Exercise · ZW-MEAN-VALUE-THEOREM-014 · ★★☆ Intermediate
For \(f(x)=e^x\) on \([0,1]\), which \(c\) satisfies \(f'(c)=f(1)-f(0)\)?
Choose one option.
Show answer
- C. \(c=\ln(e-1)\)
Explanation
We need \(e^c=e-1\), so \(c=\ln(e-1)\approx0.541\), which lies in \((0,1)\).
Review the concept: Mean value theorem
If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), there is a \(c\in(a,b)\) such that the slope of the tangent at \(c\) equals the slope of the secant line joining \((a,f(a))\) and \((b,f(b))\).
\[f'(c)=\frac{f(b)-f(a)}{b-a}\]
More practice problems: mean value theorem
- ZW-MEAN-VALUE-THEOREM-001 Find the value c guaranteed by the mean value theorem for f(x)=x^2 on [1,3].
- ZW-MEAN-VALUE-THEOREM-002 Find the value c guaranteed by the mean value theorem for f(x)=x^3 on [0,2].
- ZW-MEAN-VALUE-THEOREM-003 Find the value c guaranteed by the mean value theorem for f(x)=√x on [1,4].
- ZW-MEAN-VALUE-THEOREM-004 Which hypothesis does the mean value theorem require for f on [a,b]?
- ZW-MEAN-VALUE-THEOREM-005 Compute the slope of the secant line of f(x)=x^3 between x=1 and x=2.
- ZW-MEAN-VALUE-THEOREM-006 According to the mean value theorem, what slope must the tangent at the point c have?
- ZW-MEAN-VALUE-THEOREM-007 Find the value c guaranteed by the mean value theorem for f(x)=x^2-4x on [0,4].
- ZW-MEAN-VALUE-THEOREM-008 Find the value c guaranteed by the mean value theorem for f(x)=x^3-3x on [0,2].