Application of derivatives Questions and Answers

The graph of a quadratic function f is given f x 4x 8x 1 vertex x intercepts y intercept 5 a Find the coordinates of the vertex and the x and y intercepts Round you x y x y x y range 1 x y 5 b Find the maximum or minimum value of f The Select value of f is f x X smaller x value larger x value c Find the domain and range of f Enter your answers using interval notation domain

Find the maximum or minimum value of the function f x x 2x 5 Is this a maximum or minimum value maximum value O minimum value

Progress Values of f t are given in the following table The function appears to have a 2 4 6 8 10 f 1 138 134 127 119 106 91 a Does this function appear to have a positive or negative first derivative Second derivative first derivative and a 2 i 0 b Estimate 2 and f 8 Use difference quotients to the right second derivative

Suppose that f t is a function with f 25 3 8 and f 25 0 5 Estimate f 26 and f 28 f 26 28

Solve the right triangle shown in the figure A 34 B C 90 E b 8 29 C 56 B C 10 X 0 b

3 2 In a right triangle the relaion hypotenuse opposite adjacent is 40 points 1 In a right triangle sec 0 is equal to choose your answer choose your answer 2 In a right triangle csc is equal to sine cose 3 In a right triangle tan 0 is equal to 1 sine 1 cose 4 In a right triangle cot is equal to choose your answer

Match the right triangle definition with its trigonometric function a hyp adj b c d e f opp adj opp hyp adj opp hyp opp adj hyp Select Select Select Select Select 33 Select

Find the period and amplitude y 3 sin 4 period amplitude

Find the degree measure that corresponds to the given radian measure E00 O a 70 7 Ob 22 5 Oc 1440 HY CAM

Given side a 24 side b 28 and side c 30 find the area of triangle A 285 316 336 909

A central angle in a circle of radius 2 5 cm is subtended by an arc of length 7 cm Find the measure of in degrees Recall In a circle of radius r the length of an arc that subtends a central angle of radians is S TO O a 160 4 Ob 2 8 Od 4 8 Od 35 7

Give the quadrant and reference angle of an angle that measures O a Quadrant II Reference Angle Angle Ob Quadrant I Reference Angle O c Quadrant I Reference Angle O d Quadrant II H 6 TL Reference Angle 6

A trucking company transports goods between two cities that are 960 miles apart The company charges for each pound 0 60 per mile for the first 100 miles 0 40 per mile for the next 300 miles 0 25 per mile for the next 400 miles and no charge for the remaining 160 miles a Graph the relationship between the cost of transportation in dollars C x and mileage x over the entire 960 mile route b Find the cost as a function of mileage for hauls between 100 and 400 miles from the first city c Find the cost as a function of mileage for hauls between 400 and 800 miles from the first city a Choose the correct graph below A AC x 350 04 800 280 100 18011 0 0 0 960 280 400 180 X 1000 OB 350 AC x 800 280 0 0 0 400 180 Gab bay 960 9000 b The cost as a function of mileage for hauls between 100 and 400 miles from the first city is C x Type an expression using x as the variable C OC 350 AC X 04 0 100 0 D Seb 280 FA 800 180 400 00 10011 X x 1000 Q 11 OD AC X 350 0 0 Lada 280 4400 1180 100 jen 0 Q L Q 2006 G 1000 1 U

A trigonometric function is given Find the amplitude period and horizontal shift of the function sin x 4 y 2 sin x Oa amplitude 2 period 2 horizontal shift units to the right Ob amplitude 2 period 4 horizontal shift 2 units to the right Od amplitude 2 period 2 horizontal shift units to the left Od amplitude 2 period 2 horizontal shift units to the right

Write out the first few terms of the series The second term is The third term is The fourth term is The first term is 1 Type an integer or a simplified fraction 1 100 1 10 8 n 0 1 1000 1 10 What is the series sum Type an integer or a simplified fraction Type an integer or a simplified fraction Type an integer or a simplified fraction A The series converges The series sum is Type an integer or a simplified fraction B The series diverges A If the series converges what is the series sum Select the correct choice below and fill in any answer boxes withi your choice

