In this problem, you will use a linear approximation to approximate V493. Let an) = E The linear approximation of f(C) at w = 49 can be written in the form L(:L') = ma: + b' Compute m and b. c l | Using the linear approximation to f(:n) at w = 49, find the corresponding approximation for V493. Answer: [3 Note: the linearization of a function at J: : a is also called the linear approximation at m : a\" How is the linearization of f at a: = a related to the tangent line to the graph of y 2 ag) at m = a? Note: You can earn partial credit on this problem. In this question we will calculate the Taylor Polynomial for f (x) = Vac + 3 about ac = 1. The formula for the Taylor Polynomial of degree 3 for the function f (x ) about x = a is: T3 (a) = fla) +f' (a) + "(a) 2 ! ( 20 - a ) 2 + f"(a) 31 ( 2 - a) 3 In this case, that means we need to find f(1), f'(1), f"(1) and f" (1). f(ac) = Va + 3, so f(1) f' (2 ) = ], so f' ( 1 ) f" (2) =,so f"(1) fill (ac) = so f" ( 1 ) Therefore the Taylor Polynomial for f(x) = Va + 3 about x = 1 is: D+(x -1) +(2-1)2+(20-1)3Find the MacLaurin polynomial of degree 3 for the function x) = 62\"\". Tax) {1 +0 +83: +9 Now use this polynomial to approximate e4 \"a At noon, ship A is 180 km west of ship B. Ship A is sailing south at 20 km/h and ship B is sailing north at 40 km/h' How fast is the distance between the ships changing at 4:00 PM? E] km/h A street light is at the top of a 16 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 35 feet from the base of the pole? The tip of the shadow is moving at D ft/sec. Time for some rocket science (or AV-tech science?)! Let's go to Mars (on TV)! A television camera is positioned 4000 m from the base of the Mars Explorer rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let's assume the rocket rises vertically and its speed is 600 m/s when it has risen 3000 m. (a) How fast is the distance from the television camera to the rocket changing at that moment? (b) If the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at that same moment? (a) m/s (b) radians/s A water tank has the shape of an inverted circular cone with top radius 3m and height 5m. If water is being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when the water is 2m deep. Dan/min Evaluate the limit, using L'Hopital's Rule. Enter INF for 00, -INF for 00, or DNE if the limit does not exist, but is neither 00 nor oo. . 128$ 12:12 12 $1316 32:2 i C] NOTE: Please note that we would NOT normally say or write that a limit is "equal to DNE", but this is WeBWork so we have to make something work! Keep this in mind for written problems and use good notation. Preview My Answers Submit Answers Given that lim x) 2 0, 1img(:1:) = 0, lim Mac) 2 1, 1imp(:n) = ()0, lim (1(33) 2 oo. 1'4H1 ran $4M], (BAH! $4), Which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. Enter | to indicate an indeterminate form, INF for positive infinity, NINF for negative infinity, and D for the limit does not exist or we don't have enough information to determine the limit. (a) ggglm) pm] = C] (b) mam) q(m)l = B (c) mm) + M = C] Compute the following limit using I'Hopital's rule if appropriate. Use INF to denote co and MINF to denote -co. lim 8 sin(a) In(ac) = x-+0+Evaluate the following limit using L'Hospital's rule where appropriate. . sin(5$) 11m fit>0 tan(7a3) Answer: D Evaluate the limit, using L'Hopital's Rule. Enter INF for oo, -INF for -oo, or DNE if the limit does not exist, but is neither co nor -co. lim (1 + 3x ) 2/2 = x->0+If a function an) is continuous on [(1,1)] and differentiable on (a, b), then the Mean Value Theorem says that there is at least one number c in the interval (a, 1)) such that f'(c) : w. Find all possible value(s) for c given x) : 3:3 89: + 3, 3 5 :1: g 3. Enter your answer(s) separated by commas. czo Consider the function graphed below. Does this function satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]? ? Does it satisfy the conclusion? ? wo? ? ba At what point c is f'(c) = 4b Suppose f(:1:) is continuous on [4, 7] and ,4 g f'(w) S 2 for all a: in (4, 7). Use the Mean Value Theorem to estimate f(7) - f(4)- Answer: D g f(7) f(4) 5 [j