Application of derivatives Questions and Answers

Hence f x is continuous on R Institute of Lifelong Learning University of Delhi I Q 2 I Q 3 Theorem 1 Statement Only A function f X R is said to be continuous at a point X EX if and only if for given any E neighborhood V f x of f x there exist a 8 neighborhood V x of xo such that if x is any point of XV x then f x belongs to V f x is f xV x V f x Continuity of Functions 5 Discontinuity of a Function at a Point Let X be a non empty subset of R let f X R and let X Then the function f x is said to be discontinuous at if and only if it is not continuous at 5 1 Types of Discontinuity Let f be a function defined as f X R and let x eX and let left hand limit right hand limit limit of the function and the value of the function at xare denoted by lim f x lim f x lim f x and f x 195 respectively Then I Removable Discontinuity at x let at s The function f X R is said to have the removable discontinuity at X EX if the limit of the function exist at x but not equal to the value of the function at x i e lim f x f x X pg 8 efa II Discontinuity of the First Kind at x The function f X R is said to have a discontinuity of first kind at x if both the left hand limit and right hand limit exist at x but are not equal to each other i e lim f x lim f x 294 Institute of Lifelong Learning University of Delhi Continuity of Functions III Discontinuity of the First Kind from the left pg 9 9

If F x f xf xf x where f 1 3 f 3 6 f 1 4 f 3 5 and f 6 6 Find F F 1

If x xy y 1 find the value of y at the point where x

Let f x x 5 5 x 5 2 5 5 5 DI Then 2

Algebraically find the inverse function of f x 6 5e Graph f f and the line y x on the same screen and check whether the graphs of f and fare reflections about the li

1 0 101 and find the equation of the tangent line in slope intercept form to f z at a nice point near 0 101 1 0 101 Use linear approximation i e the tangent line to approximate Let f z Then use this to approximate 1 as follows

1 5 20 A4 1 Place each of the following numbers in the most appropriate box above 510 28 20 08 3 6 12 2 Check each box that is correct to classify the number given Natural Whole Integers 1 20 7 35 Natural Numbers 3 A 44 H B Rational V Irrational Real 3 Place each on a number line Label it with the capital letter DRAW A NUMBER LINE A 67 5 c 13 D 5 E 75 411 10 9 8 7 6 5 3 2 1 0 4 Place each on a number line Label with the capital letter DRAW A NUMBER LINE 4 B 64 c 21 12 C D 3 6 E 23 E 2 3 4 56 1 7 8 1111 11 10 9 8 7 6 54 3 2 1 0 1 2 3 4 5 6 9 10 7 8 9 10

Use linear approximation i e the tangent line to approximate 16 1 as follows Let f 2 2 The equation of the tangent line in slope intercept form to f x at 16 can be written in the form y mx bwhere m b Using this we find our approximation for 16 1 is NOTE For this last part give your answer to at least 6 significant figures or use fractions to give the exact answer

Indicate the answer choice that best completes the statement or answers the question Which function is an odd function Oa f x 2x b f x x 4 Oc f x x 9 Od f x x 4x

1 Suppose that the height of a skydiver t seconds after jumping from an airplane is given by s t 4500 20t 16t2 feet neglecting air resistance The skydiver pulls the ripcord after 10 seconds of free fall to deploy the parachute after he has fallen Write a velocity function for the skydiver t How fast is the skydiver falling the instant before he pulls the ripcord ft sec miles per hour Calculate the skydiver s average velocity over the time interval from 5 seconds to 10 seconds What is the skydiver s acceleration function 2 A car moves along the x axis so that at any time t t 0 the position is given by s t 4t 18t 15t 1 Write a velocity function for the car Write an acceleration function for the car What is the velocity of the car at t 2 What is the velocity of the car at t 4 What is the average velocity of the car on the time interval from 4 to 6 When is the car at rest circle

Helium is pumped into a spherical balloon at a rate of 3 cubic feet per second How fast is the radius increasing after 2 minutes Note The volume of a sphere is given by V 4 3 TT Rate of change of radius in feet per second

The linear approximation at z 0 to f x where B 1 5 I is L x A Br

Suppose xy 4 and dt 11 dy dt 1 Find dx dt when a 3

If 5x 2x xy 4 and y 4 21 find y 4 by implicit differentiation

da 9 sect tan dt Now substitute and simplify to get a trig integral in terms of t The trig integral should no longer involve a square root 81 da A 81 sect tan dt t Part 2 Evaluating the Trig Integral From previous integration techniques we know that St tan t sec t dt sec t sec t dt sec t tan t 2 The final result would then be In sec t tan t 2 Use this to evaluate the trig integral from Part 1 and then substitute back to get the final result in terms of x To substitute back in terms of x you would use sec t tan t

