Mean Value Theorem

The Mean Value Theorem says that if a function is continuous on a closed interval and differentiable on the open interval, then at least one tangent slope matches the average rate of change over the interval. In practice, this means we set \(f'(c)\) equal to the slope of the secant line from \(a\) to \(b\). These problems focus on finding all values of \(c\) that satisfy the conclusion of the Mean Value Theorem.

Notes

Notes of Mean Value Theorem

Practice Problems

Find all numbers c that satisfy the conclusions of the mean value theorem.

\(\textbf{1)}\) \( f(x)=x^2 \, \) on \( \, [-3,2] \)

 

\(\textbf{2)}\) \( f(x)=x^3+2x+4 \,\) on \( \, [0,2] \)

 

\(\textbf{3)}\) \( f(x)=\sin x \,\) on \( \, [0,\pi] \)

 

\(\textbf{4)}\) \( f(x)=x^2 \,\) on \( \, [-1,1] \)

 

\(\textbf{5)}\) \( f(x)=x^3 \) on \([1,3]\)

 

\(\textbf{6)}\) \( f(x)=x^2+4x \) on \([-2,4]\)

 

\(\textbf{7)}\) \( f(x)=x^3-3x \) on \([-2,2]\)

 

\(\textbf{8)}\) \( f(x)=\sqrt{x} \) on \([1,9]\)

 

\(\textbf{9)}\) \( f(x)=\ln{x} \) on \([1,e^2]\)

 

\(\textbf{10)}\) \( f(x)=e^x \) on \([0,\ln 4]\)

 

\(\textbf{11)}\) \( f(x)=\frac{1}{x} \) on \([1,4]\)

 

\(\textbf{12)}\) \( f(x)=x^4 \) on \([-1,1]\)

 

\(\textbf{13)}\) \( f(x)=x^2-6x+1 \) on \([0,5]\)

 

\(\textbf{14)}\) \( f(x)=\cos{x} \) on \([0,\pi]\)

 

\(\textbf{15)}\) \( f(x)=x+\frac{4}{x} \) on \([1,4]\)

 

\(\textbf{16)}\) \( f(x)=\sqrt{x+1} \) on \([0,8]\)

 

\(\textbf{17)}\) \( f(x)=x^3+x^2 \) on \([-1,2]\)

 

\(\textbf{18)}\) \( f(x)=x^3-6x^2+9x \) on \([0,4]\)

 

\(\textbf{19)}\) \( f(x)=\frac{x}{x+1} \) on \([1,3]\)

 

\(\textbf{20)}\) \( f(x)=x^{2/3} \) on \([-1,1]\)

 

See Related Pages\(\)

\(\bullet\text{Mean Value Theorem Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(emathhelp.net)}\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail of Graphs of First and Second Derivatives
\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

 

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