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I have an assignment where I should determine $a$ and $b$ so that the following function is continuous at $x=0$:

$$f(x)=\begin{cases} 2+\ln(1+x), & x>0\\ x^2+ax+b, & x\le 0 \end{cases}$$

I can do that just by setting $x = 0$ and calculating $$ 2+\ln(1+x)=x^2+ax+b$$

The answer is that $b=2$ and $a$ could be anything. Here comes the second part of the assingment, which I do not understand:

Determine, by calculating left and right limit of the differential quotient, all the values on the real parameters $a$ and $b$ so that $f$ is differentiable at the point $x=0$.

How do I do that, and what is the differential quotient? And what do I google if I want to search for this kind of problems?

egreg
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rablentain
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2 Answers2

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You want to determine when $\displaystyle f^{\prime}(0)=\lim_{h\to 0}\frac{f(0+h)-f(0)}{h}$ exists by evaluating the one-sided limits

$\displaystyle\lim_{h\to 0^{+}}\frac{f(0+h)-f(0)}{h}$ and $\displaystyle\lim_{h\to 0^{-}}\frac{f(0+h)-f(0)}{h}$ and seeing when these are equal.

user84413
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  • And can I do that by doing this: $lim_{x\to0}\frac{d}{dx}(2+\ln(1+x))=lim_{x\to0}\frac{d}{dx}(x^2+ax+b)$? – rablentain Aug 12 '14 at 19:44
  • You want to use $\frac{f(0+h)-f(0)}{h}=\frac{2+\ln(1+h)-b}{h}$ when $h>0$, and $\frac{f(0+h)-f(0)}{h}=\frac{(h^2+ah+b)-b}{h}$ when $h<0$. – user84413 Aug 12 '14 at 20:15
  • Is it just a coincidence that I get the right answer with my calculation then? – rablentain Aug 13 '14 at 06:26
  • @theva Your calculation will usually give the right answer. You have to be careful, though: if $f(x)=x^2+1$ for $x<0$ and $f(x)=x^3$ for $x\ge0$, you might be led by your method to conclude that $f^{\prime}(0)=0$. – user84413 Aug 13 '14 at 20:50
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The differential quotient is most likely the difference quotient: $\frac{y_2 - y_1}{x_2 - x_1}$.

The problem wants you to find $a$ and $b$ so that the function is differentiable at $x = 0$. In other words, you want find $a$ and $b$ so that $\lim_{h\rightarrow 0^-}\frac{f(0 + h) - f(0)}{h} = \lim_{h\rightarrow 0^+}\frac{f(0 + h) - f(0)}{h}$.

As for what to google, try "piecewise differentiability".

blakeoft
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