Spring 2014: bcc project calculus part1


BCC Project Calculus deal-out 1      Spring 2014

 

 

1. Show that the equation  has accurately 1 existent stem.

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2. Research the capacity and elevate the graph  . On the selfselfidentical graph draw     and assimilate their manner.

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3. Perceive the summit on the method  , closest to the summit (2,6)

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4. Research the capacity and portray the graph  

5. At 2:00 pm a car’s accelerateometer reads 30 mph. At 2:10 pm it reads 50 mph. Show that at some space betwixt 2:00 and 2:10  the acceleration was accurately 120 mi/h2.

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6. Perceive the size of the rectpredilection of the largest area that has  its disingenuous on the x-axis and its other two vertices aloft the x-axis and untruthful on the parabola  .

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7. Find  , if  

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8. Demonstrate the identity

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9. Fond tanh(x) = 0.8. Perceive other 5 values of hyperbolic capacitys: sinh(x), cosh(x), sech(x), csch(x), coth(x).

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10. Research and elevate the graph: h(x) = (x + 2)3 - 3x - 2

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11. Perceive the reckon c that satisfies the misentry of the Mean Value Theorem.

                               f(x) = 5x2 + 3x + 6 

                                         xΠ [-1, 1]

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12. Evaluate the name:    a)           ;          b) 

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13. Portray the incurvation.     y =  x/(x2+4)

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14. Perceive the area of the largest rectpredilection that can be inscribed in a proper tripredilection following a while legs of prolixitys 5 cm and 6cm if two causes of the rectpredilection lie adesire the legs.

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15. The perpendicularity of a tripredilection is increasing at a admonish of 1 cm/min while the area of the tripredilection is increasing at a admonish of 2 cm sq. per tiny.  At what admonish is the disingenuous of the tripredilection changing when the perpendicularity is 10 cm and the area is 100 cm sq?________________________________________________________________________

 

 

16. Use a direct advance or differentials to veneadmonish the fond reckon:    

            tan440 ________________________________________________________________________

 

17. Demonstrate the identity: cosh 2x = cosh2x + sinh2x

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18. Demonstrate that the formula for the derivative of tangent hyperbolic inverse :

 

          

 

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19. Show that the equation   has accurately one existent stem.

Find the interims on which F(x) is increasing or decreasing. Perceive national zenith and incompleteness of F(x). Perceive the interims of angularity and the inflexion summits.  

 

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20. Suppose that    for all values of x.

Show that   .

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21. Suppose that the derivative of a capacity f(x) is :    

        On what interim is f (x) increasing?

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22. Explore and criticise the forthcoming three capacitys.

a) Perceive the perpendicular and dull asymptotes.                I.       

     b) perceive the interims of acception or abate.               II.                                

     c) perceive national zenith and incompleteness values.                III.                                                             

     d) perceive the interims of angularity and the inflexion  summits.

     e) use the counsel from deal-outs a) to d) to portray the graph

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23. A share of wire 10 m desire is cut into two shares. One share is predilection into a balance and the other is predilection into an equilateral triangle. How should the wire be cut so that the entirety area enclosed is a) zenith ?  b) incompleteness ?

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24. Perceive the summit on the method   that is closest to the derivation.

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25. Portray the graph of 

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26. Find  f(x)  if 

 

 

 

27. Express the name as a derivative and evaluate:       

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28. The magnitude of the cube is increasing at the admonish of  10 cm3/min.  How firm is the exterior area increasing  when the prolixity of the border is  30 cm.

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29. Evaluate dy, if   , x = 2, dx = 0.2

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30. Perceive the parabola   that passes through summit (1,4) and whose tangent methods at x = 1 andx = 5 bear excels 6 and -2 respectively.

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31. Cobalt-60 has  a half society of 5.24 years.  A)  Perceive the bulk that debris from a 00 mg case following 20 years. B) How shortly gain the bulk of 100 mg waning to 1 mg?

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 32. Perceive the summits on the figure    where the tangent method has  excel 1.

 

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 33. Suppose that a population of bacteria triples entire hour and set-outs following a while 400 bacteria.

(a) Perceive an indication for the reckon n of bacteria following t hours.

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(b) Veneadmonish the admonish of augmentation of the bacteria population following 2.5 hours. (Round your rejoinder to the unswerving hundred.)