1 Assume that f x is a function on the interval 0 1 that has a Fourier sine series on 0 given by 00 give the value of ba 1 12 f x sin nx 71 1 Use a Fourier sine series to find a solution to This solution will have the form 3v x f x 0 x n v 0 v n 0 v x b sin nx n 1

The bar graph below shows that as costs changed over the decades a group of people devoted less of their budget to groceries Percentage of Total Spending on Food Percent of Total Spending 30 24 18 12 6 23 17 8 6 1950 1970 1990 2010 In 1950 the group spent 23 of their budget on food This has decreased at an average rate of approximately 0 28 per year since then Find a linear function in slope intercept form that models the percentage of total spending p x by the group x years after 1950 The equation p x models the given description Use integers or decimals for any numbers in the expression

da sin z 12 COS 12 Now substitute and simplify to get a trig integral in terms of t The trig integral should no longer involve a square root Tz cos 1 dt 1 14 dt 1 144m da Part 2 Evaluating the Trig Integral From previous integration techniques we know that cos t dt Use this to evaluate the trig integral from Part 1 and then substitute back to get the final result in terms of a To substitute back in terms of a you would use sin t cos t The final result would then be sin t cos t 2

2 Evaluate each limit Use algebraic methods if possible If not use graphs or tables x 7x 12 x 9 2K 4K a limx 31 X 8 b limx 8 2

a Are the events A and C independent Justify your answer b Are the events A and D independent Justify your answer c Compute 1 P AB and P AB Are they equal d Compute 1 P AB and P AB Are they equal 26 A large corporation has spent considerable time developing employee performance g scales to evaluate an employee s job performance on a regular basis so major adjustments e made when needed and employees who should be considered for a fast track can be isolat eys to this latter determination are ratings on the ability of an employee to perform to his or pabilities and on his or her formal training for the job

Which function is always increasing O x 1 O x 1 O x 2x 1 Ox 2x

Which inequality has a solution set of 5 00 0 1 4 0 1 4 0 1 4 02 124

2 O 2 2 O 1 3 O 2 2 0 2 T Where is f x increasing Give your answer in interval notation

Which function has an x intercept of 12 and a y intercept of 3 Oy 4x 3 Oy 3x 12 O y 12x 3 Oy z 3

Given f x 3x 4 and g x 5x 2 find 3f x 2g x O 25 9x 12 19x 16 A

Given f x 3x 4 and g x 5x 2 find 2f 4 O 6x 8 O 16 O 32 O 2 3x 4

The goal of this problem is to evaluate sin cos da using substitution 1 Because the cosine term has an odd power the first step is to rewrite cos5 x cos x cos x cos x cos x where n 0 2 Next replace the cos x term with 3 Now do a substitution with u and du dx 4 With this substitution the integral after multiplying out into a sum of powers of u becomes Odu Do not put a negative sign in front of the integral 5 Now antidifferentiate in terms of u and substitute back to get the final result siz C sin x cos5 x dx

2 31 3 9 x 34 5x 1 0 04 2 4 37 e x 2 et 1

Solve for a in the given matrix equation 1 2 3 det 1 x 1 26 2 3 4 Choose the correct answer a x 2 Ob z O c C x 1 Od 2 13 e None of the options

1 Place the phases of the business cycle in order Recession Trough Peak Expansion How long do each business cycle last

6 The battery charge of a cell phone is tracked with a function over the course of one day Graph a possible function below that correctly represents the domain and range of the situation Battery charge 150 125 100 75 50 25 8 4 0 25 4 8 12 16 20 24 28 Time hours

9 Which of the following functions has a domain given by x Rx 4 A f x x 4 B g x x 16 C h x 9 x D j x 2 3x 12

02 01 MC The two way frequency table given shows the results from a survey of students who attend the afterschool program Takes Art Class Doesn t Take Art Class Total 140 Plays a Sport Doesn t Play a Sport 45 Total 45 Does the data show an association between taking an art class and playing a sport O There is a strong positive association O There is a strong negative association O There is a weak positive association 225 O There is a weak negative association

Next question The table below gives the annual sales in millions of a product year 1998 1999 2000 2001 2002 2003 2004 sales 138 192 234 264 282 288 282 What was the average rate of change of annual sales a Between 1998 and 1999 54 Get a similar question You can retry this question below b Between 1998 and 2006 millions of dollars year millions of dollars year 2005 2006 264 234

ir Question 5 3 2 1 3 2 My 2 3 2 3 4 S Increasing on the interval s a The function graphed above is Decreasing on the interval s

