Lim f (x) e. Find the largest possible domain and its discontinuity points. Why is it discontinuous at those points (if any)?

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1 THESE ARE SAMPLE QUESTIONS FOR EACH OF THE STUDENT LEARNING OUTCOMES (SLO) SET FOR THIS COURSE. SLO 1: Understand and use the cncept f the limit f a functin i. Use prperties f limits and ther techniques, like L Hppital s rule, t determine the eistence r nt f the limit f a functin at a given value; ii. Understand the definitin f cntinuity f functins 1. Frm a functin given in a graph determine the discntinuity pint indicating which prperties f cntinuity fail;. Given a piece-wise functin defined by frmulas determine the pints at which the functin is discntinuus. + 1, 4 1. Cnsider f ( ) = Calculate the fllwing limits. If the limit des, > 4 3 nt eist indicate why nt. a. ( ) Lim f 4 Lim f b. ( ) 3 c. () d. + Lim f Lim f () 0 e. Find the largest pssible dmain and its discntinuity pints. Why is it discntinuus at thse pints (if any)?. Cnsider g() = 4 +, 1 + Calculate the fllwing limits. If the limit 5(4 ), 1 < 4 des nt eist indicate why nt. a. Lim g() 4 b. Lim g() c. Lim g() d. Lim g() 1 e. Find the largest pssible dmain and its discntinuity pints. Which f the cnditins fr cntinuity is nt satisfied? Classify the discntinuity pints as remvable r nt remvable. 3. The functin g () is defined n the clsed interval [-1,3]. Calculate the fllwing limits. In case it des nt eist write DNE 1

2 5 0 y=g() a. b. c. d Lim g() 1 Lim g() Lim g() 4 Lim g() 3 e. Find the pints f discntinuity f this functin. Indicate what f the cnditins fr cntinuity is nt satisfied. Classify the discntinuity pints as remvable r nn-remvable 4. Give eamples with sequences t shw why limits f the frms "0" "0", " ", "0" " ", " " " " are undetermined undetermined. " " 5. Give eamples with sequences that have the limits f the type described belw. Calculate each f them. " 3." " ", "4" "0 + ", "1. " "5 ", "0+ " " ", 3 ".5", 5 "5+ ", 6. Calculate the fllwing limits. Yu can use cntinuity prperties, dminance, L Hppital s rule, r sequences. Start by identifying the type f limit ( "0+ ", "5" " ", " "+" " etc.) " " + 3 a. Lim 3 3 b. Lim 0

3 c. Lim d. Lim 0 + Ln() e. Lim f. Lim sin() 0 g. Lim 0 h. sin() Lim SLO : Be able t prvide eamples and cuntereamples dealing with imprtant results discussed in this curse, and specially t understand the necessity f the cnditins fr sme f them: Give an eample f a discntinuus functin with a remvable/nn-remvable discntinuity; Give an eample f a functin whse limit des nt eist at a pint. Give an eample f a functin that is cntinuus at a pint but nt differentiable at that pint. 7. Prvide a. The graph f a functin with a remvable discntinuity b. The algebraic epressin f a functin that has a remvable discntinuity at =-5. c. An eample f a functin that is cntinuus at pint but nt differentiable at that pint. 8. Sketch a. The graph f a functin that is defined at a pint but it is discntinuus at that pint b. The graph f a functin whse limit eists at a pint but it is discntinuus at that pint. SLO 3: Understand and interpret the cncept f the derivative: c. Graphically, as the slpe f the tangent line at a pint; d. Given the graph f a functin find the equatin f the tangent line at a pint (include the units) iii. Given the graph f a functin find the average rate f change n a clsed interval (include the units) iv. Use infrmatin abut the first and secnd derivative t btain infrmatin abut the riginal functin; interpret the units f the derivative. 3

4 9. Let T (t) represent the temperature f an egg, in Fahrenheit degrees, t minutes after it is placed in a refrigeratr with cnstant temperature f 45 degrees. Its graph is given belw t HtimeL a. Find the average rate f change f temperature during the first 40 minutes. Include the units. b. Indicate the units fr T(t), T ʹ (t), T ʹ ʹ (t) c. Determine the sign f T ʹ (0), T ʹ ʹ (0) d. Accrding t the prblem situatin, what shuld be LimT(t) t e. Estimate T ʹ (10), and use this value t estimate when the egg will rich the refrigeratr s temperature. f. Will ever T ʹ (t) = 0? g. Slve dt(t) = 0.5 and interpret yur answer in the cntet f the prblem. dt h. Slve and interpret T(t) = 60 i. Yu knw that T(0) = 53, T ʹ (0) = 1, T ʹ ʹ (0) = 0.. Use this infrmatin t estimate T(3) 10. Frm a piecewise defined functin determine whether r nt the functin is differentiable at the pint(s) where the pieces jin. First, yu need t guarantee that the functin is cntinuus at that pint. Then, determine if the derivative eists. a. f () = + 3, < 5, b. h() =, < 0 (Ln) 1, 0 c. g() =, < 1 +1, 1 4

