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MATH 154 Lecture 24: Optimization & Related Rates

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lemongiraffe176

6 Nov 2018

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MATH 154 verified notes

MATH 154 Lecture 21: Optimization & Closed Interval Method

MATH 154 Lecture 24: Optimization & Related Rates

MATH 154 Lecture 25: Related rates problems

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If the function cos has derivative sin, find the derivative of cos(x3-2x) Problem: a colony of ants maintains a trail network to connect their nest to food sources. Maintaining the trails takes energy, so the colony tries to keep the smallest possible trail length. So we"re looking for a value of x that minimizes the function. We can assume d is much smaller than d. To solve this, find all critical points for each function & evaluate each function at these critical points & the endpoints; the smallest value is the minimum. We want to know when x = 10. We know that (cid:4666)(cid:4667)=(cid:2870), so we can say ((cid:4666)(cid:4667))=(cid:4666)(cid:4666)(cid:4667)(cid:4667)(cid:2870) Eg. the radius of a tumour expands at a rate of k; determine the rate of growth of its volume when the radius is 1cm. We know =, & we want to know dv/dt.

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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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