MATH114 Integration and Differentiation

Exercises Week 1

[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

• State 3 substantially different examples of uniformly continuous functions and 3 substantially different examples of continuous but not uniformly continuous functions. Provide a precise definition of the functions, with domain and codomain etc., but no detailed proof required. They should not be taken from the lecture notes.

• Consider $h:(0,10)\to ℝ$, with $h(x)=x^3$. For given $x_0\in (0,10)$ and $\epsilon >0$, find $\delta >0$ in order to prove that $h$ is continuous at $x_0$. Can you choose $\delta$ depending only on $\epsilon$ but not on $x_0$?

• Define an interesting step function, draw its graph and compute the area under the graph.

• Prove Lemma 1.2.9.

Written Assessment

• Prove Lemma 1.2.10. [4]

• Consider a set $A\subset ℝ$ which need not be an interval.

  • What would be the best definition of an indicator function ${\mathrm{𝟏}}_A$ for the set $A$ and why? [1.5]

  • Provide a non-trivial example of such an $A$ and ${\mathrm{𝟏}}_A$ and draw the corresponding graph. [1.5]

  • How would you define the integral of ${\mathrm{𝟏}}_A$? Would you like to make special assumptions on $A$? Which ones and why? [2]

  • Use step (3) to compute the integral of your example function ${\mathrm{𝟏}}_A$ in (2), providing all details of your computation. [1]

Quiz

• Uniform continuity I. Which of the following defines a well-defined uniformly continuous function?

  • $f:[0,1]\to ℝ,f(x)=\frac{1}{x}$

  • $g:[0,\mathrm{\infty })\to ℝ,g(x)=x^2$

  • $h:[0,1]\to ℝ,h(x)=\sqrt{x}$

  • $i:[0,2]\to ℝ,i(x)=\tan (x)$

  • $j:ℝ\to ℝ,j(x)=x^2$

• $\epsilon$-$\delta$ criterion. Consider the function $f:(-10,10)\to ℝ$, defined by $f(x)=x^4$. Given $\epsilon >0$, which of the below is the greatest possible $\delta >0$ that we can choose in order to fulfill the definition of continuity of $f$ on all of $(-10,10)$?

  • $\delta =0$

  • $\delta ={\epsilon }^{1/4}$

  • $\delta =\frac{1}{4}{\epsilon }^4$

  • $\delta =\frac{\epsilon }{10000}$

  • $\delta =\frac{\epsilon }{4000}$

• Uniform continuity II. Which of the following statements is true?

  • Every uniformly continuous function is continuous.

  • Every continuous function is uniformly continuous.

  • Every differentiable function is uniformly continuous.

  • Every continuous function is a polynomial.

  • Every uniformly continuous function is a polynomial.

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

  • The sum of two step functions is a step function.

  • A step function is bounded.

  • A step function is continuous.

  • The product of two step functions is a step function.

  • The difference of two step functions is a step functions.

• Integral of step functions. What is the value of the integral of the step function

  $$f:[0,10]\to ℝ,f(x)=2.5\cdot {\mathrm{𝟏}}_{[0,2]}+\pi \cdot {\mathrm{𝟏}}_{[2,4]}+0.5\cdot {\mathrm{𝟏}}_{[4,8]}+10\cdot {\mathrm{𝟏}}_{[8,8.1]}\mathrm{?}$$

  • $6+\pi$

  • $3.5+\pi$

  • $3.5+2\pi$

  • $8+2\pi$

  • $6+4\pi$