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101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

We know that the value of the derivative of a function f at a point x represents the slope of the tangent line to the graph of f at the point (x, f (x)) If that slope is positive, then the tangent line rises as x increases from left to right, hence so does the curve (we say that the function is increasing) If instead the slope of the tangent line is negative, then the tangent line falls as x increases from left to right, hence so does the curve (we say that the function is decreasing) We summarize: On an interval where f > 0 the graph of f goes uphill On an interval where f < 0 the graph of f goes downhill See Figure 31 With some additional thought, we can also get useful information from the second derivative If f = ( f ) > 0 at a point, then f is increasing Hence the slope of the tangent line is getting ever greater (the graph is concave up) The picture must be as in Figure 32(a) or 32(b) If instead f = ( f ) < 0 at a point then f is decreasing Hence the slope of the tangent line is getting ever less (the graph is concave down) The picture must be as in Figure 33(a) or 33(b) Using information about the first and second derivatives, we can render rather accurate graphs of functions We now illustrate with some examples

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097514 097876 098201 098494 098754 098982 099183 099355 099501 099625 099725 099806 099870 099918 099952 099976 099990 099997 099999 100000

Going back to the constraint equation solved for y7 the corresponding y dimension is y = 84 -4(14) = 84 - 56 = 28

04554 04122 03707 03309 02929 02569 02228 01910 01613 01340 01090 00865 00664 00489 00341 00218 00123 00055 00014 00000

Of course this is a simple and familiar function, and you know that its graph is a parabola But it is satisfying to see calculus confirms the shape of the graph Let us see how this works First observe that f ( x) = 2x We see that f < 0 when x < 0 and f > 0 when x > 0 So the graph is decreasing on the negative real axis and the graph is increasing on the positive real axis Next observe that f ( x) = 2 Thus f > 0 at all points Thus the graph is concave up everywhere Finally note that the graph passes through the origin We summarize this information in the graph shown in Figure 34

-52695 -50832 -48830 -46693 -44429 -42043 -39541 -36931 -34221 -31416 -28525 -25556 -22517 -19416 -16262 -13063 -09829 -06568 -03289 00000

First observe that f ( x) = 3x 2 Thus f 0 everywhere The function is always increasing Second observe that f ( x) = 6x Thus f ( x) < 0 when x < 0 and f ( x) > 0 when x > 0 So the graph is concave down on the negative real axis and concave up on the positive real axis Finally note that the graph passes through the origin We summarize our findings in the graph shown in Figure 35

16 17 18 19 20

001506 001799 002124 002486 002884

A box with a square end 14 inches on a side and length 28 produces the maximum volume within the girth and length restrictions

We see that g ( x) = 1 + cos x Since 1 cos x 1, it follows that g ( x) 0 Hence the graph of g is always increasing Now g ( x) = sin x This function is positive sometimes and negative sometimes In fact sin x is positive when k < x < ( k + 1) , k odd and sin x is negative when k < x < ( k + 1) , k even So the graph alternates between being concave down and concave up Of course it also passes through the origin We amalgamate all our information in the graph shown in Figure 36

03309 03707 04122 04554 05000

Sketch the graph of h( x) =

46693 48830 50832 52695 54414

46 47 48 49 50

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