3.Consider the sequences of functions f₁: [-π, π] → R,sin(n²x)An(2)nf pointwise as(i) Find a function ƒ : [-T,π] → R such that fnn∞. Further, show that fn →f uniformly on [-π,π] as n → ∞.[20 Marks](ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]?Justify your answer.[10 Marks]

Answered: Image /qna-images/answer/633f8bcd-0f64-4b5f-83f1-2209b001f01b.jpg

Answered: 3. Consider the sequences of functions…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

See similar textbooks

Similar questions

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.
What is the purpose of the Intermediate Value Theorem?
2. a. f(x) = (x+2)/(x +4). Use the Mean Value Theorem on the interval [0, 4] to find all values of x in the interval (0, 4) such that the value of the first derivative of f at those points is (f(b) – f(a))/(b – a). b. For the interval [-5, 0], fb) - fa))(b — а)-GG0) — f-5)(0 — (-5)) = (½ – 3)/(0 + 5) = -½ but, since the first derivative of f(x) is always positive, nowhere on that interval does the first derivative of f(x) = -½. Thus the Mean Value Theorem does not hold for f(x) on this interval. Why?
4.* (a) Let S = {1, 2, 3, 4}. Define F: SN by F(x) = x² for each x = S. What is the range of the function F and what is F(S)? How do these two sets compare? Now let A and B be sets and let f: A B be an arbitrary function from A to B. (b) Explain why f(A) = range(f). (c) Define a function g: A f(A) by g(x) = f(x) for all x in A. Prove that the function g is a surjection.
Let f be a function such that f(-1) = 14 and f(8) = 2. Which of the following conditions ensures that f(c) = 8 in the open interval (–1,8) ? -8 (A) f(x) dx exists (B) f' is defined for all values of x in the closed interval [-1,8] (C) f is defined for all values of x in the closed interval [-1,8]. (D) ƒ is decreasing on the closed interval [–1,8]. O A
3) Consider the following functions: f(x)=x√3-x; 3 g(x) = x³ - 6x² + 3; h(x) = = 2 x² - 9 .2 x² + 1 a) Use the limit definition of the derivative to find f'(a), g'(x), and h'(1). b) Why is h continuous at x = 1? Where are f and g continuous? Use part (a). c) Find all locations where the tangent line is horizontal for f and g. d) Find the equation for the tangent line for h at x = 1.
Which of the following is the correct limit definition for the derivative of g(x) = x² + x? O (x+h)² + (x+h) - (h² + h) lim h→0 O O O lim h→0 lim h→0 X (x + h)² + (x + h) - (x² + x) h lim h→0 (x+h)² + (x + h) - (h² + h) h (x+h)² + (x + h) - (x² + x) X
6. Sketch: f(x) = [x+ 2] -1, on [-2,2]
3. Consider the sequences of functions fn: [-T, π] → R, sin(n²x) n(2) n (i) Find a function f : [-T, π] R such that fnf pointwise as n∞. Further, show that f uniformly on [-T,π] as n→ ∞. [20 Marks] (ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7,π]? Justify your answer. [10 Marks]
Suppose the function f is continuous and has domain (-∞,00) and has all of the following characteristics. Use the information given to answer the questions that follow. f(0) = ft(5) = 0 f'(x) > 0 on the intervals (-o, 0) and (5, ∞) ft(x) 0 on the interval (1, co) f"(x) < 0 on the interval (-o, 1) a. On what interval(s) is f increasing? а. b. For what value(s) of x does f have a local maximum? b. c. On what interval(s) is ƒ concave up? с. d. For what values(s) of x does f have any inflection point(s)? d. e. Use the information to sketch the graph of f.
The graph of f is shown in the figure. 6. 4 х 14 -4 -6+ 2-
Let 1 f(x) = 2 9, - 2 < x < 5 - The domain of ƒ¯'is the interval [A, B| where A = and B =

SEE MORE QUESTIONS

Recommended textbooks for you

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

College Algebra (MindTap Course List)

Algebra

ISBN:

9781305652231

Author:

R. David Gustafson, Jeff Hughes

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS