Final Exam 1311&0183-2 (1) PDF

‫‪Final Exam‬‬ ‫]‪[0183‬‬ ‫خالص حتيايت لكم ابلنجاح والتوفيق‬  ‫‪Final Exam‬‬ ‫]‪[1311‬‬ ‫خالص حتيايت لكم ابلنجاح والتوفيق‬ ‫‪2‬‬ Choose the correct answer: 𝟓 𝟏 𝟏 1- The solution set of the e...

‫‪Final Exam‬‬ ‫]‪[0183‬‬ ‫خالص حتيايت لكم ابلنجاح والتوفيق‬  ‫‪Final Exam‬‬ ‫]‪[1311‬‬ ‫خالص حتيايت لكم ابلنجاح والتوفيق‬ ‫‪2‬‬ Choose the correct answer: 𝟓 𝟏 𝟏 1- The solution set of the equation 𝒙 + = 𝟐𝒙 − is 𝟒 𝟐 𝟐 3 4 4 3 e) No answer of them a) { } b) { } c) {− } d) {− } 4 3 3 4 2- The solutions of the equation 𝒙𝟐 + 𝟒𝒙 + 𝟑 = 𝟎 is a){−1, −3} b) {1,3} c) {1, −3} d){−1,3} e) No answer of them 𝟑𝒙+𝟐 3- The solution of the inequality ≥ 𝟏 is 𝒙−𝟏 𝟑 a)(−∞, − 𝟐) ∪ (−𝟏, ∞) b) (−∞, − 𝟑𝟐] ∪ (𝟏, ∞) c)(−∞, − 𝟑𝟐) d)(−𝟏, ∞) e) No answer of them 4- The solution of the inequality |𝟑𝒙 − 𝟕| ≥ 𝟓 is 𝟐 𝟐 a) (−∞, 𝟑] ∪ [𝟒, ∞) b)(𝟑 , 𝟒) c)(−∞, 𝟐𝟑) ∪ (𝟒, ∞) d)(−∞, 𝟐𝟑) ∪ (𝟒, ∞) e) No answer of them 5- The domain of the function 𝒇(𝒙) = √𝟗 − 𝒙 is a)(−∞, 9) b)(−∞, −9] c)(−∞, −9) d){∞, 9} e) No answer of them 𝒙+𝟏 6- The domain of the function 𝒇(𝒙) = √ is 𝒙−𝟐 a)(−∞, 𝟏] ∪ (𝟐, ∞) b)(−∞, −𝟏) ∪ [𝟐, ∞) c)(−∞, −𝟏] ∪ (𝟐, ∞) d)𝑹 − {𝟐} e) No answer of them √𝟒+𝒙 7- The domain of the function 𝒇(𝒙) = is 𝟏−𝒙 a)(−𝟒, 𝟏] ∪ [𝟏, ∞) b)[−𝟒, 𝟏) ∪ (𝟏, ∞) c)(−𝟒, 𝟏] ∪ (𝟏, ∞) d)(−𝟒, 𝟏) e) No answer of them 3 8- The domain of the function 𝒇(𝒙) = (𝟐𝒙𝟐 + 𝟑𝒙 + 𝟏) + √𝒙 is a) 𝑅 − {0} b) 𝑅 c) (0, ∞) d)[0, ∞) e) No answer of them 𝒙𝟐 (𝒙−𝟏) 8- The function 𝒇(𝒙) = is 𝒙𝟒 +𝟓 a) Even b) Odd c) Even and odd d) Neither even nor odd e) No answer of them 9- If 𝒈(𝒙) = −𝒙𝟐 + 𝟓, 𝒇(𝒙) = 𝟐𝒙 + 𝟑 then (𝒇𝒐𝒈)(𝒙) is a)−2𝑥2 + 13 b)−2𝑥2 − 13 c) 2𝑥 2 + 13 d) 2𝑥 2 − 13 e) No answer of them 10- If 𝒇(𝒙) = 𝒙𝟐 − 𝟏𝟔, 𝒈(𝒙) = √𝒙 then (𝒇𝒐𝒈)(𝒙) is a) 𝑥 − 4 b) 𝑥 + 16 c) 16 − 𝑥 d) 𝑥 + 4 e) No answer of them 𝟐𝒙𝟑 +𝟕𝒙𝟐 −𝟐𝒙+𝟏 11- The limit value of 𝐥𝐢𝐦 ( ) is 