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UAE-Math-Grade-12-Vol-1-SE-718383-ch4
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CHAPTER 4
Applications of Differentiation
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306_319_ADVM_G12_S_C04_L09_v2_684362.indd Page 317 11/6/18 7:15 PM f-0198
is AED 2 million per mile. Find the point on the boundary
3
3
4
19. f(x) = x − 4x + 2 QC: OSO/OVY T1: OSO 2 November 6, 2018 15:33 Review Exercises 4-86 Unsaved...
20. f(x) = x − 3x − 24x
of swampland and dry land to which the highway should
2
21. f(x) = xe −4x 22. f(x) = x ln x be built to minimize the total cost.
23. f(x) = x − 90 24. f(x) = (x − 1) 2∕3 50. If a muscle contracts at speed v, the force produced by the
2
x 2 muscle is proportional to e −v∕2 . Show that the greater the
25. f(x) = x x speed of contraction, the less force produced. However, the
x + 4 26. f(x) = √ x + 2 power produced by the contracting muscle is proportional
2
2
............................................................ to ve −v∕2 . Determine the speed that maximizes the power.
51. A soda can in the shape of a cylinder is to hold 16 fl oz. Find
In exercises 27–30, find the absolute extrema of the given func- the dimensions of the can that minimize the surface area of
tion on the indicated interval. the can.
2
3
2
27. f(x) = x + 3x − 9x on [0, 4] 52. Suppose that C(x) = 0.02x + 4x + 1200 is the cost of man-
′
ufacturing x items. Show that C (x) > 0 and explain in busi-
′′
28. f(x) = √ x − 3x + 2x on [−1, 3] ness terms why this has to be true. Show that C (x) > 0 and
3
2
explain why this indicates that the manufacturing process
29. f(x) = x 4∕5 on [−2, 3] is not very efficient.
2 −x
30. f(x) = x e on [−1, 4] 53. The diagram shows a football field with hash marks H feet
............................................................ apart and goalposts P feet apart. If a field goal is to be tried
from a (horizontal) distance of x feet from the goalposts, the
In exercises 31–34, find the x-coordinates of all local extrema. angle gives the margin of error for that direction. Find x
to maximize .
3
2
2
31. f(x) = x + 4x + 2x 32. f(x) = x − 3x + 2x
4
2
5
33. f(x) = x − 2x + x 34. f(x) = x + 4x − 4x
2
5
............................................................
35. Sketch a graph of a function with f(−1) = 2, f(1) =−2, P
′
′
f (x) < 0 for −2 < x < 2 and f (x) > 0 for x < −2 and x > 2. H
θ
′
′
36. Sketch a graph of a function with f (x) > 0 for x ≠ 0, f (0)
′′
′′
undefined, f (x) > 0 for x < 0 and f (x) < 0 for x > 0.
x
In exercises 37–46, sketch a graph of the function and com-
pletely discuss the graph.
54. In the situation of exercise 53, sports announcers often say
4
4
37. f(x) = x + 4x 3 38. f(x) = x + 4x 2 that for a short field goal (50 ≤ x ≤ 60), a team can im-
4
39. f(x) = x + 4x 40. f(x) = x − 4x 2 prove the angle by backing up 5 yards with a penalty. De-
4
1
termine whether this is true for high school (H = 53 and
3
1
1
41. f(x) = x 42. f(x) = x P = 23 ), college (H = 40 and P = 18 ) or pros (H = 18 1
2
2
x + 1 x − 1 3 2 2
1
and P = 18 ).
2
2
43. f(x) = x 44. f(x) = x 2
x + 1 x − 1 55. The charge in an electrical circuit at time t is given by
2
2
Q(t) = e −3t sin 2t coulombs. Find the current.
3
45. f(x) = x 46. f(x) = 4
x − 1 x − 1 56. If the concentration x(t) of a chemical in a reaction
2
2
............................................................ changes according to the equation x (t) = 0.3x(t)[4 − x(t)],
′
find the concentration at which the reaction rate is a
2
47. Find the point on the graph of y = 2x that is closest to (2, maximum.
1).
57. Suppose that the mass of the first x meters of a thin rod is
2
48. Show that the line through the two points of exercise 47 is given by m(x) = 20 + x for 0 ≤ x ≤ 4. Find the density of
2
perpendicular to the tangent line to y = 2x at (2, 1). the rod and briefly describe the composition of the rod.
49. A city is building a highway from point A to point B, which 58. A person scores f(t) = 90∕(1 + 4e −0.4t ) points on a test af-
is 4 miles east and 6 miles south of point A. The first 4 miles ter t hours of studying. What does the person score with-
′
south of point A is swampland, where the cost of building
out studying at all? Compute f (0) and estimate how many
Copyright © McGraw-Hill Education Copyright © McGraw-Hill Education
the highway is AED 6 million per mile. On dry land, the cost
points 1 hour of studying will add to the score.
317
317
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