MATH114 Integration and Differentiation

Exercises Week 2

[2.5ex]

Workshop questions that you cannot solve during the workshop should be regarded as additional training material and solved at home. Afterwards please compare with the model solutions. There are two marks for every correctly answered quiz question. Submission deadline for written assessment: Wednesday 16:00 in your tutor’s homework box. Submission deadline for quizzes: Wednesday 23:59 on Moodle.

Workshop

• True or false? Think for 1 minute, then resolve the quiz and discuss briefly.

  • Upper and lower approximating step functions of a continuous function on a compact interval may converge to different limits.

  • A uniformly continuous function is automatically continuous.

  • A bounded function on a compact interval is automatically uniformly continuous.

• Prove Corollary 1.3.2. (Hint: you might like to use ideas from the proof of Proposition 1.3.1.)

• Compute ${\int }_0^2\sin (x)dx$ approximately using the integral of upper and lower step functions with up to $n=3$ bisections. Explain in detail what you are doing and what is happening. (Hint: a little picture might be helpful to start with and see what you are doing.)

• Given two convergent series ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ and ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n$ such that $a_n\le b_n$ for all $n\in \mathbb{N}$. Is it true that ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n\le {\sum }_{n=1}^{\mathrm{\infty }}{b}_n$? If yes, then prove it; if no then provide a counter-example.

Written Assessment

• Prove Proposition 2.1.4: If a series is convergent/divergent then so is any other series formed from it by altering only finitely many summands.

  Make sure to reference all statements/results from the lecture notes (or potential external resources) that you use in your proof. [4]

• Write an essay explaining the construction of the integral of a continuous function.

  Instruction: Make your own choice what exactly you would like to say. Use your own words, examples and illustrations to support the theory. Be mathematically correct and precise. 200-300 words, excluding mathematical symbols or pictures. Clearly reference all resources that you use. Handwritten answer is fully sufficient. You will be assessed as follows:

  • mathematical correctness [2]

  • originality of examples, illustrations [2]

  • mathematical style and presentation [2]

  Grammatical mistakes shall not affect your score as long as every sentence is understandable. Scores in each rubric: [2] - very good, only minor critique; [1] - ok, possibly several weaknesses; [0] - extremely poor or no response at all. Disobeying the word limit or the requested topic may lead to [0] for the whole essay.

Quiz

• Combining functions. Which of the following statements is false?

  • The sum of two piecewise continuous functions is piecewise continuous.

  • The sum of two uniformly continuous functions is uniformly continuous.

  • The product of two uniformly continuous functions is uniformly continuous.

  • The product of two piecewise continuous functions is piecewise continuous.

  • The product of two bounded functions is bounded.

• Maximum and Co. Given two continuous functions $f,g:I\to ℝ$ on a compact interval $I$. Which of the following relations is true?

  • ${\max }_{x\in I}(f(x)+g(x))\le {\max }_{x\in I}f(x)+{\max }_{x\in I}g(x)$.

  • ${\max }_{x\in I}(f(x)+g(x))\ge {\max }_{x\in I}f(x)+{\max }_{x\in I}g(x)$.

  • ${\max }_{x\in I}(f(x)+g(x))={\max }_{x\in I}f(x)+{\max }_{x\in I}g(x)$.

  • ${inf}_{x\in I}(f(x)+g(x))={inf}_{x\in I}f(x)+{inf}_{x\in I}g(x)$.

  • ${sup}_{x\in I}(f(x)+g(x))={sup}_{x\in I}f(x)+{sup}_{x\in I}g(x)$.

• Approximating step functions. Let $f:I\to ℝ$ be a continuous function and $I$ compact. Let ${\mathrm{\ell }}_n^{(f)}$ and $u_n^{(f)}$ be the integrals of the lower and upper, respectively, approximating step function of $f$ after $n$ bisections. Then which of the following statements is true?

  • ${({\mathrm{\ell }}_n^{(f)})}_{n\in \mathbb{N}}$ forms a decreasing sequence.

  • ${(u_n^{(f)})}_{n\in \mathbb{N}}$ forms an increasing sequence.

  • ${(u_n^{(f)})}_{n\in \mathbb{N}}$ and ${({\mathrm{\ell }}_n^{(f)})}_{n\in \mathbb{N}}$ converge to the same limit point.

  • ${(u_n^{(f)})}_{n\in \mathbb{N}}$ is constant and bounded below.

  • , for all $n\in \mathbb{N}$.

• Computing approximating step functions I. Consider the function $g:[-1,1]\to ℝ,g(x)=x^3.$ What are the values of the integrals of the approximating steps functions after $n=2$ bisections, ${\mathrm{\ell }}_2^{(g)}$ and $u_2^{(g)}$?

  • $-\frac{1}{2}$ and $\frac{1}{2}$.

  • $-\frac{1}{3}$ and $\frac{1}{3}$.

  • $-\frac{2}{3}$ and $\frac{2}{3}$.

  • $0$ and $\frac{2}{3}$.

  • $0$ and $0$.

• Computing approximating step functions II. Consider the function $f:[-\pi ,\pi ]\to ℝ,f(x)={\cos }^2(x).$ What are the values of the integrals of the approximating steps functions after $n=3$ bisections, ${\mathrm{\ell }}_3^{(f)}$ and $u_3^{(f)}$?

  • $\pi$ and $\pi$.

  • $0$ and $2\pi$.

  • $\frac{\pi }{2}$ and $\pi$.

  • $\frac{\pi }{2}$ and $\frac{3\pi }{2}$.

  • $\pi$ and $2\pi$.