Functions Questions and Answers

inal Exam and Surname 1 Evaluate the indefinite integral of f x x5 5 dx 8118 2 Evaluate lim xe using the L Hospital s rule 3 Differentiate f x cos x cos x cosh x 4 Differentiate f x ln 7x 2x 5 Evaluate the indefinite 6 Prove the sta Std No

Find the following f 3 if f x x 3x 2x 2 A 191 OB 108 O C 189 R 119 D

a p is true q is true and r is true p is false q is true and r is true If p is true q is true and r is true what is the truth value of p q r False True If p is false q is true and r is true what is the truth value of p q Ar False True

Let p q and r represent the following simple statements p I work hard q It is snowing outside r It is Tuesday Write the symbolic statement in words r qAp Choose the correct answer below OA If it is not Tuesday then it is not snowing outside and I work ha B If it is Tuesday then it is not snowing outside and I do not work OC If it is Tuesday then it is snowing outside and I work hard D If it is not Tuesday then it is snowing outside and I work hard

p The stove is hot q The soup is cold r The food is done llowing Write the following statement in its symbolic form The stove is hot and the soup is not cold and the food is The symbolic form is

wo or more sides indicate that the marked sides are of equal length Is this triangle scalene isosceles or equilateral Equilateral Isosceles Scalene Is this triangle acute obtuse or right Obtuse Acute Right Comme

A 55 X B 99 88 0 y 0 Simplify your answers C 10 A B X y C

Determine the sum of the measures of the interior angles of the Hexagon The sum of the measures of the interior angles of a hexagon is

Consider the following compound inequality 18 6 Answer y 3 3 23 6 Step 1 of 2 Solve the inequality and express your answer in interval notation Use decimal form for numerical values

Of the given values of x identify those at which the function is continuous f x x 6 x 3 x 2 6 0 3 2 OA 6 0 3 2 OB 6 0 OC 3 2 OD 6 3 2

Find all points where the function is discontinuous B A x 1 B x 2 OC x 2 x 1 Nono X

178 170 362 154 348 138 230 on 13 Sep 12 Average price of up in US chies Nov 12 Dec 12 Month ck to expand image Mar 11 Apr 11 May 11 An 11 MI1 Aug 11 Sep 11 Question 1 of 8 The average price in cents per pound of sugar observed during the period of September 2012 to September 2013 in most U S cities is displayed in the line graph provided What is the approximate increase in the price in cents per pound during the period of November 2012 to March 2013 Select a Choice Below A B C About 1 5 cents About 2 5 cents About 4 4 cents Skip

Is the given function continuous at the given values of x x 4 if x 5 4 x ifx 5 f x x 4 x 5 Is the function continuous at x 4 Select the correct choice below and if necessary fill in the answer box to complete your ch OA Yes the point is defined lim f x exists and lim f x Simplify your answer X 4 X 4 OB No the function is not continuous at x 4 If the function continuous at x 5 Select the correct choice below and if necessary fill in the answer box to complete your ch OA Yes the point is defined lim f x exists and lim f x Simplify your answer X 5 X 5

the angle as acute right obtuse straight or none of these angles Choose the correct an Straight Acute Obtuse O Right None of the abov

Choose the correct answer below Straight Right Acute O Obtuse None of these angles

Use the figure Which of the following is DBG n GF BG 4DBF BG B

Use the figure to find CBD n FBA Which of the following is CBD n FBA BD 4CBA B 0 DO B JE C

An angle whose measure is less than 90 is called a n

Find the derivative of the function 5 7 dy y 3 x N X

Fill in the blank with an appropriate word When two straight lines intersect the nonadjacent angles formed are called When two straight lines intersect the nonadjacent angles formed are called angles ang

me demand for a certain portable USB battery charger is given by D p 4p 4p 5 where p represents the price in dollars Find the rate of change of demand with respect to price Find and interpret the rate of change of demand when the price is 15 a The rate of change of demand with roopor

Find the given derivative P dx d dx 9x 9x IN 2 N W 00 8 XNIW 00

1 point Which of the following functions are onto Recall that surjective is another word for onto 2 7 6 a 3 R 41 00 A R R H 25 41 X B Z B B XE B 41 Y B Z

