Math 1413 Test 2 Review 1 Math 1431 Exam 2 Review Material Covered: Chapters 1 and 2 - Know your instructor’s name. - Know your lab’s TA name, lab’s time and room number. - Test: 50 minutes; CASA Testing reservation required. - Test is part multiple choice and part free response (with parts). You will have the multiple choice grade right away, but the free response will take some time to grade. - No formulas are provided. No calculators allowed. - You will be given scratch paper. - For the free response you’ll be given stapled pages asking for your name, lab info, etc to do all your free response work on. Show all your work here and not on the scratch paper. - Do the practice test (20 attempts allowed) for more practice and extra credit. 1. Evaluate the following limits, if they exist. a. 05lim 2 1xxxx b. 2312limxxxxMath 1413 Test 2 Review 2 c. 222310lim4xxxx d. 042limxxx 2. Find the following limits, if they exist. Recall: 0sin( )limxax abx b and 01cos( )lim 0xaxbx a. 0sec(5 )lim2xxx b. 05limtan(9 )xxxMath 1413 Test 2 Review 3 c. 201cos(2)lim7xxx d. 012lim6xcos xsin x 3. The function f is given below along with its graph. Find the following limits, if they exist. 227 21 2 21 2<x,xfx x , xx, x a. 2limxfx b. 2limxfx c. 2limxfx What type of discontinuity occurs at x = -2? at x = 2?Math 1413 Test 2 Review 4 4. Find the points of discontinuity (if any). Classify each as removable, jump or infinite. a. 3241fxx xx b. 3282xfxxx 33 2 233 2 2()( )()( )a b aba abbab abaabb 5. Determine if the following function is continuous at x = -2. If the function is not continuous, state the type of discontinuity. 252725)(2xxxxxxf - Step 1: Is f(-2) defined? - Step 2: Check to see if 2lim ( )xfxexist. 2lim ( )xfx must equal 2lim ( )xfx. Does 2lim ( )xfxexist? - Step 3: Does 2lim ( )xfx= f(-2)? i.e. Compare #1 and #2 above. Hence, f(x) is: CONTINUOUS at x = -2 or DISCONTINUOUS at x = -2 Type: JUMP REMOVABLE INFINITEMath 1413 Test 2 Review 5 6. Find the values of a and b such that f (x) is differentiable. 2210, 2()6, 2ax xfxxxbx You must first check continuity at x = 2. f(2) 2lim ( )xfx 2lim ( )xfx Next, work on differentiability. 'f 'fMath 1413 Test 2 Review 6 7. Find the derivative of the following functions using the definition. Recall: hxfhxfh)()(lim0 25fxx x 8. Find the derivative of the following: a. 21() sec tan2fx x xxMath 1413 Test 2 Review 7 b. 23() 3 1fx x c. 23() 4( )fxxxx d. 2)1)(1()(xxxxfMath 1413 Test 2 Review 8 e. )164(sin)(24 xxxf 9. Find 102433433xxxdxd. 10. Find 2xdydxfor 22fxxcosx. 11. Find dxdyusing implicit differentiation for 227xyxyMath 1413 Test 2 Review 9 12. Find the equation of the tangent line to the curve 1xfxtanx at x =0. 13. Write the equation of the normal line to sin cos 0xy at the point ,44.Math 1413 Test 2 Review 10 14. Suppose we are given the data in the table about the functions f and g and their derivatives. Find 2'(2) () () ()hifhxfxgx . Extra Review Questions for Lab Find the derivative of the following functions using the definition. Recall: hxfhxfh)()(lim0 a. 24fxx b. () 2gx x x 1 2 3 4()fx 3 214'( )fx 1 423()gx 2 143'( )gx 4
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