Calculus Questions

Estimate the value of lim 1 0 x 4 21 lim 21 24 Use a table of values to estimate the value of the limit If you have a graphing device use it to confirm your result graphically x 1 23 lim x 1 X 1 10 1 9 t 2 lan 3x x 0 tan 5x 22 lim U 24 lim x 0 3 9 5 X

Consider the contour diagram for the function z f x y below 2 1 2 0 Z 0 1 O 1 only O II only O III only O 1 and II II and III Z 2 2 Au P 3 V Z 4 4 O 5 Z 6 6 X Based on the contour diagram which of the following statements are true 1 The directional derivative of fat Pin the direction of u is positive II The directional derivative of fat Pin the direction of v is positive III The directional derivative of fat Qin the direction of w is negative

roblem 7 The following limit represents the derivative of some function fat some number a Find such an fand a lim h 0 25 h 5 h A x x a 25 h B f x x a 25 C f x x a 5 D f x x h 5 a E f x x h a 25 F f x x h a 5 G f x a 5th ann

5 Determine the intervals of increasing myopenmath com Determine the intervals of decreasing If an interval does not exist write DNE Separate intervals using a union U Determine the intervals for which the function is continuous Determine the interval for which the function is differentiable Determine the behavior at x 4 continuous and not differentiable corner Determine the behavior at x 3 discontinous gap and not differentiable Determine the behavior at x 1 contiuous and not differentiable cusp Determine the intervals for which the function is concave upward Determine the intervals for which the function is concave downward 4 Determine if there are any points of inflection 2 0

Let h a 5 x and let f be the inverse function of h Notice that h 1 7 f 7

Let f and g be inverse functions The following table lists a few values of f g and f 30 2 f x g x f x 19 9 2 10 g 2 13 12 12

Problem 2 Evaluate the following limit lim h 0 71 5 x h V71 5x h

23 If f x x x 10 and f 1 2 find f 1

Given f x x 4 x 3 x 2 a f 0

16 sin xy cos x y

roblem 1 For what value of the constant c is the following function continuous at x 5 f x 1 5 if x 5 and x 0 if x 5 x 5 C

12 esin y x y 14 tan x y 2xy 1 16 sin xy cos x y

14 tan x y 2xy 1

Find the secant of ZF E 24 D 10 F Simplify your answer and write it as a proper fraction improper fraction or whole num sec F

X 3 2 16 pts Consider f x 5 X 6 and g x X 4 the results if possible and describe their domains a f g x Find the rules of the new functions below simplify

10 xe x y 12 e sin y x y

12 e sin y x y

Step 2 The second integration by parts Step 1 has shown that Jeta s sin 5x dx The integral on the right is not much different than the one you started with Let s try using integration by parts again on the integral on the right u el cos 52 e cos 52 dr 4z 5 Fill in the table below with the appropriate terms for setting up the integration by parts for the integral du e cos 5x da dx V dv dx Now substitute all of this into the Integration by Parts formula giving e cos 5x dx 0 0 da

10 xe x y

5 22 Find dy dx by implicit differentiation 5 x 4xy y 4 7 x x y 5 6 2x xy y 2 8 x xy y 1

All fractions are to be written in lowest terms NO DECIMAL NUMBERS ALLOWED Answers need to be submitted as ONE file 1 Expand and Simplify the following a 3x x 6 5x 2x 4 3x 18x10x 20 1 4x 2x 6 Marks b 3xy 5x 2y 1 2xy 3y 2 7x 15x y Gxyt 3xy Gry axy 14x y x y 12vy 7xy

2 Solve the following equations a 9x 13 103 9x 103 15 9x 90 X 10 3 11 16A 32 29 15n 3 25 07 14 31 12 multiply borth vides by 12

53 56 Find y and y 53 y cos sin 30

f x 2x 17 5x k x 4 x 4

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

4 5 6 7 3 Transter Certificate T C Character Certificate C C Migration Certificate Bonafied Certificate Caste Certificate Aadhar Card Photo Copy

Hello there we sincerely apologize for any inconvenience caused by the delay in processing your payment We have already processed payments up to the 23rd of August We understand that you may have been eagerly awaiting the remaining payments and we want to assure you that they will be processed very soon We appreciate your patience and support as we work to resolve this matter Thank you for your understanding and please do not hesitate to reach out to us if you have any further concerns or questions

