0 001 0 0001 sin 3x X e f g h 37 Use the preceding two exercises to conjecture guess the value of the following limit lim positive real value sin ax x 0 x for a a

ercises 65 74 Use the graph of f to determine intervals where fis creasing and where f is decreasing 65 3 1 T 321 2 23 2 3 23 X

a A radioactive material decays according to the formula w 7 64 e 0 05 t 1 Find the half life of the material 2 How long will it take until the substance to decay to 25 of its initial amount

A Lorenz curve is given by y L x where 0 x 1 represents the lowest fraction of the population of a society in terms of wealth and 0 y 1 represents the fraction of the total wealth that is owned that fraction of the society Use this information to complete parts a through g Click the icon to view an example of a Lorenz curve a A Lorenz curve y L x is accompanied by the line y x called the line of perfect equality Explain why this line is given this name Choose the correct answer below OA The lowest 50 of households own 60 of the wealth OB The lowest 50 of households own 100 of the wealth OC The lowest p of households owns exactly 1 p of the wealth for 0 p 100 D The lowest p of households owns exactly p of the wealth for 0 sp 100

Directions Use the function f x below to answer the following questions Part 1 Intervals Find the open interval s on which f is increasing decreasing 1 Increasing 2 Decreasing f x 7x 4 Answer Formats If necessary use INF and INF to denote and o If the function is not increasing decreasing for any interval enter NONE If there is more than one interval type your answer as a comma separated list For example a b c d Part 2 Relative Extremum Find the x values s where f has a relative maximum relative minimum

Find an equation of the tangent line to the given function at a 2 y x 4 6 f x x X

7 Determine the values of the trigonometric functions of a where sin a 25 tan Type a simplified fraction and cos a 24 25

Use the given figure and any appropriate relationship s among circular functions to find 3 sin 4 0 1 14 4 4 1 0 Ay 4 4 11x 0 1 7x 6 5x 4 1 0 43 2 oo 4x sin 3 Type an integer or a fraction Type an exact answer using radicals as neede

Question Given the function f x x 2 find the total area between f x and the x axis over the interval 5 1 Provide your answer below

Determine the point on the graph of the unit circle that corresponds to 3x 3 findcos andsin 2 2 and state which functions are undefined for The point on the unit circle is pimplified fraction 3x 2 Then 3x 2 1 0 2 Ay 0 1 0 1 0 0 1 x o

Consider the graph shown in the figure which gives the velocity of an object moving along a line Assume time is measured in hours and distance is measured in miles The areas of three regions bounded by the velocity curve and the t axis are also given Complete parts a through e Using the values shown in the graph the displacement of the object over the interval 0 6 is 4 miles c What is the total distance traveled by the object over the interval 2 10 Using the values shown d What is the displacement of the object over the interval 0 10 Using the values shown in the graph the displacement of the object over the interval 0 10 is 6 miles e Describe the position of the object relative to its initial position after 10 hours Select the correct choice and if necessary fill in the answer box to complete your choice the graph the total distance traveled by the object is 25 miles A After 10 hours the object is OB After 10 hours the object is FOOD mi in the negative direction from its initial position mi in the positive direction from its initial position returned to its initial position 11 2 4 15 16 10 8

Perform a first derivative test on the function f x x 5x 2 5 5 a Locate the critical points of the given function b Use the first derivative test to locate the local maximum and minimum values c Identify the absolute maximum and minimum values of the function on the given interval when they exi

Consider the curve y f x shown below Find f 7 and f 7 V y f x S y x 2 tangent line