AP Calculus BC / Unit 5: Analytical Applications of Differentiation
5.1 Using the Mean Value Theorem
Notes
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $c$ in $(a,b)$ has $$f'(c)=\frac{f(b)-f(a)}{b-a}$$
- The tangent slope equals the secant slope somewhere.
Practice
For $f(x)=x^2$ on $[0,4]$, the MVT value $c$ is:
Secant slope 4; $2c=4$.
Which function fails the MVT hypotheses on $[-1,1]$?
It isn't differentiable at 0.
A car's position goes from 0 to 120 miles in 2 hours. The MVT says:
Instantaneous rate equals the average rate somewhere.
Ready for more? Take the Unit 5 test.