Note: Here's an example of how you might solve a tangent line approximation problem. A rectangular field has one side along a straight riyerr and the other three sides are fenced. There are exactly 100 feet of fencing available. We measure the river side to be 40 feet and compute the area of the field to be 1200 square feet. But our measurement is not exactly accurate. Use the tangent line approximation to find a linear relationship between small variations for the length of the river side (from 40) and the resulting {approximate} variations in area. About honI big is the field if the side along the river is actually 40.3 feet long? 'lOO-x Solution: In this problem, you're measuring the river side that will be your x. {Sb label the picture with it, hot 40.) The other direction will be 1\"" ' " . The area of the eld is what you'll be approximating here. The area function isA = J'Tx) =xliugij. and f'{x]=50 x. The tangent line approximation will give a linear relationship: A is approximately equal to n40} + f\"{40){x}40= 1200 + 100:) 40. Finally, you can use this to approximate the area if the river side is actually 40.3 feet long: 11403) is approximately equal to 1200 + 10(3) 2 1203 square feet. Now it's your turn. In each of the problems that follow, remember to identify what you're approximating, what the known constants are, and what the variables are. Proceed to next page. 2. D. If :14} = :i and Fiat] = 2, rpm). You measure a cubic container and find it to be about 10 cm on each side. From this, you conclude that it holds 1000 CC. However. your measurement is accurate only to within :0.1 cm. [So you can he Sure that the side is between 9.9 and 10.1cm.) What are I, x), and a in this problem? Use the tangent line approximation to estimate the error in your value of 1000 for the volume. Give the percent error in your measurement of the length of the side. Give the percent error in your estimate of the volume, Ifa right triangle has legs 6 and 8, its hypotenuse is 10. The triangle will be inscribed within a circle with area 25-1. (The hypotenuse will be the diameter of the circle.) Suppose one leg of the triangle is known to be exactly 6, but the other leg is known to be 8 with an error of :11. what are x, ffx), and a in this problem? B. Use a tangent line approximation to estimate the area of the circumscribed circle. C. Now consider the sphere that just contains the triangle (so the hypotenuse Is the diameter of the sphere). Use a tangent line approximation to estimate the volume of this sphere. 4. Suppose the point 3'4 is on the curve sinx , Sly =C, where C is a constant. Use y the tangent line approximation to find the y-coordinate of the point on the curve with x- coordinate 3 180 *. Be sure to show all your work. 5. Use tangent line approximation to derive an estimate for (1+ x)" , when x is near 0, and n is any real number.6. Use tangent line approximation to estimate $2390 (to seven decimal places), recognizing that 7* = 2401. Show all your work. 7. Suppose your friend gives you the following problem: There is some function p(x) that represents profit p (in dollars) as a function of the number of units sold x. Your friend doesn't know what the function p(x) is explicitly, but does know that p'(x) = 2 for all x. Your friend also knows that p(200) = 3175.24. Now, your friend has enough information to approximate p(203) using tangent line approximation, but your friend really wants to know how accurate that approximation will be. The question to you then is: What can you tell your friend about the accuracy of the tangent line approximation for p(203), given the above information about the function p? (You may explain yourself mathematically, or by presenting graphs, or just by discussing in complete sentences.)