Exercise · ZW-MEAN-VALUE-THEOREM-001 · ★☆☆ Basic
Find the value c guaranteed by the mean value theorem for \(f(x)=x^{2}\) on \([1,3]\).
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- D. \(2\)
Explanation
The slope of the secant is \(\dfrac{f(3)-f(1)}{3-1}=4\). Solve \(f'(c)=4\) with \(f'(x)=2 x\) and the solution inside \((1,3)\) is c = 2.
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-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].
- ZW-MEAN-VALUE-THEOREM-009 If f'(x)=0 for all x in (a,b) and f is continuous on [a,b], what can be said about f?