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13 Nov 2019

Let f(x) = 1 x and g(x) = 1 x if x > 0 4 + 1 x if x < 0 Show that fâ'(x) = g'(x) for all x in their domains. Can we conclude from the corollary below that f â g is constant? If fâ'(x) = g'(x) for all x in an interval (a, b), then f â g is constant on (a, b); that is, f(x) = g(x) + c where c is a constant.

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If U and V are positive constants, find all critical points of the function. (Enter your answers as a comma-separated list.) F(t) = Ue t + Ve -t t = Find all critical points of f(x) = 20 x 2 e - 0.3x and classify them as local minima of ,maxima. x = (smaller value) This critical point is a . x = (larger value) This critical point is a . Consider f(x) = x 8 (7 - x) 9. Find f '(x) and simplify your answer. f '(x) = Find all critical points of f(x) and then use the first derivation test to determine local maxima and minima. f(x) has a at x = (smallest value) Local Maximum OR Minimum OR Neither f(x) has a at x = Local Maximum OR Minimum OR Neither f(x) has a at x = (largest value) Local Maximum OR Minimum OR Neither Find the inflection points of f(t) = t 4 + t 3 - 18t 2 +8. Give exact answer. t = (smallest value) t = (largest value) Sketch a possible of y = f(x), using the given information about derivation y' = f'(x) and y" = f"(x). Assume that the function is defined and continuous for all real x. Let f be a function f(x) > 0 for all x. Let g = 1/f. If f is increasing in an interval around x 0, what about g? g is increasing in an interval x 0. g is zero in an interval around x 0. g is decreasing in an interval around x 0. We cannot tell whether g is increasing or decreasing in an interval around x 0. g is constant in an interval around x 0. If f has a local maximum at x 1, what about g? g has a local maximum at x 1. g has both a local minimum and a local maximum at x 1. We cannot tell whether g has a local minimum or a local maximum at x 1. g has an inflection point at x 1. g has a local minimum at x 1. If f is a concave down at x 2, what about g? g is increasing at x 2. g is flat at x 2. We cannot tell whether g is concave up of concave down at x 2. g is concave down at x 2. g is concave up at x 2. You are given the graph of the second derivation function f" below. Which of the given have x-values that are inflection points of the function f? (Select all the apply.) A B C D E F G H I none of the above

How large should we take n in order to guarantee that the Trapezoidal and Midpoint Rule approximations for 2 1 x dx 1 are accurate to within 0.00002? SOLUTION If f(x) = 1 x , then fâ'(x) = , and fâ''(x) = . Since 1 â¤ x â¤ 2, we have 1 x â¤ 1, so |fâ''(x)| = â¤ 2 13 = We take K = 2, a = 1, and b = 2. Accuracy to within 0.00002 means that the size of the error should be less than 0.00002. Therefore we chose n so that 2 3 12n2 â¤ 0.00002 Solving the inequality for n, we get n2 > 2 3 12(0.00002) or n > 1 0.00012 â 91.29 Thus n = (rounded up to the nearest integer) will ensure the desired accuracy. For the same accuracy with the Midpoint Rule we choose n so that 2 3 24n2 < 0.00002 which gives the following. (Round your answer up to the nearest integer.) n > 1 0.00024 â

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