AP Calculus BC / Unit 2: Differentiation: Definition and Fundamental Properties
2.3 Estimating Derivatives of a Function at a Point
Notes
With a table, estimate $f'(a)$ with the slope between nearby points, ideally on both sides of $a$ (a symmetric difference).
- $\displaystyle f'(a)\approx\frac{f(b)-f(c)}{b-c}$ for points $c<a<b$ close to $a$.
- From a graph, estimate the slope of the tangent line.
Practice
A table gives $f(2)=5$ and $f(4)=11$. The best estimate of $f'(3)$ is:
$\displaystyle \frac{11-5}{4-2}=3$.
To estimate $f'(a)$ from a table, you should use points that are:
Closer points give a better approximation of the tangent slope.
$t$ (hours): 0, 1, 3; $T(t)$ (°C): 20, 24, 30. Estimate $T'(2)$.
$\displaystyle \frac{30-24}{3-1}=3$.
Ready for more? Take the Unit 2 test.