Question g/h/i Consider the function f : R → R defined by(b)(a)By using definition, explain whether f is differentiable on R. Deducethe continuity of f on R.(c)(d)Evaluatewith the partitionLet f' be the first derivative of f computed in (a). Is is Riemannintegrable on [-1, 1]? Explain.1f'(2) Use the formulae(e).(f)(g)(h)(i)-∞.f(x) =2 1f'(x)Hint: (1) Each (sub)interval is given bynΣ1Pn = {1, fill in, the blankWIdx by using the definition of Riemann integral1(n + i - 1)²,2}, ne N.fill in, the blank ], 1≤i≤n.1732n 8n² 48n³++0n.5n11375 '' ਫੈਨ ਲੱਦ ਲੰਬਰ ()Σ++2n 8n² 48n3+0(n + i)²i=1Determine the range f(A) where A = (a, 0] for some fixed a < 0, a ‡Determine the supremum sup(ƒ(A)\{0}), if exists.Is f(A) open in R? Explain.Is f(A) connected? Explain.Is f(A) compact? Explain.XnFor each nN, consider n = with x € R+ = {x € R | x > 0}.Define the sequence {gn} by gn(x) = f(xn), x € R+. Find the limit functionof {n} if exists.

Basic concepts and definition of compactness, connectness and limits of a function will be used in…

Answered: (g). (h) (i) Is f(A) connected?…

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

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
Assume that f : (0, 1) → R is a differentiable function with the derivative f' increasing on (0, 1). Prove that f' is continuous on (0, 1).
Let f(t) = et^2 be an integrable function defined on the closed interval [-x, x] on ℝ. If F is the anti-derivative of f on [-x, x], prove that F'(x) = f(x) for all x in [-x, x].
Show that f(z) =xy+iy is everywhere continuous but is not analytic
Let ƒ(x, y) =(x2-y2)/(x2+y2) for(x, y) ≠ (0, 0). Is it possible to define ƒ(0, 0) in a way that makes ƒ continuous at the origin? Why
Assume the second derivatives of ƒ are continuous throughout the xy-plane and ƒx(0, 0) = ƒy(0, 0) = 0. Use the given information and the Second Derivative Test to determine whether ƒ has a local minimum, a local maximum, or a saddle point at (0, 0), or state that the test is inconclusive. ƒxx(0, 0) = -9, ƒyy(0, 0) = -4, and ƒxy(0, 0) = -6
Consider the function f defined on [0,∞), f(x)=(x^r)sin(1/x), for x≠0 and f(x)= 0, where r>0. Determine the range of r in which a) f is continuous on [0,∞), b) f is differentiable on [0,∞), c) f' exits and is differentiable on [0,∞).
Show that f(x,y)=exyx is differentiable at point (1,0) using the definition of differentiability.
Let f(x, y) = { cosy. sinx, x ≠ 0cosy, x = 0Is f continuous at (0,0)? Is f continuous everywhere?
Suppose that a nonnegativefunction y = ƒ(x) has a continuous first derivative on [a, b] . LetC be the boundary of the region in the xy-plane that is bounded below by the x-axis, above by the graph of ƒ, and on the sides by the lines x = a and x = b. Show that ∫ƒ(x) dx = - ∮C y dx.
Let f(x, y) = x2y4 sin(1/ (sqrt(x2+y2)) where (x, y) cannot be (0, 0) and f(0, 0) = 0 Is f continuous in origo?
Suppose that a differentiable function ƒ(x, y) has the constant value c along the differentiable curve x = g(t), y = h(t); that is, ƒ(g(t), h(t)) = c for all values of t. Differentiate both sides of this equation with respect to t to show that ∇ƒ is orthogonal to the curve’s tangent vector at every point on the curve.

SEE MORE QUESTIONS

Recommended textbooks for you

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

SEE MORE TEXTBOOKS

(g). (h) (i) Is f(A) connected? Explain. Is f(A) compact? Explain. with x € R+ = {x € R | x>0}. n For each nN, consider n = Define the sequence {gn} by gn(x) = f(xn), x R4. Find the limit function of {gn} if exists.