1 Worksheet # 1: Applications of Limits Problem 1. Sensitivity of Measurements: Suppose that f is a function of x. If x = x0 + ∆x, then we define ∆f = f (x0 + ∆h) − f (x0) and ∆f /∆x measures how much the value of f changes when x is changed. With GPS, radio signals give us h up to a certain measurement error (See Fig. 2 and Fig. 3). The question is how accurately can we measure L. To decide, we find ∆L ∆h . In other words, these variables are related to each other. We want to find how a change in one variable affects the other variable. The planet Quirk is flat. GPS satellites hover over Quirk at an altitude of s = 20, 000 km (see Fig. 3 above). See how accurately you can estimate the distance L from the point directly below the satellite to a point on the planet surface knowing the distance h from the satellite to the point on the surface in two cases. (The letter h is for hypotenuse.) 2 We see that p p 2 2 L(h) = h − s = h2 − (20000)2 a. Use a calculator or a spread sheet to compute ∆L/∆h for h = h0 ± ∆h = 25, 000 ± ∆h, and ∆h = 1, 10−1, 10−2. Write an estimate of L in the form |L(h0 + ∆h) − L(h0)| = |∆L| ≤ C|∆h| choosing the simplest integer C that works for all six cases (the first case is filled in for you, double check it). ∆h -1 1 -0.1 0.1 -0.01 0.01 ∆L -1.667 ∆L/∆h 1.667 b. Do the same for h = 20, 001 ± ∆h, and ∆h = 1, 10−1, 10−2. (the first case is filled in for you, double check it). ∆h -1 1 -0.1 0.1 -0.01 0.01 ∆L -200.002 ∆L/∆h 200.002 c. Compute the limit: p p 2 2 (h0 + ∆h) − s − h20 − s2 ∆L 0 L (h0) = lim = lim ∆h→0 ∆h ∆h→0 ∆h d. Now go back to part a. above with h0 = 25, 000. How does L0(h0) compare with your choice of C in part a.? e. For h0 = 20, 001 how does L0(h0) compare to your choice of C found in part b. above? 3 Problem 2. Lots of Trig Limits Consider the unit circle: A C θ O D Determine the following limits: area of sector AOB θ→0 area of ∆ AOD (a) lim area of sector AOB θ→0 area of ∆ COB (b) lim area of sector AOB θ→0 area of ∆ AOB (c) lim length of segment AB θ→0 length of segment CB (d) lim length of segment AB θ→0 length of arc AB (e) lim length of segment AB θ→0 length of segment AD (f) lim B