𝒙→∞ 𝟒𝒙𝟑 −𝒙 7 b) Limit does not exist 1 1 e) No answer of them a) c) d) 4 4 2 𝒔𝒊𝒏 𝟗𝒙 12- The limit value of 𝐥𝐢𝐦 ( ) is 𝒙→𝟎 𝒙 a) Limit does not exist b) 9 c) 0 d)−9 e) No answer of them −𝒙𝟐 , 𝒙𝟏 a) -2 b) 2 c) Limit does not exist d) −1 e) No answer of them 𝟏 14- The limit value of 𝐥𝐢𝐦 𝒙𝟑 𝒔𝒊𝒏 is 𝒙→𝟎 √𝒙 a) 1 b) Limit does not exist c) 0 d) 3 e) No answer of them 4  𝒙 𝐈𝐟 𝒙 ≠ 𝟎 15- If 𝒇(𝒙) = { |𝒙| 𝐭𝐡𝐞𝐧 𝐥𝐢𝐦 𝒇(𝒙) is 𝒙→𝟎 𝟏 𝐈𝐟 𝒙=𝟎 a) 1 −1 1 d) Limit does not exist e) No answer of them b) c) 2 2 16- The solution set of the equation 𝒙𝟐 + 𝟏𝟐𝒙 = −𝟑𝟔 is a) {6} b) {−12,6} c) {−17,6} d){12, −6} e) No answer of them 𝟓 17- The solution set of the equation 𝒙 − 𝟐𝒙 = −𝟏 is 𝟒 3 −3 4 4 e) No answer of them a) { } b) { } c) {− } d) { } 4 4 3 3 𝐬𝐢𝐧(𝟕𝒙) ,𝒙 ≠ 𝟎 18- The value of k that make the function 𝒇(𝒙) = { 𝒙 continuous is 𝒌 , 𝒙=𝟎 a) −𝟕 b) 𝟕 c) 𝟎 𝟏 e) No answer of them d) 𝟕 𝟏 𝐬𝐢𝐧 𝒙 𝟑 19- The limit value of 𝐥𝐢𝐦 is 𝒙→𝟎 𝟑𝒙 1 b) 9 1 d) 3 e) No answer of them a) c) 9 3 𝟑𝒙−𝟔 20- The solution of the inequality > 𝟐 is 𝒙−𝟓 a)(−∞, −𝟒] ∪ (𝟓, ∞) b)(−∞, 𝟒) ∪ (𝟓, ∞) c)(−∞, −𝟏] ∪ [𝟓, ∞) d)[𝟒, 𝟓) e) No answer of them 21- The inverse of 𝒇(𝒙) = 𝟐𝒙 − 𝟑 is 𝑥+3 𝑥−3 𝑥−2 d) x + 3 e) No answer of them a) b) c) 2 2 3 5  𝟐 𝒙 ≥ −𝟏 22- The function of 𝒇(𝒙) = { 𝒙 the function is 𝒙 + 𝟐 𝒙 < −𝟏 a) Continuous b) not continuous as lim 𝑓(𝑥) does not exist 𝑥→−1 c) ) not continuous as 𝑓(−1) ≠ lim 𝑓(𝑥) 𝑥→−1 d) not continuous as 𝑓(−1) not defined 𝟐 𝒙≤𝟓 ( ) 23- The function of 𝒇 𝒙 = { 𝒙 the value of A is 𝒙+𝑨 𝒙>𝟓 a) −20 b) 20 c) 5 d) 15 e) No answer of them 𝒙+𝟏 24- The inverse of 𝒇(𝒙) = is 𝒙−𝟏 1+𝑥 1+𝑥 c) −𝑥 + 1 2𝑥 e) No answer of them a) b) d) 𝑥 𝑥−1 𝑥−1 25- The domain of the function 𝒇(𝒙) = √𝟐 + 𝒙𝟐 is a) 𝑅 b) (−1,1) c) [−1,1] d) (1, ∞) e) No answer of them 𝟑 26- The domain of the function 𝒇(𝒙) = √𝒙𝟐 − 𝟒 is a) 𝑅 b) 𝑅/{2, −2} c) 𝑅/{3} d) 𝑅/{6} e) No