Exercise · ZW-DERIVATIVE-020 · ★★★ Advanced
Let \(f(x)=x^2\) if \(x\le1\) and \(f(x)=ax+b\) if \(x>1\). For which \(a,b\) does \(f'(1)\) exist?
Choose one option.
Show answer
- D. \(a=2,\ b=-1\)
Explanation
From the left the quotient tends to \(2\); from the right, for it to have a limit, we need \(a+b=1\) (joining at \(f(1)=1\)), and then it tends to \(a\). So \(a=2\) and \(a+b=1\), giving \(b=-1\). With \(a=1,\ b=0\) the pieces join but the slopes differ.
Review the concept: Derivative
The derivative of \(f\) at \(a\) is the limit of the difference quotient as \(h\to0\). It measures the instantaneous rate of change and is the slope of the tangent line at \((a,f(a))\).
\[f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\]
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