Define the sequence of functions on 0 1 as h0x x -12 and

Question

Math 104: Homework 8-

Part 1: Curve sketching

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

For questions 1-3, 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

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

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

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

Draw f(x) and draw a strip of width e = 1/4 around f(x). If f_n → f uniformly, then there exists an N such that n > N implies that f_n 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

(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. Consider the function g₀(x) = |x| on R. For n ∈ N, define g_n(x) = |g_n-1(x) -2^1-n|.

(a) Draw g₀(x), g₁(x), g₂(x), and g₃(x).0

(b) Prove that the functions g_n converge uniformly to a limit g on R.

Part 2: a choice of question

In this part, you can either do question 5 or 6. Question 5 is based on the last midterm problem, and is good practice if you had trouble with this. If you felt confident with the midterm problem, you can try question 6 instead.

5. (a) Consider the function f(x) = 1 - x on (0, 1) and define a sequence of functions as

Sketch f₁(x), f₂(x), and f₃(x), and prove that f_n does not converge uniformly to f.

(b) Now suppose that f(x) = x - x² instead. Sketch f₁(x), f₂(x), and f₃(x) and prove that in this case fn converges uniformly to f.

6. Define the sequence of functions on [0, 1] as h₀(x) = |x -1/2| and h_n(x) = |h_n-1(x) - 3^-n|. Draw several of the h_n. Prove that h_n converges uniformly to a limit h. Where is h differentiable?