Math 131 Exam 4 (Final Exam) F04M

Transcription

1 Math 3 Exam 4 (Final Exam) F04M3.4. Name ID Number The exam consists of 8 multiple choice questions (5 points each) and 0 true/false questions ( point each), for a total of 00 points. Mark the correct anser on the anser card. Only the anser on the card ill be graded.. We begin ith 20g of a certain radioactive isotope. As it decays, the amount!þ!)> remaining at time > (hrs) is E œ!/ Þ What is its rate of decay at time > œ )? ( Round your anser to 2 decimal places. ) A)!Þ" g/hr B)!Þ$$ g/hr C)!Þ$) g/hr D)!Þ) g/hr E)!Þ(' g/hr F)!Þ) g/hr G)!Þ)* g/hr H)!Þ* g/hr I)!Þ*) g/hr J) "Þ!$ g/hr ( ( "!! " 2. If ' ÐBÑ.B œ!ß ' ÐBÑ.B œ ß and ' ÐBÑ.B œ (ß then hat is ' ÐBÑ.B? A) 0 B) 9 C) D) 7 E) 6 F) 6 G) 23 H) 3 I) 8 J) 27

2 F04M If C œ sin ÐBÑ ÐB Ñtan ÐBÑ, hat is the slope of the tangent line to the graph at Ð!ß!Ñ? " A) B) C) D) E) F) 2 G)! H) " I) J) 4. For a certain gas held inside a container, the pressure and volume are related by the equation TZ œ "Þ Use a differential to estimate the change in pressure hen the volume $ $ decreases from 0 cm to *Þ& cm. ( Round your anser to 4 decimal places. ) A)!Þ!! 7 B)!Þ!! 42 C)!Þ!" 2 D)!Þ!" 48 E)!Þ!!&! F)!Þ!!&$ G)!Þ"'($ H)!Þ"'"! I)!Þ&(" J) 5.3"

3 $ ( 5. At time >, the velocity of a point moving along a line is > Ð> "Ñ m/sec. At time >œ ", the point is at position!. What is its position hen >œ!? " " " A) &! m B) m C) m D) m E) 2 m " & " F) m G) " m H) m I) m J) m 8 F04M3.4.3 ( B + ) &È cos B 6. If 0ÐBÑ œ ln Ð Ñß hat is 0 Ð"Ñ? A)! B) ln(2) C) tan Ð"Ñ D)! " " " & & & E) " tan Ð"Ñ F) sin Ð"Ñ G) sin Ð"Ñ " È & cosð"ñ & sinð"ñ H) I) ln Ð Ñ J) lnðtan())

4 7. Find the shaded area: F04M3.4.4 $ & ( A) B) $ C) $ D) E) & $ F) ln G) ln Ð Ñ H) I) J) ln(3) " 2Ä! 2 $2 8. lim Ð Ñ is hich of the folloing? B $B A) 0 Ð$ Ñ here 0ÐBÑ œ ln Ð Ñ B) 0 Ð0Ñhere 0ÐBÑ œ B C) 0 Ð2Ñhere 0ÐBÑ œ $B D) 0 ÐÑhere 0ÐBÑ œ $B E) 0 Ð!Ñ here 0ÐBÑ œ B B F) 0 ÐÑ here 0ÐBÑ œ 2 ln Ð$B Ñ G) 0 Ð$Ñ here 0ÐB) œ B B H) 0 Ð!Ñ, here 0ÐBÑ œ $B " I) 0 Ð Ñ here 0ÐBÑ œ J) 0 Ð$Ñhere 0ÐBÑ œ $ B $B

5 È F04M What is the slope of the tangent line to cos C $ B œ at the point Ð"ß Ñ? A) B) C) D) E) 2 " F) " G) H) I) J) tangent is vertical! " $ $ 0. What is the maximum value of Ð) Ñ œ ) sin Ð ) Ñ on the interval Ò!ß Ó? " È$ È$ & È$ ' A) B) C) D) E) & " È $ & È $ " ' $ ' F) G) " H) I) J) F04M3.4.6