Find the values of x for which the given geometric series converges Also find the sum of the series as a function of x for those values of x n 3 n 0 x 4 Find the values of x for which the given geometric series converges Type an inequality or a compound inequality Use integers or fractions for any numbers in the inequality

A surveyor is attempting to measure the distance across a river at points A and B on opposing sides of the river She marks points A and C on one side of the river and notes that the distance between them is 500 feet Using her equipment she measures the angles formed by the triangle created when she marks the locations and finds that LBAC measures 107 5 and ZACB measures 22 3 see diagram Find the distance across the river from point A to point B Round your solution to 2 decimal places 500 feet 22 3 107 5 river O al 246 95 feet Ob 2322 62 feet Od 205 06 feet d 1012 35 feet B

Determine if the geometric series converges or diverges If a series converges find its sum O A 9 Select the correct choice below and fill in any answer boxes in your choice 9 3 0 9 Simplify your answer OB The series diverges 5

Write out the first few terms of the series The first term is 1 10 n 0 What is the series sum Type an integer or a simplified fraction

If t has the terminal point find the value of tan t ON 13 10 a Ob 3 7 3VF 7 O c O d V 3 3 P 3 y on the unit circle in the second quadrant 4

The number of hours of daylight y for day x x 1 to 365 in Anchorage Alaska can be 1 4 12 4 What is the minimum number of hours of daylight in Anchorage modeled by y 7 0 sin a 5 4 hours b 7 0 hours c 12 4 hours 2 365 X

Oa 484 25 feet Ob 456 21 feet Oc 426 32 feet Od 375 43 feet 113 the marks the distance to the point where the angebween these distances is 113 what is apran Note that the answers are rounded to 2 450 ft

The voltage V produced by an ac generator at time t in seconds V t 220 sin 120n t 360 360 V t Find the period of O a 1 60 O b O d T 60 Od 1 360 T 110 seconds seconds seconds seconds

If t has the terminal point P y on the unit circle in the second quadrar find the value of tan t Oa 1 1 20 3 O b 3 7 Od V 3 d 7

Convert the angle measure from degrees to radians Round your answer to three decimal places 177 3 Trad

For the cost function C x O O Cave x Cave x Cave I 2000 X 2000 x 10 10 3x 6 2 3 2000 10x 3x find Cave x

Given a revenue function calculate the average revenue when a 1 answer R x 1 000 x 3 Give your answer to the nearest dollar Do not include the dollar symbol in your

K The function f is defined as follows f x 3 X if x 0 if x20 a Find the domain of the function b Locate any intercepts A c Graph the function d Based on the graph find the range B Ay M d Based on the graph find the range Select the correct choice below and fill in the answer box es to complete your choice OA The range does not have any isolated values It can be described by Type your answer in interval notation B The range has at least one isolated value It can be described as the union of the interval s Use a comma to separate answers as needed OC The range consists exclusively of one or more isolated values It can be described as and the set Points 0 5 of 1 D

ilon t is defined as follows 2x 5 if x 1 4x 1 if x 1 a Find the domain of the function b Locate any intercepts c Graph the function d Based on the graph find the range f x C Choose the correct graph below OA B Ay O C d Based on the graph find the range Select the correct choice below and fill in the answer box es to complete your choice OA The range consists exclusively of one or more isolated values It can be described as O Use a comma to separate answers as needed B The range has at least one isolated value It can be described as the union of the interval s and the set Use a comma to separate answers as needed OC The range does not have any isolated values It can be described by

Given an average cost function Cave what is the correct form for the cost function C x Select the correct answer below O C x 3x x 0 5e0 5x O C x 3x x 2e0 5x O C x x 11x 723 xe0 5x C x x x e0 5z I 203 1 22 10 57

Given a cost function C x as C x 120 2x calculate the marginal average cost when x 5 use that to estimate the total cost when x6 and calculate the absolute error between your result and the exact result What is the absolute error between your result and the exact result Provide your answer below

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f x 4 VA Hole x intercept s Domain LEB HAZ y intercept Range REB

Given a revenue function R x 1 000 x x calculate the average revenue when x 3 Give your answer to the neare dollar Do not include the dollar symbol in your answer

Question Given a cost function C x as C x 20 x Provide your answer below 1 calculate the marginal average cost when x 2 and use this to approximate the total cost when x 3 Give your answer to one decimal place

