Bcc project calculus part 1

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BCC Project Calculus part 1      Spring 2014



1. Show that the equation  has exactly 1 real root.



2. Research the function and build the graph  . On the same graph draw     and compare their behavior.



3. Find the point on the line  , closest to the point (2,6)


4. Research the function and sketch the graph 

5. At 2:00 pm a car’s speedometer reads 30 mph. At 2:10 pm it reads 50 mph. Show that at some time between 2:00 and 2:10  the acceleration was exactly 120 mi/h2.


6. Find the dimensions of the rectangle of the largest area that has  its base on the x-axis and its other two vertices above the x-axis and lying on the parabola  .


7. Find  , if 


8. Prove the identity


9. Given tanh(x) = 0.8. Find other 5 values of hyperbolic functions: sinh(x), cosh(x), sech(x), csch(x), coth(x).



10. Research and build the graph: h(x) = (x + 2)33x – 2



11. Find the number c that satisfies the conclusion of the Mean Value Theorem.

                               f(x) = 5x2 + 3x + 6
Π [-1, 1]



12. Evaluate the limit:    a)           ;          b)



13. Sketch the curve.     y =  x/(x2+4)



14. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 5 cm and 6 cm if two sides of the rectangle lie along the legs.


15. The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm sq. per minute.  At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm sq? ________________________________________________________________________



16. Use a linear approximation or differentials to estimate the given number:    

            tan440 ________________________________________________________________________


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




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





19. Show that the equation   has exactly one real root.

Find the intervals on which F(x) is increasing or decreasing. Find local maximum and minimum of F(x). Find the intervals of concavity and the inflection points. 



20. Suppose that    for all values of x.

Show that   .


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

        On what interval is f (x) increasing?


22. Explore and analyze the following three functions.

a) Find the vertical and horizontal asymptotes.                I.      

     b) find the intervals of increase or decrease.               II.                                

     c) find local maximum and minimum values.                III.                                                             

     d) find the intervals of concavity and the inflection  points.

     e) use the information from parts a) to d) to sketch the graph


23. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is a) maximum ?  b) minimum ?


24. Find the point on the line   that is closest to the origin.


25. Sketch the graph of


26. Find  f(x)  if




27. Express the limit as a derivative and evaluate:       



28. The volume of the cube is increasing at the rate of  10 cm3/min.  How fast is the surface area increasing  when the length of the edge is  30 cm.



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



30. Find the parabola   that passes through point (1,4) and whose tangent lines at x = 1 and x = 5 have slopes 6 and -2 respectively.




31. Cobalt-60 has  a half life of 5.24 years.  A)  Find the mass that remains from a 00 mg sample after 20 years. B) How soon will the mass of 100 mg decay to 1 mg?




 32. Find the points on the figure    where the tangent line has  slope 1.





 33. Suppose that a population of bacteria triples every hour and starts with 400 bacteria.

(a) Find an expression for the number n of bacteria after t hours.


(b) Estimate the rate of growth of the bacteria population after 2.5 hours. (Round your answer to the nearest hundred.)


34. Find the n-th derivative of the function  y = xe-x


35. Find the equation of the line  going  through the point (3,5),  that cuts off the least area from the first quadrant.

36. Research the function and sketch its graph.   Find all important points, intervals of increase and decrease

         a)  y =

          b)  y =  


37. Use Integration to find the area of a triangle with the given vertices:  (0,5), (2,-2), (5,1)


 38. Find the volume of the largest circular cone that can be inscribed into a sphere of radius R.


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


40.  For what values of the constants  a and b  the point (1,6) is the point of inflection for the curve

41.  If 1200 sq.cm of material is available to make  a box with a square base and an open top, find the largest possible volume of the box.

42. Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate

         is the distance between the cars increasing two hours later?

43.   A man starts walking north at 4 ft/s from a point . Five minutes later a woman starts walking south at 5 ft/s from a point

500 ft due east of . At what rate are the people moving apart 15 min after the woman starts 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 fast is the distance between the ships changing at 4:00 PM?


46.  Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2 degrees per minute . How fast is the length of the third side increasing when the angle between the sides of fixed length is 60 ?


47. Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How fast is

the distance between the people changing after 15 minutes?

48.  The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm.

         Use differentials to estimate the maximum error in the calculated area of the disk.


49.   Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m.


50.  Use a linear approximation (or differentials) to estimate the given number:   2.0015


51. Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers that satisfy the conclusion of the Mean Value Theorem.


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


53.  Suppose that f(x) and g(x) are continuous on [a,b] and differentiable on (a,b). Suppose also that f(a)=g(a) and f’(x)< g’(x) for a<x<b prove that f(b)<g(b). [Hint: apply the Mean Value Theorem for the function h=f – g. ]


54. Show that the equation     has at most 2 real roots.


55 – 58.   (a) Find the intervals on which is increasing or decreasing.

        (b) Find the local maximum and minimum values of f(x).

        (c) Find the intervals of concavity and the inflection points.



    56.         for    

     57.        for  



(a) Find the vertical and horizontal asymptotes.

(b) Find the intervals of increase or decrease.

(c) Find the local maximum and minimum values.

(d) Find the intervals of concavity and the inflection points.

(e) Use the information from parts (a)–(d) to sketch the graph

of  f(x)





62.   , for    


Find the limits:









67. A stone is dropped from the upper observation deck of a Tower, 450 meters above the ground.

(a) Find the distance of the stone above ground level at time t.

(b) How long does it take the stone to reach the ground?

(c) With what velocity does it strike the ground?

(d) If the stone is thrown downward with a speed of 5 m/s, how long does it take to reach the ground?



68.  What constant acceleration is required to increase the speed of a car from 30 mi/h to 50 mi/h in 5 seconds?


A particle is moving with the given data. Find the position of the particle.


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


70.    ,   



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