Draw the sequence of functions

Assignment Help Mathematics
Reference no: EM131081571

Math 104: Homework 8- Curve sketching

At this point in the class, being able to sketch functions and determine their properties is an important skill, which will greatly help in understanding concepts such as continuity, differentiability, and uniform convergence. Therefore, this week's homework is mainly devoted to this topic.

In the following questions, no detailed proofs are required, although you will need to provide some discussion in words about what is going on. To begin, I would like you to try and draw the graphs by hand. There are many ways to do this, such as looking at the behavior as x → ±∞, calculating a few specific points and drawing a line through them, using calculus, or searching for zeroes of the function.

After this, you can confirm your results using a plotting program. There are many free ones available, such as Gnuplot (www.gnuplot.info), which runs on Windows, Mac, and Linux.

1. Consider the function

1144_Figure.png

defined on the interval [0, ∞). Draw f(x).

(a) Draw f(x/2), f(x/3), and f(x/4), and explain how the shapes of these curves relate to f(x).

(b) Draw 2 f(x), f(x + 1/2), f(x) - 1/2 and explain how the shapes of these curves relate to f(x).

(c) Draw | f(x) - 1/2|. Is this function continuous? Is it differentiable everywhere?

(d) Draw f(x2) and f(x)2.

2. Consider the sequence of functions

fn(x) = nx2/1 + nx2

defined on the interval [0, ∞).

(a) Begin by considering f1(x). How does it behave as x → ∞? How does it look close to x = 0? Use these facts to draw f1(x).

(b) Show that fn(x) = f1(√nx). By considering question 1(a), use this fact to draw several of the fn(x).

(c) It can be shown that fn converges pointwise to a function f defined on [0, ∞) as

311_Figure1.png

Draw f(x) and draw a strip of width ε = 1/4 around f(x). If fn → f uniformly, then there exists an N such that n > N implies that fn lies wholly within this strip. Use the graph to explain in words why no such N exists, so that fn does not converge uniformly to f.

3. Consider the sequence of functions defined on R as

916_Figure2.png

(a) Draw the sequence of functions f0(x), f1(x) and f2(x). Which of the functions are continuous at x = 0? Which of them are differentiable at x = 0?

(b) Consider the functions fn on the interval [-1/2, 1/2], and define f(x) = 0. By considering a strip of width e around f(x), explain why fn will converge uniformly to f on this interval.

4. Plot the functions

  • f1(x) = x2(x - 1)(x - 2)
  • f2(x) = |f1(x)|
  • f3(x) = x/1+x2
  • f4(x) = |x| + |x - 2|
  • f5(x) = |x| - 2|x - 1| + |x - 2|

For each function, write down any values of x where it is not differentiable.

5. Consider the function g0(x) = |x| on R. For n ∈ N, define gn(x) = |gn-1(x) -21-n|.

(a) Draw g0(x), g1(x), g2(x), and g3(x).

(b) Optional for the enthusiasts. Prove that the functions gn converge uniformly to a limit g on R.

Reference no: EM131081571

Questions Cloud

Calculate company after-tax weighted average cost of capital : During the past three years, Tysseland Communications Limited (TCL) has been constrained by the high cost of capital to fund for many of its investments. Recently, capital costs have been declining steadily causing the company to consider a major pro..
Do you think these costs should be included in gdp : Do you think these costs should be included in GDP?
Generate perpetual after-tax cash flows : A firm is considering a project that will generate perpetual after-tax cash flows of $19,000 per year beginning next year. The project has the same risk as the firm’s overall operations and must be financed externally. Equity flotation costs 16 perce..
Operating leverage at financial break-even level of output : A project has the following data: Price/Unit=$45. Variable Cost/Unit=$36. Fixed costs=$20,000. Required return-8%. Initial investment=$100,000. Life=4 yrs. Ignoring tax effects, what are the following: acct break-even; cash break even; financial brea..
Draw the sequence of functions : Math 104: Homework 8. Draw the sequence of functions f0(x), f1(x) and f2(x). Which of the functions are continuous at x = 0? Which of them are differentiable at x = 0
Mutual funds are such popular investment vehicles : Why do you think mutual funds are such popular investment vehicles? Why doesn’t management simply diversify its operations? Isn’t that what management is supposed to do?
Should the old riveting machine be replaced by the new one : Mississippi River Shipyards is considering the replacement of an 8-year-old riveting machine with a new one that will increase earnings before depreciation from $24,000 to $48,000 per year. The new machine will cost $82,500, and it will have an estim..
Discuss how widely-held beliefs about health and wellness : Discuss how widely-held beliefs about health and wellness in that culture may contrast or compare with beliefs in your own culture.
Consider each transaction separately,not cumulatively : Consider each transaction separately,not cumulatively.

Reviews

Write a Review

Mathematics Questions & Answers

  Questions on ferris wheel

Prepare a Flexible Budget Gator Divers is a company that provides diving services such as underwater ship repairs to clients in the Tampa Bay area.

  Logistic map

This assignment has two question related to maths. Questions are related to bifurcation cascade and logistic map.

  Finding the probability of cards

This assignment has questions related to probabiltiy.

  Systems of ode

Find all the xed points, and study their stability and Draw the phase portrait of the system, as well as the graphs of the solutions in all relevant cases.

  Derive the boolean expression

Derive the Boolean Expression and construct the switching circuit for the truth table stated

  System of equations

Evaluate which equations are under-identified, just-identified, and over-identified.

  Linear programming problem

Linear programming problem consisting of only two constraints with one objective function.

  Find the natural domain

Find the natural domain of the given functions.

  Introduction to numerical methods

Compute the coecients of the polynomials using the term recurrence relation.

  Chart of the topological manifold

De?nition of smoothness of functions on a smooth manifold is chart independent and hence geometric.

  Mathematics in computing

Questions related on mathematics in computing.

  Complex problems

Complex problems

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd