MAT135H1 Lecture Notes - Lecture 28: Maxima And Minima, Thx

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13 Mar 2019

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MAT135H1 Lecture Notes - Lecture 27: Linearization, Thx

MAT135H1 Lecture Notes - Lecture 28: Maxima And Minima, Thx

MAT135H1 Lecture Notes - Lecture 29: Maxima And Minima, Inflection

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Connection t " f " f " f " 70 co between the derivative function and the. > 0 then interval on is t an function itself. increasing decreasing on the interval i on an interval then the graph off is concave concave up down. P is less than or equal to is larger than or equal to the. For or any tip ) f function f is undefined the point c p , tips ) A critical of value on f p in a point is the graph of value is called critical f the domain of point of is also tips , at the f where function . In addition called a critical point . a critical point p . critical and defined on points an interval. If interval critical is is a a point . and has not is x - A a an then local maximum of endpoint or the flat.

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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

4. (30 pts) EXPLANATION IS NOT REQUIRED, AND NO PARTIAL CREDIT WILL BE GIVEN. (I) The entire) graph of a function f is shown in the figure below. (a) Find the x-coordinates of all critical points of f (or write NONE). ANSWER : critical point (s) at x = (b) Find the x-coordinates of all local minima of f (or write NONE). ANSWER : local min (s) at x = (c) Find all values of x at which fattains its global minimum (or write NONE ANSWER : global min (s) at x = (d) Find the interval (or intervals) on which the derivative of fis increasing. ANSWER : derivative off is increasing on (II) The figure below shows the graphs of f, f', and f". Which curve is which? Fill in the blanks: A = -; B= 2; CD- 4. EXPLANATION IS NOT REQUIRED, AND NO PARTIAL CREDIT WILL BE GIVEN. (III) A function f' (derivative of f) is given f'(x) = (x - 2) e*. The following questions are about the function f. (a) Find all critical points of f (or write NONE). ANSWER : critical point (s) at x = (b) on what interval (or intervals) is f increasing ? ANSWER : f is increasing on (c) Find the point or points where f has a local maximum (or write NONE). ANSWER : local max at x = (d) Find the point or points where f has a local minimum. ANSWER: local min (s) at x = (e) Find f "(x), the second derivative of f. ANSWER : f''(x) = (f) Identify any inflection pionts. ANSWER : inflection point (s) at x = (g) Determine the intervals on which the function is concave up or concave down. ANSWER: fis CONCAVE UP on fis CONCAVE DOWN on

6. (14 pts) EXPLANATION IS NOT REQUIRED, AND NO PARTIAL CREDIT WILL BE GIVEN. The graph of f'(derivative of f) on the interval [-6, 7] is shown in the figure. y f(x) y = f'(x) -6 -5 -4 -3 2 L - 2 3 4 5 7 Use the given graph of f' to answer the following questions about f: (a) (2 pts) on what interval (or intervals) is f decreasing? (b) (3 pts) Find the critical points of f. (c) (2 pts) which critical points (or point) correspond to local maxima ? (d) (2 pts) which critical points (or point) correspond to neither local maxima nor local minima ? (e) (2 pts) on what interval (or intervals) is f concave up? (f) (3 pts) At what point (or points) does f have an inflection point ?

MAT135H1 Lecture Notes - Lecture 28: Maxima And Minima, Thx

MAT135H1 Lecture Notes - Lecture 27: Linearization, Thx

MAT135H1 Lecture Notes - Lecture 28: Maxima And Minima, Thx

MAT135H1 Lecture Notes - Lecture 29: Maxima And Minima, Inflection

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