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)3 - 3x - 2;11. Find the number c that satisfies the conclusion of the Mean Value Theorem.;f(x) = 5x2 + 3x + 6;x? [-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.;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 =;c);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;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.;55.;56. for;57. for;58.;(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);59.;60.;61.;62., for;Find the limits;63.;64.;65.;66.;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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