Pilot Name 3 nhal LIGH gear shofe RIC 46GH THIRRJE zgertzen SZIG 2127 GITEZ PE Total Privious ODO Meter Current ODO Meter 10511 339929 9956 203231 58745 16134 252054 45459 171405 5018 los 1 339924 9959 203237 58856 16 34 252093 45595 171439 5035 11063 11070 130877 130900 219256 219258 Case J 1 O 2 5 2 2 1 1 1 ODO Meter Difference 0 3 6 39 736 27 17 73 7 23 2 GPS KM 2 37 107 56498 46 26907 63 47585 08 43815 03 130 27 17 8 5 GPS Total Km 3 21048 55 24906 8 22897 89 17186 55 33945 53 28172 08 15487 83 37369 9 25664 8

Find by implicit differentiation dy dx dy dx 5 sin x cos y sin x cos y

S No Vechile No 1 RJ29PA1284 CHC Geejgarh RJ29PA1283 PHC Rahuwas RJ29PA1286 CHC Paparda RI29PA1289 PHC Bhandarage 2 3 4 5 6 7 00 8 RJ14PE6665 11 RI29PA1290 PHC Lotwara RJ14PE6667 Location R 29PA1285 PHC Guda Katla CHC Lalsot 9 RJ29PA1287 PHC Santha RI29PA1742 10 RJ29PA1288 PHC Khedla 12 RJ29PA1744 CHC Mahwa CHC Manpur Sikrai CHC Mandawar 13 R129PA1720 PHC Sainthal Morden Emergency Service Privious ODO Meter 11042 340103 10246 203393 62075 17039 255256 48765 Pilot Name zibar RRARDOT Ayol RIGH VEGH disay Sakata GUES E Current ODO Meter 110 50 11334 131812 219441 340105 10251 203399 62177 77050 2 55338 48831 171782 171782 5272 5278 11336 13 8 2 219441 Case 1 1 1 1 6 1 5 3 1 ODO Meter Difference 8 2 S 6 102 77 66 0 6 2 0 O Date 1 9 2023 GPS KM 7 O S O 102 17 83 64 LD 1 0 GPS Total Km 21557 0 25067 41 23171 56 O 17210 92 59770 27769 5 50647 46995 34272 2540 s 15931 380 25

7 52 Find the derivative of the function 7 f x 2x 5x 4 9 f x 5x 1 11 g t 8 f x x5 10 f x 1 21 1 13 f 0 cos 0 15 g x e 17 y x e 3x 19 f t e sin bt 21 F x 4x 5 x 2x 5 12 F t 21 14 g 0 cos 16 y 5 x 18 f 1 sin 20 A r r

8 The Taylor Series of 1 x x2 x3 x4 for 1 x 1 1 x Knowing that 0 51dx 0 693147180 1 x a Truncate the series to the first 4 terms then calculate the definite integral based on these four terms b What is the absolute error c What is the relative error d Determine the accuracy of the approximation obtained the number

1 04 Points NOTES Find y and y y ASK YOUR TE DETAILS y e e3e

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

Find by implicit differentiation dy dx dy dx x x y y 6

Let x 2x cos 2x f x 4x cos 2x 4x sin 2x f 5 20 cos 10 100 sin 10 f x 4 cos 2x 16x sin 2x 18x cos 2 5 80 sin 10 196 cos 10

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

Find the domain of the following functions Write the answer in interval notation Note If the answer includes more than one interval write the intervals separated by the union symbol U If needed enter o as Infand co Inf A 4 1 e Domain B 1 4t Domain

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

Find the limit If needed enter Inf for co Inf for o or dne if the limit does not esist lim 1 023 1 100

Find the limit If needed enter Inf for Inf for or dne if the limit does not esist co 8 4 8 lim 00 2 9 8

Find the limit If needed enter Inf for Inffor o or dne if the limit does not esist lim e 00 cos 4x

Evaluate the following limits If needed enter Inf for Inf for or dne if the limit does not esist 5 A lim 2100 4 5 0 0

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

2 Let y 4x 7x 4 Find the differential dy when 4 and dx 0 2 Find the differential dy when x 4 and da Sam 0 4

If z 8 and dy dt 5 when 2 and y Zensk dy dt 2 what is when 2 and y 2

find x cos x 8x sin x 4x Find 2 4x tan x 4x 2 tan x 2 sec x f x 42 ta sec