Write a function called get square that takes in a single numeric value as an argument called x squares that value and returns the result Run the cell by either clicking on the Run button in the menu bar or by hitting Return Shift at the same time on your keyboard Graded Cell The cell below will be graded Remove the line raise NotImplementedError before writing your code YOUR CODE HERE

12 U nc dereceden ger el katsay l bir P x polinomunda P 1 P 3 P 4 3 t r P x polinomunun sabit terimi 3 oldu una g re P 2 ka t r B 2 C 3 E 6

Soru 5 4 puan B Soru 25 Cevap 4 Which of the following is the tangent line to the curve of f x y 7x 4 y x 4 y x 4 y 3x 5 2x 3x 4 Sonraki S at x 1

Soru 8 4 puan For the function f x A B 3x Soru 25 Cevap which of the following are the intervals on which f is increasing 2 2 0 Sonraki Soru 0 0 20 0 2 U

4 Find the integral surface of the equation p q x pz passing through the curve I x 0 z 4y

Which of the following is the limit of the function f x lim 818 lim IX lim PIX 2 2x 3 lim 400 X 2x 5 2x 3 2x 2x 3 2 5 9 2r 5 2 3 5x 5x 52 52 e 2 5 e3 5 1 2x 5 2 3 5x 0 at infinity

Let f n and g n asymptotically positive functions and assume that limn o x g n Prove that if f n g n then Ln f n O Ln g n

Show that if there are 28 students in the class then at least two will have last names that begin with the same letter For the toolbar press ALT F10 PC or ALT FN F10 Mac 10nt Y Y

T 7 a b O LLL Find the radius of convergence of the power series for the following functions expanded about the indicated point 2 1 about z 3 i about z 0 COS Z 22 4 c d sin sin about 2 i z about 2 i Solution a Removable singularity at z 1 poles at i 1 R 3 to i nearest singularity b singularities at 7 2 are removable c removable at z 0 poles at 2 R ni 2 TV d 2 sin z simple poles at z 0 R T

LLL Find the isolated singularities of the following functions and de ermine where they are removable essential or poles Determine the order of any pole and find the principal part at each pole a z 2 1 d tan z sin g Log 1 COS Z b e sin h Log z 1 f 22 4 i e 1 e2 1 Solution a J R J Groves with g 2 We have that z 1 is a pole of order 2 since 21 1 and g 2 is analytic near z 1 Similarly we have z 1 is a pole of order 2 since 2 1 with h 2 2 1 and h 2 is analytic near 1 a Double poles at 1 principal parts 1 4 z 1 e A Kumjian 8 2 sim 2 1 2k 1 k0 13 The function is analytic on the punctured plane D C 0 It has an isolated singularity at z 0 For 0 we have k 0 24 1 1 k 1 2k 1 22k 1 2 1 2k 1 22k 1 k 1 Hence 0 is an essential singularity for the function since there are an infi nite number of nonzero terms in the Laurent series with negative exponents e J R J Groves The function 2 sin has an isolated singularity at z 0 Since

Set Remark Check the result by differentiating the formula in the pre ceding exercise with respect to the parameter Solution VII 2 2 and integrate R Le f z and therefore dx IT x a 20 P I 1 2 a along the contour in Figure VII 2 2 Residue at a double pole at 2 ia where by Rule 2 dr x 1 0 1 d 1 2 Res 52 2 2 10 lim 12 1 ia 22 a iadz 2 ia 2 ia Integrate along 1 and let Roc This gives R da 2 dz 1 17 z S R Integrate along 72 and let Roo This gives 1 0 2 i Remark From Exercise VII 2 1 we have the formula 1 1 7 1 z ia z ia da 0 x a 2 1 2 da Lx 1 9 4 zdr 427 1 1 1 2 d 5 R a Using the Residue Theorem and letting Roo we obtain that that 4 3 R 75 75 TR 0 R zia dr x a 77 a 0 It is allowed to differentiate both sides in the formula because for every compact interval with a 0 both the integrand and its derivative are con tinuous and the primitive of the derivative is uniformly bounded by a a independent function h r such that fh x on every such like interval We start by differentiate the left hand side in the formula with respect to the parameter a a jz dr 2a 2 0 1 6 and differentiate the right hand side in the formula in the same way da 1 I 0 4a3 dr 2a a dr Loo 2

