(1) Suppose f(r) is differentiable at r = a. Then f'(a) exists and is the slope of the tangent line to f(r) at r = a. The tangent line is given by L(x) = f(a) + f'(a)(x - a). (2) L(x) is called the Linear Approximation of f(r). This means that near r = a, we have that f(x) ~ L(I). Note also that L(a) = f(a). (3) Let y = f(x). Then da = du = f'(x). We define dr to be an independent variable and define the differential dy to be dy = f'(x)dx. So if dx * 0, then dy = f'(x)dx can be written in the familiar form a dy = f'(x). (4) We define the change in f, denoted Af, from r = To to I = To + Ar by Af = f(ro + Ax) - f(To).(1] Let y = .133. Find d-y and A3; at 3: = 1 with d9: = A9: = 1. Sketch the graph of y = 11:3 showing rig and 3-3; in the picture. (2] (a) Find the local linear approximation of f(:r:) = \\J 1 + :r: at 9:0 2 0. Use your approxima- tion formula to approximate 1.! 0.9 and v\" 1.1. ([1] Graph 3' and its tangent line at In together, and use the graphs to illustrate the rela- tionship between the exact values and the approximations of v0.9 and 1.! 1.1. (3] N ewton's Law of Gravitation shows that if a person weighs in pounds on the surface of the earth, then his or her weight at. distance :r: from the center of the earth is iii-TH L410 PROJECT 1 CALCULUS I 3 2 . mR W (9:) = 2 :2: where R = 3960 miles is the radius of the earth. (a) Show that weight lost at an altitude h miles above the earth's surface is AW z .UUD.'th