Spring 2014: bcc project calculus part1


BCC Project Calculus divorce 1      Spring 2014

 

 

1. Show that the equation  has correspondently 1 veritable commencement.

______________________________________________________________________________________

 

2. Research the character and set-up the graph  . On the corresponding graph draw     and collate their proceeding.

______________________________________________________________________________________

 

3. Meet the aim on the direction  , closest to the aim (2,6)

______________________________________________________________________________________

4. Research the character and color the graph  

5. At 2:00 pm a car’s urgeometer reads 30 mph. At 2:10 pm it reads 50 mph. Show that at some season betwixt 2:00 and 2:10  the succor was correspondently 120 mi/h2.

______________________________________________________________________________________

6. Meet the dimensions of the rectdispose of the largest area that has  its worthiest on the x-axis and its other two vertices overhead the x-axis and false on the parabola  .

______________________________________________________________________________________

7. Find  , if  

______________________________________________________________________________________

8. Confirm the identity

______________________________________________________________________________________

9. Absorbed tanh(x) = 0.8. Meet other 5 values of hyperbolic characters: sinh(x), cosh(x), sech(x), csch(x), coth(x).

______________________________________________________________________________________

 

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

______________________________________________________________________________________

 

11. Meet the number c that satisfies the omission of the Mean Value Theorem.

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

                                         xΠ [-1, 1]

______________________________________________________________________________________

 

12. Evaluate the condition:    a)           ;          b) 

______________________________________________________________________________________

 

13. Color the flexion.     y =  x/(x2+4)

______________________________________________________________________________________

 

14. Meet the area of the largest rectdispose that can be inscribed in a suitable tridispose delay legs of protractions 5 cm and 6cm if two edges of the rectdispose lie concurrently the legs.

________________________________________________________________________

15. The tallness of a tridispose is increasing at a blame of 1 cm/min suitableness the area of the tridispose is increasing at a blame of 2 cm sq. per specific.  At what blame is the worthiest of the tridispose changing when the tallness is 10 cm and the area is 100 cm sq?________________________________________________________________________

 

 

16. Use a straight way or differentials to admire the absorbed number:    

            tan440 ________________________________________________________________________

 

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

________________________________________________________________________

 

 

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

 

          

 

________________________________________________________________________

19. Show that the equation   has correspondently one veritable commencement.

Find the interims on which F(x) is increasing or decreasing. Meet national acme and narrowness of F(x). Meet the interims of angularity and the bend aims.  

 

________________________________________________________________________

20. Suppose that    for all values of x.

Show that   .

______________________________________________________________________________________

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

        On what interim is f (x) increasing?

______________________________________________________________________________________

22. Explore and excite the forthcoming three characters.

a) Meet the upright and insipid asymptotes.                I.       

     b) meet the interims of enlargement or retrench.               II.                                

     c) meet national acme and narrowness values.                III.                                                             

     d) meet the interims of angularity and the bend  aims.

     e) use the notification from divorces a) to d) to color the graph

______________________________________________________________________________________

23. A concern of wire 10 m covet is cut into two concerns. One concern is determination into a clear and the other is determination into an equilateral triangle. How should the wire be cut so that the aggregate area enclosed is a) acme ?  b) narrowness ?

______________________________________________________________________________________

24. Meet the aim on the direction   that is closest to the commencement.

______________________________________________________________________________________

25. Color the graph of 

______________________________________________________________________________________

26. Find  f(x)  if 

 

 

 

27. Express the condition as a derivative and evaluate:       

_____________________________________________________________________________________________

 

28. The book of the cube is increasing at the blame of  10 cm3/min.  How accelerated is the exterior area increasing  when the protraction of the aspect is  30 cm.

_____________________________________________________________________________________________

 

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

_____________________________________________________________________________________________

 

30. Meet the parabola   that passes through aim (1,4) and whose tangent directions at x = 1 andx = 5 own arises 6 and -2 respectively.

_____________________________________________________________________________________________

 

 

31. Cobalt-60 has  a half activity of 5.24 years.  A)  Meet the body that remains from a 00 mg illustration behind 20 years. B) How shortly conciliate the body of 100 mg rottenness to 1 mg?

