Question 1

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Consider the function f(x) = e* near x = 0. Find the linear approximation error when using the linear approximation to estimate e 0.45 ( a.) -0.1183 O b.) 0.1183 O C.) 1.45 O d.) -1.45Identify all of the global and local extrema of the graph. a.) 2 is a global maximum of f(x) at x = - 2 and x = 4. O -7 is a global minimum of f(x) at x = 1. b.) 2 is a global and local maximum of f(x) at x = - 2 and X =4. O -7 is a global and local minimum of f(x) at x = 1. c.) 2 is a global and local maximum of f(x) at x = - 2 and X = 4. O -7 is a global and local minimum of f(x) at x = 1. -5 is a local minimum and it occurs at x = 5. d.) 2 is a global maximum of f(x) at x = - 2 and x = 4. O -7 is a global minimum of f(x) at x = 1. -5 is a local minimum and it occurs at x = 5.Determine if the requirements for Rolle's theorem are met by the function f(x) = 5x + ~ on the interval . 5 . If so, find the values of c in 51 5 guaranteed by the theorem. a.) f(x) is continuous on any interval not including 0; therefore, it is continuous on 5 5. Also, f(x) is not differentiable on 5. 5) O Therefore, the conditions of Rolle's theorem have not been met. b.) f(x) is continuous on 5. 5 and f(x) is differentiable everywhere except where x = 0, so f(x) is differentiable on O 5 5. When evaluated, f 5 = 26 and f(5) = 26. Therefore, f(a) = f(b) and the conditions of Rolle's theorem are met. The value guaranteed by Rolle's theorem is c = 1.c.) f(x) is continuous on any interval not including 0; therefore, it is continuous on 5 5 . Also, f(x) is differentiable everywhere except where x = 0, so f(x) is O differentiable on 51 5 . When evaluated, f5 = 2 and f(5) = 26. Therefore, f(a) # f(b) and the conditions of Rolle's Theorem are not met. d.) f(x) is continuous and differentiable on any interval not including 0; therefore, it is continuous on 51 5 and differentiable on 5 5 . When evaluated, f = 26 and O f(5) = 26. Therefore, f(a) = f(b) and the conditions of Rolle's theorem are met. The values guaranteed by Rolle's theorem are c = - 1 and C = 1.Determine if the conditions of the mean value theorem are met by the q - ll h 2 :l J. function fIIE-QII = T on [-4, 3]. If so, find the values of cin If: - 4, 133) guaranteed by the theorem. a.) The function is NOT continuous on [-4, 3], and therefore, the mean value theorem does not apply for this function on the given interval. b.) fl? l is continuous on [-4, 53-] and differentiable on the interval i 4, 73). The values guaranteed by the mean value theorem are .3 = t3 and .3 = 0. fix 12' is continuous on [- l, 3] and differentiable on the interval The values guaranteed by the mean value theorem are ,- fF c=J+Vio. d.) ft! i is continuous on [- l, 3] and differentiable on the interval i -l, 3). The values guaranteed by the mean value theorem are , , ,_ = - :3: v" 'l . .I From the graph of ft}. 2|, determine the graph of f'liita It. a.) 3 2.5 2 1.5 .5 O -3 -2.5 -2 -1.5 -1 O 5 1.5 2 2.5 3 X -.5 -1.5 -2 -2.5 -3b.) 3 2.5 2 1.5 O -3 -2.5 -2 -1.5 -1 O 1.5 2 2.5 3 X .5 1.5 -2 -2.5 -3