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j=1 j=1 k=1 3. For a ER, a 70, the function f defined on (lal,00) by...
Suppose that the piecewise function J is defined by f(2)= {**** -1<<3 - 3x2 + 2x...
Suppose that the piecewise function J is defined by f(2)= {**** -1<<3 - 3x2 + 2x + 23, 2> 3 Determine which of the following statements are true. Select the correct answer below: O f() is not continuous at I = 3 because it is not defined at I = 3. Of() is not continuous at 2 = 3 because lim f(x) does not exist. f() is not continuous at I = 3 because lim f() f(3). ->3 f(x) is...
(1 point) For the function f(x) = e2x + e- defined on the interval (-4, o),...
(1 point) For the function f(x) = e2x + e- defined on the interval (-4, o), find all intervals where the function is strictly increasing or strictly decreasing. Your intervals should be as large as possible. f is strictly increasing on f is strictly decreasing on (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10)) whenever r is near c on the left Find and classify all local max's and min's. (For the purposes...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
Problem 5 (10 points). Consider the function f(x) = L. (1-4e+") dt, defined in (-3, +00)....
Problem 5 (10 points). Consider the function f(x) = L. (1-4e+") dt, defined in (-3, +00). Find the local max/min of f(x).
4 -2 2. The function f is defined on the closed interval [-4,9]. The graph of...
4 -2 2. The function f is defined on the closed interval [-4,9]. The graph of f consists of a semicircle, a quarter circle, and three linear segments, as shown in the figure above. Let g be the function defined by g(x) = 3x + f(t) dt. (a) Find g(8) and g'(8). (b) Find the value of x in the closed interval (-4,9] at which g attains its maximum value. Justify your answer. (c) Find lim f'(x), or state that...
Sketch the graph of the function f(r) with the following characteristics: lim f(x) = -00 lim...
Sketch the graph of the function f(r) with the following characteristics: lim f(x) = -00 lim f.) = -1 -2 0-0 lim f(x) =0 lim f(3) = -1 lim f (r) = 2 1-2
1. Suppose the a function g(x) is defined according to the formula f(c) 3(x + 2)...
1. Suppose the a function g(x) is defined according to the formula f(c) 3(x + 2) +2 for – 3 <x< -2 (x+2)+1 for-2<x< -1 (+2)+1 for - 1<x<1 2 for r=1 for > 1 y 3+ 21 11 1 -2 1 2 (a) Compute f(a) for each of a = -2, -1,0,1,2. (b) Determine lim f(x) and lim f(x) for each of a = -2,-1,0,1,2. (c) Determine lim f(a) for each of a = -2,-1,0,1,2. If the limit fails...
4. Let f be a differentiable function defined on (0, 1) whose derivative is f'(c) =...
4. Let f be a differentiable function defined on (0, 1) whose derivative is f'(c) = 1 - cos (+) [Note that we can confidently say such an f exists by the FTC.) Prove that f is strictly increasing on (0,1). 5. Let f be defined on [0, 1] by the following formula: 1 x = 1/n (n € N) 0, otherwise (a) Prove that f has an infinite number of discontinuities in [0,1]. (b) Prove that f is nonetheless...
(20 points) Sketch the graph of the function f(x) which satisfies the following conditions. Using interval...
(20 points) Sketch the graph of the function f(x) which satisfies the following conditions. Using interval notation list all intervals where the function fis decreasing, increasing, concave up, and concave down List the x-coordinates of all local maxima and minima, and points of inflection. Show asymptotes with dashed lines and give their equations. Label all important points on the graph. 1 a f(x) is defined for all real numbers b. f'(x) = c. f"(x) = (x-1) d. f(2)= 2 e...
Sketch a graph on the right side of the problem of a single function that has these properties. 5) (a) defined fo...
Sketch a graph on the right side of the problem of a single function that has these properties. 5) (a) defined for all real numbers (b) increasing on (-3,-1) and (2, oo) (c) f '(x) < 0 on (-00-3) and (-1,2) (d)f"x)>0 on (0, 00) (e) concave down on (-oo, -3), (-3, o) 6) (a) defined for all real numbers (b) increasing on (-3, 3) (c) decreasing on (, -3) and (3, ) (d) fix) <0 on (0, (e) f(x)>...
Suppose that the piecewise function J is defined by f(2)= {**** -1<<3 - 3x2 + 2x + 23, 2> 3 Determine which of the following statements are true. Select the correct answer below: O f() is not continuous at I = 3 because it is not defined at I = 3. Of() is not continuous at 2 = 3 because lim f(x) does not exist. f() is not continuous at I = 3 because lim f() f(3). ->3 f(x) is...
(1 point) For the function f(x) = e2x + e- defined on the interval (-4, o), find all intervals where the function is strictly increasing or strictly decreasing. Your intervals should be as large as possible. f is strictly increasing on f is strictly decreasing on (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10)) whenever r is near c on the left Find and classify all local max's and min's. (For the purposes...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
Problem 5 (10 points). Consider the function f(x) = L. (1-4e+") dt, defined in (-3, +00). Find the local max/min of f(x).
4 -2 2. The function f is defined on the closed interval [-4,9]. The graph of f consists of a semicircle, a quarter circle, and three linear segments, as shown in the figure above. Let g be the function defined by g(x) = 3x + f(t) dt. (a) Find g(8) and g'(8). (b) Find the value of x in the closed interval (-4,9] at which g attains its maximum value. Justify your answer. (c) Find lim f'(x), or state that...
Sketch the graph of the function f(r) with the following characteristics: lim f(x) = -00 lim f.) = -1 -2 0-0 lim f(x) =0 lim f(3) = -1 lim f (r) = 2 1-2
1. Suppose the a function g(x) is defined according to the formula f(c) 3(x + 2) +2 for – 3 <x< -2 (x+2)+1 for-2<x< -1 (+2)+1 for - 1<x<1 2 for r=1 for > 1 y 3+ 21 11 1 -2 1 2 (a) Compute f(a) for each of a = -2, -1,0,1,2. (b) Determine lim f(x) and lim f(x) for each of a = -2,-1,0,1,2. (c) Determine lim f(a) for each of a = -2,-1,0,1,2. If the limit fails...
4. Let f be a differentiable function defined on (0, 1) whose derivative is f'(c) = 1 - cos (+) [Note that we can confidently say such an f exists by the FTC.) Prove that f is strictly increasing on (0,1). 5. Let f be defined on [0, 1] by the following formula: 1 x = 1/n (n € N) 0, otherwise (a) Prove that f has an infinite number of discontinuities in [0,1]. (b) Prove that f is nonetheless...
(20 points) Sketch the graph of the function f(x) which satisfies the following conditions. Using interval notation list all intervals where the function fis decreasing, increasing, concave up, and concave down List the x-coordinates of all local maxima and minima, and points of inflection. Show asymptotes with dashed lines and give their equations. Label all important points on the graph. 1 a f(x) is defined for all real numbers b. f'(x) = c. f"(x) = (x-1) d. f(2)= 2 e...
Sketch a graph on the right side of the problem of a single function that has these properties. 5) (a) defined for all real numbers (b) increasing on (-3,-1) and (2, oo) (c) f '(x) < 0 on (-00-3) and (-1,2) (d)f"x)>0 on (0, 00) (e) concave down on (-oo, -3), (-3, o) 6) (a) defined for all real numbers (b) increasing on (-3, 3) (c) decreasing on (, -3) and (3, ) (d) fix) <0 on (0, (e) f(x)>...
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