4.4 The mean value theorem  (Page 5/7)

Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample.

Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample.

When are Rolle’s theorem and the Mean Value Theorem equivalent?

If you have a function with a discontinuity, is it still possible to have $f^{\prime }\left(c\right)\left(b-a\right)=f\left(b\right)-f\left(a\right)?$ Draw such an example or prove why not.

$y=\text{sin}\left(\pi x\right)$

$y=\frac{1}{x^3}$

$y=\sqrt{4-x^2}$

$y=\sqrt{x^2-4}$

$y=\text{ln}(3x-5)$

[T] $y=3x^3+2x+1$ over $[\mathrm{-1},1]$

[T] $y=\text{tan}\left(\frac{\pi }{4}x\right)$ over $\left[-\frac{3}{2},\frac{3}{2}\right]$

[T] $y=x^2\text{cos}\left(\pi x\right)$ over $[\mathrm{-2},2]$

[T] $y=x^6-\frac{3}{4}{x}^5-\frac{9}{8}{x}^4+\frac{15}{16}{x}^3+\frac{3}{32}{x}^2+\frac{3}{16}x+\frac{1}{32}$ over $[\mathrm{-1},1]$

$f(x)=x^3$

$f(x)=\text{sin}(\pi x)$

$f(x)=\text{cos}(2\pi x)$

$f(x)=1+x+x^2$

$f(x)={(x-1)}^{10}$

$f\left(x\right)={\left(x-1\right)}^9$

$f\left(x\right)=\left|x-\frac{1}{2}\right|$

$f(x)=\frac{1}{x^2}$

$f(x)=\sqrt{\left|x\right|}$

$f\left(x\right)=\lfloor x\rfloor$ ( Hint : This is called the floor function and it is defined so that $f(x)$ is the largest integer less than or equal to $x.)$

$y=e^x$ over $[0,1]$

$y=\text{ln}(2x+3)$ over $\left[-\frac{3}{2},0\right]$

$f(x)=\text{tan}(2\pi x)$ over $[0,2]$

$y=\sqrt{9-x^2}$ over $[\mathrm{-3},3]$

$y=\frac{1}{\left|x+1\right|}$ over $[0,3]$

$y=x^3+2x+1$ over $[0,6]$

$y=\frac{x^2+3x+2}{x}$ over $[\mathrm{-1},1]$

$y=\frac{x}{\text{sin}(\pi x)+1}$ over $[0,1]$

$y=\text{ln}(x+1)$ over $[0,e-1]$

$y=x\text{sin}(\pi x)$ over $[0,2]$

$y=5+\left|x\right|$ over $[\mathrm{-1},1]$

Show that the equation $y=x^3+3x^2+16$ has exactly one real root. What is it?

Find the conditions for exactly one root (double root) for the equation $y=x^2+bx+c$

Find the conditions for $y=e^x-b$ to have one root. Is it possible to have more than one root?

[T] $y=\text{tan}(\pi x)$ over $\left[-\frac{1}{4},\frac{1}{4}\right]$

[T] $y=\frac{1}{\sqrt{x+1}}$ over $[0,3]$

[T] $y=\left|x^2+2x-4\right|$ over $[\mathrm{-4},0]$

[T] $y=x+\frac{1}{x}$ over $\left[\frac{1}{2},4\right]$

[T] $y=\sqrt{x+1}+\frac{1}{x^2}$ over $[3,8]$

At 10:17 a.m., you pass a police car at 55 mph that is stopped on the freeway. You pass a second police car at 55 mph at 10:53 a.m., which is located 39 mi from the first police car. If the speed limit is 60 mph, can the police cite you for speeding?

Two cars drive from one spotlight to the next, leaving at the same time and arriving at the same time. Is there ever a time when they are going the same speed? Prove or disprove.

Show that $y={\text{sec}}^{2}x$ and $y={\text{tan}}^{2}x$ have the same derivative. What can you say about $y={\text{sec}}^{2}x-{\text{tan}}^{2}x?$

Show that $y={\text{csc}}^{2}x$ and $y={\text{cot}}^{2}x$ have the same derivative. What can you say about $y={\text{csc}}^{2}x-{\text{cot}}^{2}x?$