🔬 Tutorial problems epsilon \epsilon

🔬 Tutorial problems epsilon \(\epsilon\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\epsilon\).1#

Consider the function

\[\begin{split} f(x)=\left\{\begin{array}{l} \frac{1}{x-1} \text { if } x \neq 1 \\ 0 \text { if } x=1 \end{array}\right. \end{split}\]

(a) Does \(\lim _{x \rightarrow 5} f(x)\) exist? If so, what is it? Establish the validity of your answer formally using an epsilon-delta argument. If it exists, does it equal \(f(5)\) ? Is this function continuous at the point \(x=5\) ?

(b) Does \(\lim _{x \rightarrow 1} f(x)\) exist? If so, what is it? Establish the validity of your answer formally using an epsilon-delta argument. If it exists, does it equal \(f(1)\) ? Is this function continuous at the point \(x=1\) ?

(c) Is this function continuous on \(\mathbb{R}\)? Justify your answer.

Use examples in the lecture notes to establish \(\epsilon\)-\(\delta\) argument.

\(\epsilon\).2#

Consider the function

\[\begin{split} f(x)=\left\{\begin{array}{l} x^{2} \text { if } x \neq 0 \\ -50 \text { if } x=0 \end{array}\right. \end{split}\]

(a) Does \(\lim _{x \rightarrow 0} f(x)\) exist? If so, what is it? Try and establish the validity of your answer formally using an epsilon-delta argument. If it exists, does it equal \(f(0)\) ? Is this function continuous at the point \(x=0\) ?

(b) Is this function continuous?

\(\epsilon\).3#

Consider the function

\[\begin{split} f(x)=\left\{\begin{array}{l} x^{2}+1 \text { if } x<2 \\ x+3 \text { if } x \geqslant 2 \end{array}\right. \end{split}\]

(a) Is this function continuous at the point \(x=2\) ? Justify your answer.

(b) Is this function continuous? Justify your answer.

\(\epsilon\).4#

Use either the product rule or the quotient rule to find the first derivative of each of the following functions.

(a) \(y=x^{3}\left(x^{2}+4\right)\);

(b) \(y=(x-3)\left(x^{2}-5 x+7\right)\);

(c) \(y=\frac{\left(x^{3}+6 x^{2}-2\right)}{x^{4}}\);

(d) \(y=\left(x^{4}+3 x\right) x^{-6}\);

(e) \(y=5+x e^{x}\);

(f) \(y=x^{2} e^{x}+x \ln (x)\);

(g) \(y=x^{2}(1.1)^{x}\);

(h) \(y=5 e^{x} \ln (x)\);

(i) \(y=\left(x^{3}+2\right)(1.2)^{x}\); and

(j) \(y=x^{-\left(\frac{5}{6}\right)} e^{x}\).

[Shannon, 1995], p. 401

\(\epsilon\).5#

Use the chain rule to find the first derivative of each of the following functions.

(a) \(y=(3 x+4)^{3}\);

(b) \(y=\left(2 x^{2}+6\right)^{4}\);

(c) \(y=(3-2 x)^{-2}\);

(d) \(y=\left(4+\ln \left(x^{2}\right)\right)^{-1}\);

(e) \(y=5+e^{x^{2}}\);

(f) \(y=10 x+e^{\ln (x)}\);

(g) \(y=\ln \left(x+x^{\frac{3}{2}}\right)\);

(h) \(y=\frac{10}{\left(1-5 e^{0.2 x}\right)}\);

(i) \(y=\sqrt{\frac{8}{x^{-1}}}\); and

(j) \(y=100-5 e^{-0.3 x}\).

[Shannon, 1995], p. 401

\(\epsilon\).6#

Consider a function \(f: X \longrightarrow \mathbb{R}\) where \(X \subseteq \mathbb{R}\). Suppose that \(\lim _{h \rightarrow 0}\left(\frac{f(a+h)-f(a)}{h}\right)\) exists. Show that this implies that \(f(x)\) is continuous at the point \(a \in X\).

\(\epsilon\).7#

Approximate the function

\[ f(x) = \exp\left(\frac{1}{x}\right) + \ln(x) \]

around \(x = 1\) using the Taylor series.

Provide:

  • The linear approximation

  • The quadratic approximation

  • The cubic approximation

Use each approximation to estimate the value of \(f(0)\). Is this an accurate estimate?

In other words, use as many terms of Tayor series to build an approximation by a linear function, quadratic and cubic polynomials.

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