A db Fig 1 1 In this graph the points a and b are adjacent whereas b and c are non adjacent 2 Let V 1 2 3 4 and X 1 2 1 3 1 4 2 3 2 4 3 4 G V X is a 4 6 graph This graph is represented by the diagram as shown in Fig 1 2 Fig 1 2 In this graph the lines 1 3 and 2 4 intersect in the diagram and their intersection is not a point of the graph Fig 1 3 is another diagram for the graph given in Fig 1 2 Fig 1 3 3 The 10 15 graph given in Fig 1 4 is called a Petersen graph

Which of the following functions has a domain given by x ER x 4 A f x x 4 B g x x 16 C h x 9 x 2 D j x 3x 12

7 Find the domain of y A 2 x 3 x 7

19 24 Explain using Theorems 4 5 6 and 8 why the fun tion is continuous at every number in its domain State the domain no 19 F x 4 THEOREM If f and g are continuous at a and c is a constant then the following functions are also continuous at a 1 f g 2 f g f 4 fg g 2x x 1 x 1 5 c h x x if g a 0 5 THEOREM a Any polynomial is continuous everywhere that is it is continuous on R 0 0 b Any rational function is continuous wherever it is defined that is it is continuous on its domain EXAMPLE 6 On what intervals is each function continuous a f x x100 2x 7 75 x 1 x 1 3 cf b g x TAM OF x 1 x 1 User p x 2x 17 x 1 8 THEOREM If g is continuous at a and f is continuous at g a then the composite function fo q given by fog x f g x is continuous at a

oblem 2 Use the graphs of f and g to answer the following a f g 2 b c 3g 7f 2 4 4 2 4 f 8 4 Figure 1 Graph of f x and graph of g x d fog 3 e g 3 f f 2

Show that the function f x x 31 is not differentiable at 3 We have The right hand limit is f x f 3 x 3 lim X 3 f x x 3 and the left hand limit is lim f x f 3 X 3 X 3 Find a formula for f and sketch its graph f x Since these limits are not equal f 3 lim f x f 3 does not exist and f is not differentiable at 3 X 3 x 3 No Solution 2 if x 3 if x 3 4 4 a 6 2 a 3 if x 23 if x 3 2 3 4 Fill Graph Layers After you add an object to the grap can use Graph Layers to view and properties

Evaluate the function at the indicated values If an answer is undefined enter UNDEFINED x X f 9 f 5 f 0 f 2 f x f f x

The figure shows the graphs of three functions One is the position function of a car one is the velocity of the car and one is it acceleration Identify each curve and explain your choices position velocity acceleration Figure 20 0 0 Description b A

The figure shows the graphs of f f and f Identify each curve and explain your choices f 0 H f O f 70 Figure b Description a

The graph of fis given State the numbers at which f is not differentiable Enter your answers as a comma separated list y X 4 2

Let f t Vt For a 0 find f a f a

Problem 9 Consider the following matrices Problem 9 a Problem 9 b 4 033 13 15 10 B 1 4 2 4 2 3 a Let D BTC Find d21 i e find the entry in row 2 column 1 of the matrix D b Find tr 2BB c Let E B CA Find e32 i e find the entry in row 3 column 2 of the matrix E

K The velocity of a parachutist during free fall is f t 82 1 e 0 12t meters per second Answer the following questions using the graph Recall that acceleration is the derivative of velocity b What is the approximate acceleration when t 0 10 m sec2 c Approximately when is the parachutist s velocity 51 m sec 8 seconds d Approximately when is the acceleration 5 m sec CHE Ay 80 70 60 50 40 30 20 10 0 0 10 20 30 S

escribe all absolute extreme points of P r s m n c d GALER Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The absolute maximum point is There is no absolute minimum point Type an ordered pair B The absolute minimum point is is no absolute maximum point Type an ordered pair There OC The absolute maximum point is the absolute minimum point is Type ordered pairs OD There are no absolute extreme points and

13500 The value V of a particular automobile in dollars depends on the number of miles m the car has been driven according to the function V h m Suppose that h 35000 14300 and h 45000 13900 What is the average rate of change Round to the nearest thousandth of h on the interval 35000 45000 and what are the units on this value Average rate of change dollars per mile In addition to the information given in a say that h 55000 13500 Determine the best possible estimate of h 45000 Round your answer to the nearest hundredth Hint Which value do you expect to be greater h 45000 or h 55000 h 45000 and h 55000 are equal Write a sentence to describe the long term behavior of the function V h m plus another sentence to describe the long term behavior of h m Provide your discussion in practical terms regarding the value of the car and the rate at which that value is changing