VII 1 2 Calculate the residue at each singularity in the complex plane of the following functions a e 2 b tanz c Thus by definition Solution a The function e 1 has a singularity at z 0 by Laurent expansion Res tan z 2 n c The function poles By rule 2 Res e 2 0 1 b sin has isolated singularities at z The function tan z n they are simple poles COS 2 By rule 3 22 1 Res d z 1 1 el z 1 2 D 1 Res 1 Z 2122 d sin z COS 2 z z 2 nT n L dz z i 2 1 has isolated singularities at z ti they are double d Z Res 2 1 2 1 2 0 i dz z i sin z sinz n z 2 nT z i 2z z i z i z i 2z z i z i 4 1 z i 0 2 1 0

Evaluate the following integrals using residue theorem a 2 1 dz indz e 2 1 128 10 e b 2 2221dz d Sz 1 sizdz f S 2 1 2 sin z c 2 21 2 cos z Solution a Using the residue theorem and rule 1 d 2 i Bes zdz sinz 128 1dz tan z dz z 1 2 3 2 2 2mi lim sin z 2 i 1 2 i

All calculus students know that the derivative of a function f at x can be defined as f x lim f x h f x h Choose your favorite function f nonzero number x and computer or calculator Generate approxi mations f x to f x by f x 10 f x 10 n f x for n 1 2 20 and describe what happens Problem 27 from page 183 taking f x e and xo 1

f x g x lim x 5 4g x Apply limit properties to the given limit Choose the com f x g x OA lim X 5 4g x lim f x lim glx X 5 X45 OB lim X 5 f x g x 4g x lim f x fim g x 4 Ing X 5 X 5 Oc lim X 5 lim f x lim g x X 5 f x g x x 5 4g x lim 4 lim g x OD lim X 5 X 5 lim f x lim gix X 5 94X f x g x x 5 4g x Find the limit lim 4 lim gix Sex Sex x 6 wer

d the average rate of change for the function 1 X 1 between x 5 and x 0 The average rate of change is Type an integer or a simplified fract

Find the average rate of change for the following function f x 3x 2x 4 between x 1 and x 1 The average rate of change for f x over the interval 1 to 1 is

Suppose that the total profit in hundreds of dollars from selling x items is given by P x x 10x 23 Find the marginal profit at x 1 OA 200 per item B 1 400 per item C 1 000 per item D 800 per item

Find the average rate of change for the function over the given interval 3 x 2 y OA 28 O B between x 1 and x 4 1 2 O c 2 1

Find the given limit lim x 23 x 23 23 x 23

Graph the function f whose rule is given Then use the graph to evaluate the given limits 1 x if x 1 x 4 if 1sx 1 if x 1 if x 1 f x 3 2 x a Find lim f x Select the correct choice below and if necessary fill in the answer box to x 1 complete your choice A The limit is 5 B The limit does not exist and is not coor b Find lim f x Select the correct choice below and if necessary fill in the answer box to complete your choice A The limit is 1 B The limit does not exist and is not coor 00 c Find lim f x Select the correct choice below and if necessary fill in the answer box to x 1 complete your choice CIT Choose the correct graph below OA Q Q OB O D Q Q G Q Q G

Use the properties of limits to find the given limit 10x 5x 8 lim X 0 3x lim 9x 28 10x 5x 8 2 Type an integer or a simplified fraction

Find the limit lim x 6 lim 8x 6x x 6 6x x 6

15 y 4 x 2 1 17 y x 2 5 19 y 1x hy 16 y 3 x 2 18 y 3 x 4 49 20 y x 2 5 6

14 1 point What is the period of this function YA 10 2 2 U 6 00 10 12 14 16 d 7 EXCL

16 1 point Whish graph match the equation y 0 3 sin x 45 1 90 188 270 360 AA 0 90 180 270 360 X 0 90 180 270 360 90 180 270 360

3 1 point The current i in amperes A passing through a wire is given by the equation i 8 sin 0 7t 25 where is the time in seconds The current is approximately 5 A after 20 s