APPLIED PROJECT RADIATION FROM THE STARS 627 (b) For copper, the tables give a = 0.0039/ C and (a) Expand the integrand as a binomial series and use the result px = 1.7 x 10- -m. Graph the resistivity of copper of Exercise 38 in Section 5.6 to show that and the linear and quadratic approximations for -250'C = 1 = 1000 C. 7 ko + ... (c) For what values of i does the linear approximation agree with the exponential expression to within one percent? If do is not too large, the approximation T = 27 VL/g. obtained by using only the first term in the series, is often 31. If a surveyor measures differences in elevation when making used. A better approximation is obtained by using two plans for a highway across a desert, corrections must be terms: made for the curvature of the earth. (a) If R is the radius of the carth and L is the length of the T - 207 VL (1 + 143 ) highway, show that the correction is C = R SCC(L/R) - R (b) Notice that all the terms in the series after the first one have coefficients that are at most ;. Use this fact to compare this (b) Use a Taylor polynomial to show that series with a geometric series and show that L' SL' 2714 ( 1 + 1 4 3 ) =T= 20 L 4 - 31 3 2R 24R' 4 - 4K? (c) Compare the corrections given by the formulas in parts (c) Use the inequalities in part (b) to estimate the period of a (a) and (b) for a highway that is 100 km long. (Take the pendulum with L = 1 meter and 0% = 10. How does it com- radius of the earth to be 6370 km.) pare with the estimate T = 27 VL/g ? What if 0 = 42? 33. In Section 4.7 we considered Newton's method for approxi- mating a root r of the equation f(x) = 0. and from an initial approximation x, we obtained successive approximations X2. X3, . . . . where X.41 = X. - f( x.) f'( x. ) Use Taylor's Inequality with n = 1. a = x., and x = r to show 32. The period of a pendulum with length L that makes a maxi- that if f"(.x) exists on an interval / containing r, x., and x.+ 1. mum angle do with the vertical is and | f"(x) | = M. | f'(x) | > K for all x E I, then T = 4 d.x VI - k sin -x [This means that if x, is accurate to d decimal places, then .X. . where & = sin( 00) and g is the acceleration due to gravity. is accurate to about 2d decimal places. More precisely. if the (In Exercise 34 in Section 5.9 we approximated this integral error at stage n is at most 10"", then the error at stage n + I is using Simpson's Rule.) at most (M/2K) 10-2.] APPLI PROJECT Radiation from the Stars Any object emits radiation when heated. A blackbody is a system that absorbs all the radiation that falls on it. For instance, a matte black surface or a large cavity with a small hole in its wall (like a blastfurnace) is a blackbody and emits blackbody radiation. Even the radiation from the sun is close to being blackbody radiation. Proposed in the late 19th century. the Rayleigh-Jeans Law expresses the energy density of blackbody radiation of wavelength A as f(A) = STAT Luke Dodd / Photo Researchers, In where A is measured in meters, T is the temperature in kelvins (K), and & is Boltzmann's con- stant. The Rayleigh-Jeans Law agrees with experimental measurements for long wavelengths Y Graphing calculator or computer with graphing software required