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CAMCO B-12345 (SAMPLE) FOLLOWER RADIUS FOLLOWER VELOC FOLLOWER ACCEL (INCHES) (INCHES/SEC) (INCH/SEC/SEC) 12922 13029 13274 13668 14229 14984 15972 17246 18887 21010 23791 27500 27500 23791 21010 18887 17246 15972 14984 14229 13668 13274 13029 12922 12913 12922 12944 12979 13027 13087 13159

y = f (x)

Solution:

RADIUS OF CURV (INCHES)

from a need to know explicitly what the variable is The numbers a and b are called the limits of integration---the number a is the lower limit and b is the upper limit The function f is called the integrand Before we can present a detailed example, we need to record some important information about sums: I We need to calculate the sum S = 1 + 2 + + N = this goal, we write S= 1 + 2

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1240000 1250000 1260000 1270000 1280000 1290000 1300000 1310000 1320000 1330000 1340000 1350000 2250000 2260000 2270000 2280000 2290000 2300000 2310000 2320000 2330000 2340000 2350000 2360000 2370000 2380000 2390000 2400000 2410000 2420000 2430000

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2S = (N + 1) + (N + 1) + + (N + 1) + (N + 1)

(Continued)

triangle in a related rate problem look for similar triangles Don't start the problem looking for derivatives Concentrate on the defining equation for the problem The derivatives come later

Thus 2S = N (N + 1) or S= N (N + 1) 2

CAMSHAFT ANGLE (DEGREES) PRESSURE ANGLE (DEGREES) 166994 177757 188370 198818 209086 219158 229022 238664 248074 257239 266150 274797 283171 291264 299067 306573 313774 320664 327236 333483 339399 344975 350206 355085 359602 363751 367523 370909 373898 376480 378644 26426 26283 26132 25972 25804 25628 25444 25253 25054 24848 24636 24418 24193 23963 23727 23486 23241 22992 22739 22482 22222 21960 21696 21430 21162 20894 20625 20356 20088 19820 19554 -49812 -52943 -56013 -59014 -61939 -64783 -67538 -70200 -72762 -75219 -77566 -79797 -81907 -83893 -85749 -87472 -89058 -90503 -91805 -92960 -93966 -94822 -95524 -96072 -96464 -96700 -96778 -96700 -96464 -96072 -95524 -1137891 -1116689 -1093069 -1067081 -1038783 -1008235 -975503 -940658 -903776 -864937 -824224 -781727 -737536 -691748 -644462 -595780 -545807 -494653 -442427 -389243 -335217 -280464 -225103 -169255 -113041 -56582 00000 56582 113041 169255 225103

FOLLOWER ANGLE (DEGREES)

This is a famous formula that was discovered by Carl Friedrich Gauss (1777--1855) when he was a child There is also evidence that the formula was known to the ancients II The sum S = 12 + 22 + + N2 = S=

CAMCO B-12345 (SAMPLE) FOLLOWER RADIUS FOLLOWER VELOC FOLLOWER ACCEL (INCHES) (INCHES/SEC) (INCH/SEC/SEC)

RADIUS OF CURV (INCHES) 13243 13339 13447 13565 13694 13833 13982 14141 14309 14486 14673 14868 15071 15283 15503 15732 15969 16215 16470 16735 17010 17297 17596 17909 18239 18586 18955 19349 19772 20229 20727

We shall not provide the details of the proof of this formula, but refer the interested reader to [SCH2]

Your first order of business in a Fig 6-5 related rate problem is to find relationshps between the variables In this problem set up the triangle, complete with known numbers, and then label some of the distances The change in length of the hypotenuse of h s triangle is not what we are looking for It is lengths along the ground: the length fkom the light to Penny and the length of her shadow Take x as the length from beneath the light to Penny, and z as the length from beneath the light to the end of her shadow The length of her shadow is z - x Draw this triangle (refer to Figs 6-5 and 6-6)

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For our first example, we calculate the area under a parabola:

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(Continued)

Refer to Figure 48 as we reason along Let f ( x) = x 2 Consider the partition P of the interval [1, 2] consisting of k + 1 points x0 , x1 , , xk The corresponding Riemann sum is

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