answer of them 27- The 𝒇(𝒙) = 𝟐𝒙 + 𝟑 is a) injective b) bijective c) increasing d) all above e) No answer of them 28- If (𝒇𝒐𝒇−𝟏 )(−𝟑) = ⋯ a) 3 b) −3 1 1 e) No answer of them c) d)− 3 3 6  𝟐𝒙−𝟏 29- The domain of the function 𝒇(𝒙) = is 𝒙−𝒙𝟐 a) [−1,0] b) 𝑅 c) 𝑅/{0,1} d) 𝑅/{−1,0} e) No answer of them 30- The solution set of the equation 𝒙𝟑 + 𝟑𝒙𝟐 − 𝟒𝒙 = 𝟎 is a) {4,0,1} b) {−4,0,1} c) {4,0, −1} d){−4,0, −1} e) No answer of them 31- The solution of the equation 𝟑𝒙𝟐 − 𝟓𝒙 + 𝟐 = 𝟎 is ,2 2 2 −2 a) {−1, } b) {1, − } c) {1, } d) {−1, } e) No answer of them 3 3 3 3 32- The solution of the inequality |𝟐𝒙 + 𝟑| < −𝟐 is a) (−∞, −2) b) −2 c) 3 d) 𝜑 e) No answer of them 33- The solution of the inequality |𝒙 + 𝟓| > −𝟑 is a) 𝑅 − {3} b) (−∞, −3) c) (−3, ∞) d) 𝜑 e) No answer of them 𝟐−𝟑𝒙 34- The solution of the inequality 𝟏 ≤ < 𝟑 is 𝟒 a) (− 10 2 , ] b) (−∞, − 𝟑𝟐) ∪ (𝟏, ∞) c) (− 10 , − 2] d) (− 10 ,− ) 2 e) No answer of them 3 3 3 3 3 3 35- The solution of the inequality |𝟐𝒙 − 𝟑| ≤ 𝟎 is 2 3 c) 𝑅 d) 𝜑 e) No answer of them a) − b) 2 3 𝒙−𝟏 36- The solution of the inequality ≤ 𝟎 is 𝟐𝒙−𝟑 3 3 3 3 e) No answer of them a) [1, ) b) [−1, ) c) 1, ) d) [1, ] 2 2 2 2 7  √𝒙−𝟏 37- The domain of the function 𝒇(𝒙) = is 𝒙𝟐 −𝟏 a) (−∞, 1] b) [1, ∞) c) (1, ∞) d) (−1, ∞) e) No answer of them 𝐜𝐨𝐬 𝟗𝒙 38- The limit value of 𝐥𝐢𝐦 ( ) is 𝒙→𝟎 𝟑𝒙 a) Limit does not exist b) 9 c) 3 d) 0 e) No answer of them 39- The limit value of 𝐥𝐢𝐦 𝒕𝟐 𝐬𝐢𝐧√𝒕 is 𝒕→𝟎 a) 1 b) Limit does not exist c) 0 d) 3 e) No answer of them 𝟐 𝒙 𝟐 a) 0 b) 2 c) −1 d) 1 e) No answer of them 𝒅𝒚 50- If 𝒚 = 𝒙𝟑 ∙ 𝐭𝐚𝐧(𝟐𝒙) 𝐭𝐡𝐞𝐧 = is 𝒅𝒙 a) 3𝑥 2 tan(2𝑥) b)3𝑥 2 sec(2𝑥) c) 3𝑥 2 tan(2𝑥) + 2𝑥 3 sec 2 (2𝑥) d) 6𝑥 2 sec 2 (2𝑥) 9  𝒙𝟐 51- If 𝒚 = 𝐭𝐡𝐞𝐧, 𝒚′ = 𝒙+𝟏 𝟑𝒙𝟐 +𝟏 𝒙𝟐 +𝟏 𝒙𝟐 +𝟐𝒙 𝒙 e) No answer of them a) b) c) d) (𝒙+𝟏)𝟐 (𝒙+𝟏) (𝒙+𝟏)𝟐 (𝒙+𝟏)𝟐 𝒙𝟐 +𝟕𝒙−𝟑𝟎 52- 𝐥𝐢𝐦 =⋯ 𝒙→∞ 𝟗 − 𝒙𝟐 a) 0 b) 1 c) −1 d) ∞ e) No answer of them 53- The equation for the tangent line of the graph of 𝒇(𝒙) = 𝒙𝟑 + 𝟐𝒙 at the point (1,3) is a) 𝑦 = 5𝑥 − 2 b) 𝑦 = 5𝑥 + 2 c) 𝑦 = −5𝑥 + 2 d) 𝑦 = −5𝑥 − 2 e) No answer of them 54- The function 𝒇(𝒙) = 𝟐𝒙𝟑 − 𝟔𝒙 − 𝟓 is increasing at: a) (−1,1) b) (1, ∞) c) (−∞, −1) ∪ (1, ∞) d) ( −1, ∞) e) No answer of them 55- The function 𝒇(𝒙) = 𝒙𝟒 − 𝟔𝒙𝟐 − 𝟓 is concave down at: a) (−∞, 1] ∪ [1, ∞) b) (−1,1) c) (−∞, −1) ∪ (1, ∞) d) [ −1,1] e) No answer of them 56- The local maximum value of the function 𝒇(𝒙) = 𝒙𝟒 − 𝟐𝒙𝟐 + 𝟏 at: a) (1,0) b) (0,1) c) (−1,0) d) (−1,1) e) No answer of them 𝒅𝒚 57- If 𝒚 = 𝒖𝟑 & 𝒖 = 𝟑𝒙𝟐 + 𝟒 then = 𝒅𝒙 𝟐 𝟑 a) 𝟑𝒙(𝟑𝒙𝟐 + 𝟒) b)𝟏𝟖𝒙(𝟑𝒙𝟐 + 𝟒) c) 𝟏𝟖(𝟔𝒙 + 𝟒)𝟐 d) 𝟏𝟖𝒙(𝟑𝒙𝟐 + 𝟒)𝟐 e) No answer of them 𝒅𝒚 58- If 𝟓𝒚𝟐 + 𝐬𝐢𝐧 𝒚 = 𝒙𝟐 then = 𝒅𝒙 𝟐𝒙 𝒙 𝟐𝒙 𝟐𝒙 e) No answer of them a) b) c) 𝟏𝟎𝒚+𝐜𝐨𝐬 𝒚 d) 𝟓𝒚−𝐜𝐨𝐬 𝒚 𝟏𝟎𝒚−𝐜𝐨𝐬 𝒚 𝟓𝒚+𝐜𝐨𝐬 𝒚 10  𝐬𝐢𝐧(𝒙) 59- If 𝒇(𝒙) = then 𝒇′ (𝒙) … 𝒙+𝟏 𝐬𝐢𝐧(𝒙)−(𝒙+𝟏) 𝐜𝐨𝐬(𝒙) 𝐬𝐢𝐧(𝒙)−(𝒙+𝟏) 𝐜𝐨𝐬(𝒙) a) b) (𝒙+𝟏)𝟐 𝒙+𝟏 (𝒙+𝟏) 𝐜𝐨𝐬(𝒙)−𝐬𝐢𝐧(𝒙) (𝒙+𝟏) 𝐜𝐨𝐬(𝒙)−𝐬𝐢𝐧(𝒙) c) d) (𝒙+𝟏)𝟐 𝒙+𝟏 𝒅𝒚 60- If 𝒚 = 𝒖𝟑 & 𝒖 = 𝟏 + √𝒙 then =⋯ 𝒅𝒙 3(1+√𝑥) 2 3𝑥2 3(1+√𝑥) 2 c) (1+√𝑥) e) No answer of them a) b) √𝑥 d) √𝑥 2 √𝑥 2√𝑥 61- If 𝒇(𝒙) = 𝒙𝟒 + 𝟑𝒙𝟐 + 𝟏 then the second derivative 𝒇′′ (𝒙) … a) 4𝑥 3 + 6𝑥 b) 12𝑥 2 + 6 c) 24𝑥 d) 24 e) No answer of them 62- Find the tangent line for the function curve 𝒚 = 𝒙𝟐 + 𝟐𝒙 + 𝟏 at the point (1,4) a) 𝑦 − 4 = 4(𝑥 − 1) b) 𝑦 + 4 = 4(𝑥 + 1) c) 𝑦 − 1 = 4(𝑥 − 4) a) 𝑦 + 1 = 4(𝑥 + 4) 63- The function 𝒚 = 𝒙𝟐 + 𝟐𝒙 + 𝟏 has critical point at x = a) −2 b) −1 c) 1 d) 2 e) No answer of them 64- The function 𝒚 = 𝒙𝟐 + 𝟐𝒙 + 𝟏 is increasing at the interval… a) (−1, ∞) b) (−∞, −1) c) (1, ∞) d) (−∞, 1) e) No answer of them 65- The function 𝒚 = 𝒙𝟐 + 𝟐𝒙 + 𝟏 is concave up at the interval… a) (−∞, ∞) b) (−∞, 0) c) (0, ∞) d) (−2,2) e) No answer of them 11  𝒆𝒙 −𝟏 66- Compute 𝐥𝐢𝐦 , using L’Hopital Rule. 