Math 104: Homework 8 (due April 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 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

f(x) = \{ \begin{matrix}1 − |x − 1| & if 0 ≤ x ≤ 2 \\ 0 & if x > 2 \end{matrix}

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 2f(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(x^{2}) and f(x)^{2}.

2. Consider the sequence of functions

f_{n}(x) = \frac{nx^{2}}{1 + nx^{2}}

defined on the interval [0, ∞).

(a) Begin by considering f_{1}(x). How does it behave as x → ∞? How does it look close to x = 0? Use these facts to draw f_{1}(x).

(b) Show that f_{n}(x) = f_{1}(\sqrt{n}x). By considering question 1(a), use this fact to draw several of the f_{n}(x).

†If you try Gnuplot, you can define this function by typing f(x)=x>2?0:1-abs(x-1). It can then be plotted with plot [0:4] f(x).

(c) It can be shown that f_{n} converges pointwise to a function f defined on [0, ∞)

as

f(x) = \{ \begin{matrix}0 & if x = 0 \\ 1 & if x > 0. \end{matrix}

Draw f(x) and draw a strip of width ϵ = ^{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 f_{n} does not converge uniformly to f.

3. Consider the sequence of functions defined on as

f_{n}(x) = \{ \begin{matrix}x^{n} sin \frac{1}{x} & if x ≠ 0 \\ 0 & if x = 0 \end{matrix}

(a) Draw the sequence of functions f_{0}(x), f_{1}(x) and f_{2}(x). Which of the functions are continuous at x = 0? Which of them are differentiable at x = 0?

(b) Consider the functions f_{n} on the interval [−^{1}/2,^{1}/2], and define f(x) = 0. By considering a strip of width ϵ around f(x), explain why f_{n} will converge uniformly to f on this interval.

4. Plot the functions

• f_{1}(x) = x^{2}(x − 1)(x − 2)

• f_{2}(x) = |f_{1}(x)|

• f_{3}(x) = \frac{x}{1+x^{2}}

• f_{4}(x) = |x| + |x − 2|

• f_{5}(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 g_{0}(x) = |x| on . For n ∈ , define g_{n}(x) = |g_{n−1} (x) − 2^{1−n} |.

(a) Draw g_{0}(x), g_{1}(x), g_{2}(x), and g_{3}(x).

(b) Optional for the enthusiasts. Prove that the functions g_{n} converge uniformly to a limit g on .

Part 2: Additional exercises

6. Ross exercise 28.3

7. Ross exercise 28.8