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MATH 102 Study Guide - Midterm Guide: Saddle Point, Maxima And Minima

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22 Sep 2019

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Colin Macdonald

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$MATH 102 Full Course Notes$

MATH 102 Full Course Notes

Document Summary

6x + 3: find the all the tangent lines of f (x) = x3. X that pass through the points (0,2): when x = 1000, the function h(x) = 8x3. Partial solutions: (c) is the only true statement about derivatives. For (a), note that for f (x) = x4, f (0) = 0 but f (x) has a local minimum at x = 0, indeed f (x) is concave up everywhere. In general, when being asked about the classi cation of critical point via the second deriva- tive, x4 is a great test case to keep in mind. 2. lim h 0 f (x + h) f (x) h. 6(x + h) + 3 3x2 + 6x 3. 6x 6h + 3 3x2 + 6x 3 h h. = 6x 6: note that the point (0, 2) is not a point on the graph of the function y = f (x).

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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

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