🔬 Tutorial problems 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 are for additional practice. The symbol 🍹 indicates additional problems.
\(\epsilon\).1#
Consider the function
(a) Does \(\lim _{x \rightarrow 5} f(x)\) 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 \(f(5)\) ? Is this function continuous at the point \(x=5\) ?
(b) Does \(\lim _{x \rightarrow 1} f(x)\) 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 \(f(1)\) ? Is this function continuous at the point \(x=1\) ?
(c) Is this function continuous?
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\(\epsilon\).2#
Consider the function
(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?
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\(\epsilon\).3#
Find the first derivative of each of the following functions.
(a) \(y=4+2 x\);
(b) \(y=4+2 x+x^{2}\);
(c) \(y=4-\sqrt{x}\);
(d) \(y=8+x^{\frac{5}{2}}-x^{-\left(\frac{3}{2}\right)}\);
(e) \(y=x^{4}+2 e^{x}\);
(f) \(y=10 \ln (x)\);
(g) \(C=10+8 Q+\frac{2}{Q}\);
(h) \(R=10-5 Q+Q^{\frac{3}{2}}\);
(i) \(\Pi=-5+10 Q+0.5 Q^{3}\); and
(j) \(y=(1.08)^{x}+10\).
[Shannon, 1995], p. 401
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\(\epsilon\).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
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\(\epsilon\).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
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\(\epsilon\).6 🍹#
Consider the function
(a) Is this function continuous at the point \(x=2\) ? Justify your answer.
(b) Is this function continuous? Justify your answer.
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\(\epsilon\).7 🍹#
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 \(a \in X\).
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\(\epsilon\).8 🍹#
Find the ownprice elasticity of demand for each of the following demand or inverse demand functions. If possible, find the price or prices for which the demand curves will: (i) be inelastic, (ii) have unitary elasticity and (iii) be elastic.
(a) \(P=100-2 Q\);
(b) \(Q=200-0.8 P\);
(c) \(P=100 Q^{-1}\);
(d) \(Q=200 P^{-0.8}\);
(e) \(P=50 e^{-0.7 Q}\); and
(f) \(Q=\frac{150}{\ln (P)}\).
[Shannon, 1995], p. 404
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