Calculus Questions

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

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

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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

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