# A f x e 2 x approximate f 0 43 b f x x 4 x 3 x 2 x 1

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a f(x) =e2x; approximatef(0.43).bf(x) =x4-x3+x2-x+ 1; approximatef(0).cf(x) =x2cos(x)-3x; approximatef(0.18).df(x) = ln(ex+ 2); approximatef(0.25).3.3.10A car traveling along a straight road is clocked at a number of points. Thedata from the obsercations are given in the following table, where the timeis in seconds, the distance is in feet, and the speed is in feet per second.Time035813Distance0225383623993Speed7577807472aUse a Hermite polynomial to predict the position of the car and it’sspeed whent= 10 s.bUse the derivative of the Hermite polynomial to determine whether thecar ever exceeds a 55 mi/h speed limit on the road. If so, what is thefirst time the car exceeds this speed?cWhat is the predicted maximum speed for the car?3.4.30The 2004 Kentucky Derby was won by a horse named Smarty Jones in atime of 2:04.06 (2 minutes and 4.06 seconds) for the 114-mile race. Times atthe quarter-mile, half-mile, and mile poles were 0:22.99, 0:46.73, and 1:37.35.aUse these values together with the starting time to construct a freecubic spline for Smarty Jones’ race.bUse the spline to predict the time at the three-quarter-mile pole, andcompare this to the actual time of 1:11.80.cUse the spline to approximate Smarty Jones’ starting speed and speedat the finish line.3
Math 104A - Homework 3Due 7/7 2.4.6Show that the following sequences converge linearly top= 0.How largemustnbe before we have|pn-p| ≤5·10-2? 47.2.4.7aShow that for any positive integerk, the sequence defined bypn= 1/nkconverges linearly top= 0. 1
2.4.8bShow that the sequencepn= 10-nkdoes not converge quadratically, regard-less of the size of the exponentk. 2.4.9bSupposeα >1. Construct a sequence that converges to 0 of orderα.