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A girl is flying a kite The kite is moving horizontally at a height of I20 ft when 250 ft of string is out and the rate of increase in string length is 2 ft/s How fast is the kite moving in the x-direction for these conditions

Until we learn some more sophisticated techniques, the basic method that we shall use for graphing functions is to plot points and then to connect them in a plausible manner

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3 2 1 0 1 2 3

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007386 008100 008847 009630 010451

24 6 0 0 0 6 24

6- 2

08371 08778 09185 09593 10000

We plot these points on a pair of axes and connect them in a reasonable way (Figure 141) Notice that the domain of f is all of R, so we extend the graph to the edges of the picture

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3 2 1 0 1 2 3 4 5

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TABLE B4 K Cv Ca 00000 -01929 -03856 -05778 -07693 -09599 91 92 93 94 95 089182 090048 090880 091676 092436 093161 093849 094500 095114 095690 96 97 98 99 100 -11493 -13373 -15237 -17082 -18907 90 088282 10997 10594 10184 09767 09343 08912 08477 08036 07592 07143 06691 17596 17588 17564 17523 17467 17395 17307 17204 17084 16950 16800 011718 012650 013616 014613 015642 016701 017788 018905 020049 021220 022417 13229 13566 13892 14206 14508 41081 39765 38401 36989 35553 66 67 68 69 70 058750 060178 061617 063035 064442 11392 11778 12156 12524 12882 46380 45829 44722 43561 42347 61 62 63 64 65 051466 052931 054393 055851 057304 10997 47874 60 050000 Cv Ca Pt K Cv K Pt Ca

Solution: Visualize the problem and set up a right triangle with the height, horizontal direction, and string In this problem the h t e only moves horizontally, and the string is straight - idealized - but the conditions make for a problem that can be solved

y = f (x)

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1 2 3 4 5

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We plot these on a pair of axes (Figure 142) Since the definition of the function changes at x = 2, we would be mistaken to connect these dots blindly First notice that, for x 2, the function is identically constant Its graph is a horizontal line For x > 2, the function is a line of slope 1 Now we can sketch the graph accurately (Figure 143)

Referring to Fig 6-2, the problem question can be written in mathematical terms as:

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05778 11493 17082 22484 27640

Sketch the graph of f ( x) =

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We begin by noticing that the domain of f, that is the values of x for which the function makes sense, is {x : x 1} The square root is understood to be the positive square root Now we compute a table of values and plot some points

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Fig 6-2

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1 0 1 2 3 4 5 6

00840 01130 01455 01813 02199

32493 36989 41081 44723 47874

If a tree is growing in a forest, then both its height and its radius will be increasing These two growths will depend in turn on (i) the amount of sunlight that hits the tree, (ii) the amount of nutrients in the soil, and (iii) the proximity of other trees We may wish to study the relationship among these various parameters For example, if we know that the amount of sunlight and nutrients are increasing at a certain rate then we may wish to know how that affects the rate of change of the radius This consideration gives rise to related rates problems

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A toy balloon is in the shape of a sphere It is being inflated at the rate of 20 cu in/min At the moment that the sphere has volume 64 cubic inches, what is the rate of change of the radius

Go back over the problem statement and practice changing the problem statement into this mathematical statement One of the more challenging parts of any calculus problem is translating the words into mathematical statements

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