6 Î '!. Find sec B cos B.BÞ È$ È$ È " È$ $ È È $ " È $ $ $ A)! B) C) D) E) F) G) H) $ È I) J) sin 2. The curve œ Bœ > passes through Ð!ß!Ñ. CœsinÐ> sinð>ññ > What is the slope of the tangent line at Ð!ß!Ñ? A) 4 B) 3 C) 2 D) E) 0 F) " G) H) $ I) J) & F04M3.4.7

7 & 3. The function 0ÐBÑ œ +B "'!B - has an inflection point at Ðß &)!ÑÞ What are + and -? A) + œ "ß - œ * B) + œ ß - œ " C) + œ $ß - œ "&' D) + œ "ß - œ "! E) + œ!ß - œ '! F) + œ "ß - œ ) G) +œ ß-œ H) +œ $ß-œ$' I) +œ ß-œ" J) +œ ß-œ" ' " " 4. Since B.B œ ln ÐÑ, e can compute an approximate value for lnðñ by approximating the integral ' " " B.B. A very rough approximation ould be the midpoint approximation QÞ $ What is Q$? ( Round your anser to four decimal places. ) A).3325 B).3422 C).3476 D).3522 E) "Þ$& F).360 G).3678 H).3742 I).3863 J).3977 F04M3.4.8

8 5. Suppose the interval Ò"ß $Ó is divided into 8 equal subintervals and the right endpoint B is chosen from each subinterval. What is the value of lim Ð! " ÐB Ñ Ñ? 3 8 8Ä_ 3œ" " "( $& ) "* A) B) $ C) * D) $ E) $( $ "" $ &( F) G) * H) I) $ J) 3 B An investment is groing at a rate of ($/yr). Ho much does its value increase beteen the second and third years? ( Round your anser to the nearest cent. ) "!!/!Þ!"> A) $96.03 B) $97. C) $97.78 D) $98.3 E) $99.2 F) $00.45 G) $00.97 H) $0.63 I) $02. J) $02.53 F04M3.4.9

9 $B ' sin > " > 7. If JÐBÑ œ Ðsin B ).>, hat is J Ð Ñ? A) 0 B) C) D) E) $ $ $ F) " G) H) I) J) $

10 F04M A rock is dropped from the top of a cliff 650 ft. high. A camera on the ground, 50 ft. aay from the base of the cliff, stays focussed on the rock as it falls. Ho fast is the angle of elevation of the camera ) changing at the moment hen the rock is 50 feet from the ground? A) "Þ rad/sec B) 0Þ$ rad/sec C) )Þ rad/sec D) &. rad/sec E) "Þ$ rad/sec F) ". rad/sec G) $. rad/sec H) 'Þ& rad/sec I) rad/sec J) & rad/sec F04M3.4.

11 The folloing 0 questions are all true/false questions ( point each) 9. The folloing integrals are correctly listed in order of increasing size:! $ ' ' ' cos B.B cos B.B cos B.B 20. Suppose a point is moving along a straight line ith Its position at time >œ! is =Þ Then at time >, its position is = ' $ B +B / + BÄ_,B$ ", 2. If, Á!ß lim œ Þ 22. Suppose Cœ0ÐBÑis differentiable and has only one critical number, Bœ"Þ If 0 ÐBÑ! for B " and 0 ÐBÑ! for B ", then 0ÐBÑhas an absolute minimum at Bœ"Þ & $ 23. If 0 ÐBÑ œ B ÐB $Ñ ÐB "Ñ, then 0ÐBÑ has exactly to inflection points. B B $ > ( > > > 24. ' / ' /.> œ.> G, here G is a constant. F04M3.4.2

12 25. Let 0ÐBÑ œ lblþ According to the Mean Value Theorem, there is a point - 0ÐÑ 0Ð Ñ beteen and here 0 Ð-Ñœ Þ.= ". 26. ' =.= œ Þ 27. If lim B $ œ(, then 0 must be continuous at $. BÄ$ 0ÐBÑ 0Ð$Ñ 28. lim is an indeterminate form that is, e must do some manipulation and BÄ! B "ÎB perhaps use L'Hopital's ˆ Rule to determine the value of the limit.