Tables & Data

Working with Tables & Data

Part 1 of 7 — Approximating Derivatives from Tables

Estimating $f'(a)$ from a Table

When you have a table of values but no formula, estimate the derivative using:

$f'(a) approx \frac{f(b) - f(c)}{b - c}$

Choose points closest to $a$.

Worked Example

$f'(3) approx \frac{f(5) - f(1)}{5 - 1} = \frac{10 - 2}{4} = 2$

(Using symmetric difference gives better estimate than one-sided.)

Table Derivatives 🎯

Key Takeaways — Part 1

1. Use symmetric differences when possible
2. At endpoints, use one-sided differences
3. Always state units on the AP exam

Working with Tables & Data

Part 2 of 7 — Riemann Sums from Tables

Approximating Integrals from Data

When given a table with unequal subintervals, compute:

$int_a^b f(x),dx approx sum f(x_i^*) cdot Delta x_i$

Working with Tables & Data

Part 3 of 7 — MVT with Tables

Using MVT on Table Data

If $f$ is differentiable and the table shows:

Then by MVT, there exists $cin(1$ where .

Working with Tables & Data

Part 4 of 7 — IVT with Tables

Using IVT on Table Data

If $f$ is continuous and the table shows values, you can conclude that $f$ takes every value between consecutive table entries.

IVT with Tables 🎯

$f$ is continuous. $f(1) = -2$, , , .

Working with Tables & Data

Part 5 of 7 — Interpreting $f'$ from Tables of $f$ and Vice Versa

Reading $f'$ from a Table of $$

Working with Tables & Data

Part 6 of 7 — Practice Workshop

Table Workshop 🎯

Workshop Complete!

Tables & Data — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Tables & Data — Complete! ✅