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Chapter 1: Problem 65
Convert the following expressions to the indicated base. \(\ln |x|\) using base 5
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The fovea centralis (or fovea) is responsible for the sharp central vision that humans use for reading and other detail-oriented eyesight. The relative acuity of a human eye, which measures the sharpness of vision, is modeled by the function $$R(\theta)=\frac{0.568}{0.331|\theta|+0.568}$$ where \(\theta\) (in degrees) is the angular deviation of the line of sight from the center of the fovea (see figure). a. Graph \(R,\) for \(-15 \leq \theta \leq 15\) b. For what value of \(\theta\) is \(R\) maximized? What does this fact indicate about our eyesight? c. For what values of \(\theta\) do we maintain at least \(90 \%\) of our relative acuity? (Source: The Journal of Experimental Biology 203 \(3745-3754,(2000))\)
Simplify the difference quotients \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) by rationalizing the numerator. $$f(x)=\sqrt{x^{2}+1}$$
The Earth is approximately circular in cross section, with a circumference at the equator of 24,882 miles. Suppose we use two ropes to create two concentric circles; one by wrapping a rope around the equator and then a second circle that is \(38 \mathrm{ft}\) longer than the first rope (see figure). How much space is between the ropes?
Kelly has finished a picnic on an island that is \(200 \mathrm{m}\) off shore (see figure). She wants to return to a beach house that is \(600 \mathrm{m}\) from the point \(P\) on the shore closest to the island. She plans to row a boat to a point on shore \(x\) meters from \(P\) and then jog along the (straight) shore to the house. a. Let \(d(x)\) be the total length of her trip as a function of \(x .\) Graph this function. b. Suppose that Kelly can row at \(2 \mathrm{m} / \mathrm{s}\) and jog at \(4 \mathrm{m} / \mathrm{s}\). Let \(T(x)\) be the total time for her trip as a function of \(x\). Graph \(y=T(x)\) c. Based on your graph in part (b), estimate the point on the shore at which Kelly should land in order to minimize the total time of her trip. What is that minimum time?
Use the following steps to prove that \(\log _{b}(x y)=\log _{b} x+\log _{b} y\). a. Let \(x=b^{p}\) and \(y=b^{q}\). Solve these expressions for \(p\) and \(q\) respectively. b. Use property El for exponents to express \(x y\) in terms of \(b, p\) and \(q\). c. Compute \(\log _{b}(x y)\) and simplify.
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