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34. Perceive the n-th derivative of the capacity  y = xe-x

 

35. Perceive the equation of the method  going  through the summit (3,5),  that cuts off the last area from the primary quadrant.

36. Research the capacity and portray its graph.   Find all expressive summits, interims of acception and abate

         a)  y = 

          b)  y =  

         c)   

37. Use Integration to perceive the area of a tripredilection following a while the fond vertices:  (0,5), (2,-2), (5,1)

 

 38. Perceive the magnitude of the largest round cone that can be inscribed into a rank of radius R.

 

39.  Perceive the summit on the hyperbola  xy = 8  that is the closest to the summit  (3,0)

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40.  For what values of the trues  a and b  the summit (1,6) is the summit of inflexion for the incurvation 

41.  If 1200 sq.cm of embodied is advantageous to make  a box following a while a balance disingenuous and an unconcealed top, perceive the largest likely magnitude of the box.

42. Two cars set-out affecting from the selfselfidentical summit. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what admonish

         is the interim betwixt the cars increasing two hours succeeding?

43.   A man set-outs walking north at 4 ft/s from a summit . Five tinys succeeding a dowager set-outs walking south at 5 ft/s from a summit

500 ft due east of . At what admonish are the inhabitants affecting adeal-out 15 min following the dowager set-outs walking?

 

44.  At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. How firm is the interim betwixt the ships changing at 4:00 PM?

 

46.  Two causes of a tripredilection bear prolixitys 12 m and 15 m. The predilection betwixt them is increasing at a admonish of 2 degrees per tiny . How firm is the prolixity of the third cause increasing when the predilection betwixt the causes of unroving prolixity is 60 ?

 

47. Two inhabitants set-out from the selfselfidentical summit. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How firm is

the interim betwixt the inhabitants changing following 15 tinys?

48.  The radius of a round disk is fond as 24 cm following a while a zenith fallacy in size of 0.2 cm.

         Use differentials to veneadmonish the zenith fallacy in the conducive area of the disk.

 

49.   Use differentials to veneadmonish the totality of depict needed to dedicate a cheat of depict 0.05 cm inarticulate to a hemispherical dome following a while crossing 50 m.

 

50.  Use a direct advance (or differentials) to veneadmonish the fond reckon:   2.0015

 

51. Verify that the capacity satisfies the hypotheses of the Mean Value Theorem on the fond interim. Then perceive all reckons that assure the misentry of the Mean Value Theorem. 

 

52. Does there exists a capacity f(x) such that    for all x?

 

53.  Suppose that f(x) and g(x) are consistent on [a,b] and differentiable on (a,b). Suppose too that f(a)=g(a) and f’(x)< g’(x) for a<x<b demonstrate that f(b)<g(b). [Hint: dedicate the Mean Value Theorem for the capacity h=f - g. ]

 

54. Show that the equation     has at most 2 existent stems.

 

55 - 58.   (a) Perceive the interims on which is increasing or decreasing.

        (b) Perceive the national zenith and incompleteness values of f(x).

        (c) Perceive the interims of angularity and the inflexion summits.

 

    55. 

    56.         for    

     57.        for   

    58.    

 

(a) Perceive the perpendicular and dull asymptotes.

(b) Perceive the interims of acception or abate.

(c) Perceive the national zenith and incompleteness values.

(d) Perceive the interims of angularity and the inflexion summits.

(e) Use the counsel from deal-outs (a)–(d) to portray the graph

of  f(x)

 

59.    

60.    

61.    

62.   , for     

 

Find the names:

 

63.  

64.    

 

65.  

 

66.    

 

67. A stone is dropped from the conspicuous comment dress of a Tower, 450 meters aloft the basis.

(a) Perceive the interim of the stone aloft basis roll at space t.

(b) How desire does it transfer the stone to strain the basis?

(c) Following a while what fleetness does it give-a-blow-to the basis?

(d) If the stone is thrown downward following a while a accelerate of 5 m/s, how desire does it transfer to strain the basis?

 

 

68.  What true acceleration is required to acception the accelerate of a car from 30 mi/h to 50 mi/h in 5 seconds?

 

A deal-outicle is affecting following a while the fond postulates. Perceive the aspect of the deal-outicle.

 

69.   a(t)= cos(t)+sin(t),   s(0)=0,  v(0)=5

 

70.    ,