For that cost function C x 40 x C 2 use the marginal average cost when 2 to estimate the average cost when x 3 O Cave 3 19 5 O Cave 3 12 83 O Cave 3 9 5 O Cave 3 19

Question 7 15 points 7 Factories A and B are 8 0 km apart with factory B emitting eight times the pollutants into the air as factory A If the number n of particles of pollutants is inversely proportional to the square of the distance from a factory at what point between A and B is the pollution the least Paragraph V B I UA 2 v

2 Find the coordinates of the local maximum and minimum points in the following function y 3x 15x Paragraph V B I U A E COB v www

3 Sketch the graphs of the given functions by determining the appropriate information and points from the first and second derivatives a y 12x 2x b y 2x 6x 5 Paragraph v BIU A BE 00 v

l 5x 2y 13 and k x 2y 9

3 City urbanization City planners model the size of their city using 1 the function A 1 2r 20 for 0 t 50 where A 50 is measured in square miles and t is the number of years after 2010 a Compute A t What units are associated with this derivative and what does the derivative measure b How fast will the city be growing when it reaches a size of 38 mi c Suppose the population density of the city remains constant from year to year at 1000 people mi Determine the growth rate of the population in 2030

y f x Use the graphs of y f z and y g x above to the find the function value a gof 2 help numbers b gog 4 help numbers y g x

Let h x fog x 4 f x x 3 Find f x given g x x 3 help formulas

Given that f x x and g x 5x 2 calculate a fog x b go f x c fo f x d go g x P FI E A

Let f x x 2 and g x x 3 fog x I gof x help formulas help formulas

Which of the following is an equivalent expression to fog x O g f x O f x g x Of x g x Of g x

CURRENT OBJECTIVE Represent a real world application as a linear function Question A farmer notices that there is a linear relationship between the number of bean stalks n she plants and the yield Y When she plants 3 stalks each plant yields 115 ounces of beans When she plants 8 stalks each plant yields 190 ounces of beans Write the linear equation Y n that correctly represents this situation Sorry that s incorrect Try again SWITCH Y n 17 64

Example 12 Discuss the continuity of the function defined by x 2 if x 0 x 2 if x 0 Solution Observe that the function is defined at all real numbers except at 0 Domain of definition of this function is D D where D x R x 0 and D x R x 0 Case 1 If ce D then lim f x lim x 2 X C X C f x c 2 f c and hence fis continuous in D Case 2 If c E D then lim f x lim x 2 X C X C c 2 f c and hence fis continuous in D Since fis continuous at all points in the domain off we deduce that f is continuous Graph of this function is given in the Fig 5 6 Note that to graph this function we need to lift the pen from the plane of the paper but we need to do that only for thos defined Example 13 Discuss the continuity of the function f given by f x x x if x 0 if x 0 Solution Clearly the function is defined at every real number Graph of the function is given in Fig 5 7 By inspection it seems prudent to partition the domain of definition of finto three disjoint subsets of the real line 112 Let D x R x 0 D 0 and D x R x 0 MATHEMATICS X 3 2 1 The right hand limit of fat 0 is 2 4 1 1 Rationalised 2023 24 y Fig 5 6 points where the function is not Fig 5 7 Case 1 At any point in D we have f x x and it is easy to see that it is continuous there see Example 2 2 2 1 1 0 Case 2 At any point in D3 we have f x x and it is easy to see that it is continuous there see Example 6 lim f x lim x 0 0 x 0 x 0 Persiane lim f x lim x 0 x 0 x 0 X X Case 3 Now we analyse the function at x 0 The value of the function at 0 is f 0 0 The left hand limit of fat 0 is Thus lim f x 0 f 0 and hence fis continuous at 0 This means that fis x 0 continuous at every point in its domain and hence fis a continuous function Example 14 Show that every polynomial function is continuous

Use a calculator in radian mode to find an approximate value of the expression correct to five decimal places if it is defined If an answer is undefined enter UNDEFINED sin 0 25718 534 90280

Use a definite integral to find the area under the curve f x 6 x from x 0 to x 6 square units Make a sketch of the curve showing the region y 2 95 y 6 2 N y 6 2 y 6 2 6