𝒙→𝟎 𝒙 a) 0 b) ∞ c) 1 d) −1 e) No answer of them 𝒙𝟐 −𝟗 ,𝒙 ≠ 𝟑 67- Determine the values of k that make the function 𝒇(𝒙) = { 𝒙+𝟑 , 𝒙=𝟑 𝒌 continuous at x = 3 a) 3 b) 9 c) 6 d) 0 e) No answer of them 68- If 𝒇(𝒙) = (𝒙𝟑 + 𝟑)(𝒙 − 𝟏), 𝐭𝐡𝐞𝐧 𝒇′ (𝒙) = ⋯ a) 4𝑥 3 − 3𝑥 2 + 3 b) 4𝑥 3 + 3𝑥 2 + 3 c) 3𝑥 3 − 3𝑥 2 + 3 a) 4𝑥 3 + 3𝑥 2 69- The solution of the inequality |𝟐 − 𝟑𝒙| < 𝟒 is 2 2 −2 d) No answer of them a) 2 > 𝑥 > b)−2 > 𝑥 > c) 2 < 𝑥 < 3 3 3 √𝒙−𝟓 70- The limit value of 𝐥𝐢𝐦 is 𝒙→𝟏𝟔 𝒙−𝟐𝟓 1 b) 16 1 d) No answer of them a) c) 9 25 71- Find x-coordinates of the points where the tangent line to the graph of the given function is horizontal 𝒇(𝒙) = 𝒙𝟑 − 𝟑𝒙 a) 𝑥 = −1,1 b) 𝑥 = 0,3 c) 𝑥 = 0, √3 d) 𝑥 = −√3, √3 12  𝐬𝐢𝐧(𝟓𝒙) 72- At x = 0, Consider the function 𝒇(𝒙) =. Redefine f to make it 𝟑𝒙 continuous at sin(5𝑥) , 𝑥≠0 a) 𝑓 (𝑥 ) = { 3𝑥 3, 𝑥=0 sin(5𝑥) 3𝑥 , 𝑥≠0 b) 𝑓 (𝑥 ) = { 5, 𝑥=0 sin(5𝑥) 3𝑥 , 𝑥≠0 c) 𝑓 (𝑥 ) = { 3 , 𝑥=0 5 sin(5𝑥) 3𝑥 , 𝑥≠0 d) 𝑓 (𝑥 ) = { 5 , 𝑥=0 3 73- At x = 1, the function |x – 1| is a) continuous but non-differentiable b) continuous and differentiable c) discontinuous but differentiable d) discontinuous and non-differentiable 74- Find x-coordinates of the points where the tangent line to the graph of the given function is horizontal 𝒇(𝒙) = 𝒙𝟑 − 𝟏 a) 𝑥 = 0 b) 𝑥 = −1 c) 𝑥 = 1 d) 𝑥 = 3 13  𝒙+𝟏 75- Find the x-value (if any) at which the function 𝒇(𝒙) = |𝒙|+𝟒 is discontinuous. a) None, 𝑓 is continuous at every real number. b) 𝑥 = −2,2 is discontinuous at 𝑓 c) 𝑥 = −1 is discontinuous at 𝑓 d) R is discontinuous on 𝑓 𝒙+𝟐 76- Find the x-value (if any) at which the function 𝒇(𝒙) = is discontinuous. 𝒙𝟐 +𝟗 a) None, 𝑓 is continuous at every real number. b) 𝑥 = −3,3 is discontinuous at 𝑓 c) 𝑥 = −2 is discontinuous at 𝑓 d) R is discontinuous on 𝑓 𝑥 + 1, 𝑥≥0 77- At x = 0, the following function is 𝒇(𝒙) = { −𝑥 + 1, 𝑥