TOPIC 1-3: Continuity Q.#1- Q.#6 Please use the graph of function f(x) shown at right to...

Transcribed Image Text:

TOPIC 1-3: Continuity Q.#1- Q.#6 Please use the graph of function f(x) shown at right to complete the table found below. Final Answers a=0 a=3 a=5 a=8 a=10 -5 -3 3 7 9 11 Q.#1 Q.2 Q.#3 Q.#4 Q5 Q.#6 Lim f(x) Lim f(x) Lim f(x) X417 f(a) Continuous? Differentiable x+a+ x-a Q.5(a) Q.#5(b) Q.#6(a) Q.#6(b) Yes/No? Why? Yes/No? Why? x+1 if x <2 Q. #7 For the function g(x) = 3 if x 2, please find b-x2 if x>2 constant "b" that makes g(x) continuous at x=2 Q. #8-Q.# 10 Please use the function shown at the right for these questions. Q.#8 For this function, please do the following. a) Use the definition of continuity to determine algebraically if f(x) is continuous at x=-1. 3 for x-1 1-x f(x)={ 1 for x=-1 x-2 , for x>-1 4 Final Answer (yes or no) b) If a discontinuity at x=-1, classify it as removable or non-removable. If non-removable, is it jump or infinite? Final Answer (circle) No discontinuity Removable Non-removable: Jump, Inf. Page 2 of 8 Q.#9 For above function, please find and classify any other removable or non-removable discontinuities elsewhere in the function. If non-removable, classify them as jump or infinite. Hint: Normally one checks transition point(s) and the functions themselves. Q.#8 checked the transition point (x-1), so leaves only the functions themselves. Final Answers Function Continuous where applies? (Y/N) Discontinuous where applies: At x= Removable Non-removable: Jump Non-removable: Inf. a) f(x)= 3 1-x for x-1 b) f(x) = 1 for x=-1 x-2 c) x-4 for x-1 Q.# 10 Using any discontinuities in Q. #8 - Q.#9, please list the intervals of continuity for f(x). Example: If a function has removable discontinuity at x=2, its intervals of continuity are (-, 2) (2,+) or, equivalently, {x|x <2Ux>2} or {x | x2} Final Answer TOPIC 4-5: L'Hpital's Rule Q. # 11 Please evaluate Lim x-25 x5-4x-x as follows. a) Using L'Hpital's Rule b) Using earlier limit technique without L'Hpital's Rule tan(7x) Q. # 12 Please evaluate Lim- as follows. x0 sin(4x) a) Using L'Hpital's Rule b) Using earlier limit technique without L'Hpital's Rule Q. #13 Please evaluate the following limits, using L'Hpital's Rule if it applies. If not, please evaluate the limit in some other way. a) Lim x0+ sin x-x Final Final b) Lim c) Lim xex x-- d) Lim (x-1) tan x+1+ e) Lim (cos x) (1/x) x0+ f) Lim (secx- tan x) x/2 Hint: secx & tanx can be expressed in sinx &/or cosx. TOPIC 1-3: Continuity Q.#1- Q.#6 Please use the graph of function f(x) shown at right to complete the table found below. Final Answers a=0 a=3 a=5 a=8 a=10 -5 -3 3 7 9 11 Q.#1 Q.2 Q.#3 Q.#4 Q5 Q.#6 Lim f(x) Lim f(x) Lim f(x) X417 f(a) Continuous? Differentiable x+a+ x-a Q.5(a) Q.#5(b) Q.#6(a) Q.#6(b) Yes/No? Why? Yes/No? Why? x+1 if x <2 Q. #7 For the function g(x) = 3 if x 2, please find b-x2 if x>2 constant "b" that makes g(x) continuous at x=2 Q. #8-Q.# 10 Please use the function shown at the right for these questions. Q.#8 For this function, please do the following. a) Use the definition of continuity to determine algebraically if f(x) is continuous at x=-1. 3 for x-1 1-x f(x)={ 1 for x=-1 x-2 , for x>-1 4 Final Answer (yes or no) b) If a discontinuity at x=-1, classify it as removable or non-removable. If non-removable, is it jump or infinite? Final Answer (circle) No discontinuity Removable Non-removable: Jump, Inf. Page 2 of 8 Q.#9 For above function, please find and classify any other removable or non-removable discontinuities elsewhere in the function. If non-removable, classify them as jump or infinite. Hint: Normally one checks transition point(s) and the functions themselves. Q.#8 checked the transition point (x-1), so leaves only the functions themselves. Final Answers Function Continuous where applies? (Y/N) Discontinuous where applies: At x= Removable Non-removable: Jump Non-removable: Inf. a) f(x)= 3 1-x for x-1 b) f(x) = 1 for x=-1 x-2 c) x-4 for x-1 Q.# 10 Using any discontinuities in Q. #8 - Q.#9, please list the intervals of continuity for f(x). Example: If a function has removable discontinuity at x=2, its intervals of continuity are (-, 2) (2,+) or, equivalently, {x|x <2Ux>2} or {x | x2} Final Answer TOPIC 4-5: L'Hpital's Rule Q. # 11 Please evaluate Lim x-25 x5-4x-x as follows. a) Using L'Hpital's Rule b) Using earlier limit technique without L'Hpital's Rule tan(7x) Q. # 12 Please evaluate Lim- as follows. x0 sin(4x) a) Using L'Hpital's Rule b) Using earlier limit technique without L'Hpital's Rule Q. #13 Please evaluate the following limits, using L'Hpital's Rule if it applies. If not, please evaluate the limit in some other way. a) Lim x0+ sin x-x Final Final b) Lim c) Lim xex x-- d) Lim (x-1) tan x+1+ e) Lim (cos x) (1/x) x0+ f) Lim (secx- tan x) x/2 Hint: secx & tanx can be expressed in sinx &/or cosx.