_____________________________________________________________________________________________

 

 

 32. Meet the aims on the figure    where the tangent direction has  arise 1.

 

_____________________________________________________________________________________________

 

 33. Suppose that a population of bacteria triples perfect hour and initiates delay 400 bacteria.

(a) Meet an countenance for the number n of bacteria behind t hours.

_____________________________________________________________________________________________

(b) Admire the blame of enlargement of the bacteria population behind 2.5 hours. (Round your acceptance to the right hundred.)

_____________________________________________________________________________________________

34. Meet the n-th derivative of the character  y = xe-x

 

35. Meet the equation of the direction  going  through the aim (3,5),  that cuts off the meanest area from the principal quadrant.

36. Research the character and color its graph.   Find all expressive aims, interims of enlargement and retrench

         a)  y = 

          b)  y =  

         c)   

37. Use Integration to meet the area of a tridispose delay the absorbed vertices:  (0,5), (2,-2), (5,1)

 

 38. Meet the book of the largest spherical cone that can be inscribed into a globe of radius R.

 

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

 _________________________________________________________________________________________

40.  For what values of the trustworthys  a and b  the aim (1,6) is the aim of bend for the flexion 

41.  If 1200 sq.cm of embodied is available to make  a box delay a clear worthiest and an unconcealed top, meet the largest likely book of the box.

42. Two cars initiate tender from the corresponding aim. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what blame

         is the removal betwixt the cars increasing two hours succeeding?

43.   A man initiates walking north at 4 ft/s from a aim . Five specifics succeeding a dowager initiates walking south at 5 ft/s from a aim

500 ft due east of . At what blame are the race tender adivorce 15 min behind the dowager initiates 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 accelerated is the removal betwixt the ships changing at 4:00 PM?

 

46.  Two edges of a tridispose own protractions 12 m and 15 m. The dispose betwixt them is increasing at a blame of 2 degrees per specific . How accelerated is the protraction of the third edge increasing when the dispose betwixt the edges of unroving protraction is 60 ?

 

47. Two race initiate from the corresponding aim. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How accelerated is

the removal betwixt the race changing behind 15 specifics?

48.  The radius of a spherical disk is absorbed as 24 cm delay a acme untruth in extent of 0.2 cm.

         Use differentials to admire the acme untruth in the adapted area of the disk.

 

49.   Use differentials to admire the aggregate of color needed to dedicate a tempt of color 0.05 cm compact to a hemispherical dome delay crossing 50 m.

 

50.  Use a straight way (or differentials) to admire the absorbed number:   2.0015

 

51. Verify that the character satisfies the hypotheses of the Mean Value Theorem on the absorbed interim. Then meet all quantity that indemnify the omission of the Mean Value Theorem. 

 

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

 

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

 

54. Show that the equation     has at most 2 veritable commencements.

 

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

        (b) Meet the national acme and narrowness values of f(x).

        (c) Meet the interims of angularity and the bend aims.

 

    55. 

    56.         for    

     57.        for   

    58.    

 

(a) Meet the upright and insipid asymptotes.

(b) Meet the interims of enlargement or retrench.

(c) Meet the national acme and narrowness values.

(d) Meet the interims of angularity and the bend aims.

(e) Use the notification from divorces (a)–(d) to color the graph

of  f(x)

 

59.    

60.    

61.    

62.   , for     

 

Find the conditions:

 

63.  

64.    

 

65.  

 

66.    

 

67. A stone is dropped from the higher attention dispose of a Tower, 450 meters overhead the cause.

(a) Meet the removal of the stone overhead cause smooth at season t.

(b) How covet does it choose the stone to grasp the cause?

(c) Delay what celerity does it give-a-blow-to the cause?

(d) If the stone is thrown downward delay a urge of 5 m/s, how covet does it choose to grasp the cause?

 

 

68.  What trustworthy succor is required to enlargement the urge of a car from 30 mi/h to 50 mi/h in 5 seconds?

 

A divorceicle is tender delay the absorbed facts. Meet the pose of the divorceicle.

 

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

 

70.    ,