Consider the function

(a) Does $\underset{x\to 5}{lim}f(x)$ 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 $\underset{x\to 1}{lim}f(x)$ 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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Consider the function

(a) Does $\underset{x\to 0}{lim}f(x)$ 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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Find the first derivative of each of the following functions.

(a) $y=4+2x$;

(b) $y=4+2x+x^2$;

(c) $y=4-\sqrt{x}$;

(d) $y=8+x^{\frac{5}{2}}-x^{-\left(\frac{3}{2}\right)}$;

(e) $y=x^4+2e^x$;

(f) $y=10\ln (x)$;

(g) $C=10+8Q+\frac{2}{Q}$;

(h) $R=10-5Q+Q^{\frac{3}{2}}$;

(i) $\mathrm{\Pi }=-5+10Q+0.5Q^3$; and

(j) $y=(1.08{)}^x+10$.

[Shannon, 1995], p. 401

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

(a) $y=x^3(x^2+4)$;

(b) $y=(x-3)(x^2-5x+7)$;

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

(d) $y=(x^4+3x)x^{-6}$;

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

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

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

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

(i) $y=(x^3+2)(1.2{)}^x$; and

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

[Shannon, 1995], p. 401

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

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

(b) $y={(2x^2+6)}^4$;

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

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

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

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

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

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

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

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

[Shannon, 1995], p. 401

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

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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) $P=100-2Q$;

(b) $Q=200-0.8P$;

(c) $P=100Q^{-1}$;

(d) $Q=200P^{-0.8}$;

(e) $P=50e^{-0.7Q}$; and

(f) $Q=\frac{150}{\ln (P)}$.

[Shannon, 1995], p. 404

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