5 SLO 4 Find the linear apprimatin at a pint and use it t estimate the functin. Prduce the linear apprimatin frm a graph and determine if in a neighbrhd f the pint it will give an verestimate r underestimate Frm a functin defined by an algebraic epressin the student will find the linear apprimatin at a given pint and use it t estimate the riginal functin. The student has t justify whether it is an verestimate r under estimate. 11. Find the linearizatin f y = at the pint (3,1) 1 1. Use the result frm (14) t estimate the value f Underestimate? Eplain.. Is this an verestimate? 5 SLO 5 Sketch the graph f a functin r its derivative functin: Frm the graph f a functin, prduce the graphs f the first and secnd derivative functins; Frm the graph, r infrmatin, abut the first and secnd derivative f a functin they will generate the graph f the functin. Frm a given graph determine all the critical pints and indicate at which pints the functin is nt differentiable. Frm a graph f a functin r its derivative find the slutin t (all the infrmatin yu can determine) - f ( ) > 0 r undefined - f ʹ ( ) < 0 - f ʹ ( ) = 0, r undefined - f ʹ ʹ ( ) = 0, r undefined - Inflectin pints f f () - Cncavities Frm a functin defined by a frmula they will find the infrmatin t sketch its graph (dmain, cntinuity pints, increasing/decreasing, cncave up/dwn, end behavir, asympttes) 13. If y = + 1, y ʹ = 4 1, y ʹ ʹ = +, find 3 Dmain f the functin Critical pints Intervals where the functin is increasing, decreasing Intervals where the functin is cncave up/dwn Behavir f the functin tward the end pints f its dmain. Sketch the graph shwing all the previus infrmatin. 14. The graph belw is the graph f the derivative f f (). Once again, it is the graph f the derivative, nt the riginal functin. Answer the questins belw. 5

6 a. The critical pint(s) f y = f () happen at = b. The functin y = f ()in increasing n the interval(s) c. The functin y = f ()is cncave up n the interval(s) d f () d. Is decreasing n the interval(s) d e. The inflectin pints f y = f () are f. Estimate f ʹ ( 1) and f ʹ ʹ ( 1) g. Determine which f the critical pints f y=f () is lcal ma. h. Sketch the graph f y = f () passing thrugh the pint (-1,5) 15. The graph f t functin T (t) is given. 60 y = THtL a. The average rate f change f the functin n the interval [,6] is b. The average value f the functin n the interval [,6] is c. The critical pints f T (t) are d. The inflectin pints f T (t) are 6

7 e. The values f t fr which T ʹ (t) > 0 and T ʹ ʹ (t) < 0 at the same time are f. An estimated value fr the slpe f the nrmal line t the graph at the pint where t=0, is SLO 6 Use calculus techniques t slve Optimizatin prblems. Given an ptimizatin prblem the student will find the mathematical mdel fr it, and will prceed t slve it using calculus techniques (fr sme they may need t use technlgy) Related rates prblems. 16. A gardener wants t enclse 1500 square meters rectangular patch f land using the least amunt f fencing pssible. One side f the land lies alng a barn, and fr that reasn will remain unfenced. The area t be enclsed remains fied but the width, w, and length, l, vary. The amunt f fencing t be used is given as a functin f w by f (w) = w What is the minimum amunt f fencing t be used? Slve the w prblem algebraically and graphically. 17. A pster is t have an area f 180 in with 1-inch margins at the bttm and sides and a -inch margin at the tp. What dimensins will give the largest printed area? 18. a. If V is the vlume f a cube with edge f length, and the cube epands as time passes, find dv dt in terms f d dt b. A blck f ice with square base is melting and the water is being cllected in a pan. When the base f the blck is 0 inches the height is 15 inches, and we knw that at this time the edge f the base decreases at in/hur and the height decreases at 3 in./hur. Hw fast is the pan filling up at this mment? 19. Suppse il spills frm a ruptured tank and spreads in a circular pattern. If the radius f the il spill increases at cnstant rate f ft/hur, hw fast is the area f the spill 7

8 increasing when the radius is 90 feet? Predict what the area f the spill will be 30 minutes later. SL0 7 Use implicit differentiatin prperly: v. Calculate derivatives using implicit differentiatin vi. Determine the equatin f tangent lines t graphs btained frm epressins where ne variable is given implicitly as a functin f ther. 0. Find the linearizatin f y + y = at the pint (, -) 1. In the epressin zy + yz = 8, y and z depend upn t. Differentiate t find an epressin in terms f y,z, dz dt, dy dt SLO 8 Understand the cncept f the integral. Evaluate basic definite integrals 3 i. ( + ) d 1 ii. d 5 iii. 0 iv. d sin( ) d 1 v. d vi. Find the anti-derivative f y = passing thrugh the pint (4,1) 3. Calculate the area f regin enclsed by the -ais and the graph f the functin given by y = n the interval [-1,4]. Use the fundamental therem f calculus. 4. Estimate the area f the regin enclsed by = 3, = 5, y = + 8 using fur subdivisins. Feel free t use Riemann Sums r indivisibles (average f heights) 5. The rate at which water is flwing int an empty vessel after t secnds is given by R(t) =1 e t (cm 3 /min). Its graph is displayed belw Time HminL 8

9 i. Prvide an verestimate an underestimate fr the area between the -ais and the curve n the interval [0,]. What are the units f the area? ii. In the cntet f the prblem, what is the meaning f (1 e t )dt? iii. Justify why the anti derivative f R(t) is (1 e t )dt = t + e t + C. Then find the anti derivative passing thrugh the pint (0,0) iv. Use the fundamental therem f calculus t determine the amunt f water is in the vessel after 1 minute. v. At what rate the water is getting int the vessel after 1 minute? vi. Is the amunt f water in the vessel increasing after 1 minute? Why? vii. What is the sign f R'(1)? Interpret yur answer in terms f the amunt f water in the vessel. viii. It is knwn that the vessel is full after 10 secnds. What is the vlume f the vessel? i. Hw lng des it take fr the vessel t be half full? 0 9