Answers to Calculus Practical 7

Question 1 §

$$f(x)=x\ln x,f^{\prime }(x)=1+\ln x,f^{\prime \prime }(x)=\frac{1}{x}$$

$$\text{Domain}:x>0,\text{no real values for }\ln x\le 0$$

$$\text{x,y-intersects}:(1,0)$$

$$\text{Odd, even, neither}:\text{Neither}$$

$$\text{Asymptotes}:\text{No horizontal }(x\to \infty :y\to \infty ),\text{no vertical because }$$

$$\underset{x\to 0}{\lim }\left(\frac{\ln x}{x^{-1}}\right)=\underset{x\to 0}{\lim }\left(\frac{x^{-1}}{-x^{-2}}\right)\text{ by L’Hoˆpital’s rule}:\underset{x\to 0}{\lim }{x}^{-1}=-\underset{x\to 0}{\lim }{x}^{-2}=0$$

$$\therefore \text{no vertical asymptote as }x\to 0\text{, as }\frac{0}{0}\text{ is undefined.}$$

$$\text{Intervals of increase and decrease}:(e^{-1},\infty ),(0,e^{-1})$$

$$\text{Local maxima/minima}:f^{\prime }=0⟹(x,y)=(e^{-1},-e^{-1})$$

$$\text{Concavity}:\text{Concave up throughout the domain},\frac{1}{x}:x>0$$

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y=x*log(x)

Question 2 §

$$g(x)=\frac{2(x-2)}{x^2},g^{\prime }(x)=\frac{2(4-x)}{x^3},g^{\prime \prime }(x)=\frac{4(x-6)}{x^4}$$

$$\text{Domain}:x\ne 0$$

$$\text{x,y-intersects}:(2,0)$$

$$\text{Odd, even, neither}:\text{Neither}$$

$$\text{Asymptotes}:$$

$$\text{Intervals of increase and decrease}:$$

$$\text{Local maxima/minima}:$$

$$\text{Concavity}:$$

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Question 3 §

$$f(x)=,f^{\prime }(x),f^{\prime \prime }(x)$$

$$\text{Domain}:$$

$$\text{x,y-intersects}:$$

$$\text{Odd, even, neither}:$$

$$\text{Asymptotes}:$$

$$\text{Intervals of increase and decrease}:$$

$$\text{Local maxima/minima}:$$

$$\text{Concavity}:$$

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