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 rac{f(b) - f(c)}{b - c}

Choose points closest to $a$.

Worked Example

f'(3) approx rac{f(5) - f(1)}{5 - 1} = rac{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:

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

where $Delta x_i$ varies!

Table Integrals 🎯

Key Takeaways — Part 2

1. Watch for unequal subintervals — multiply each value by its own $\Delta x$
2. Trapezoidal: average the endpoints of each subinterval

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 $c in (1, 4)$ where f'(c) = rac{12 - 3}{4 - 1} = 3.

AP Tip: You MUST cite "by the Mean Value Theorem" and verify the hypotheses (continuous + differentiable).

MVT with Tables 🎯

$f$ is continuous and differentiable. $f(2) = 5$, $f(6) = 17$.

Key Takeaways — Part 3

1. MVT + tables is a very common AP pattern
2. Always state the theorem by name and verify conditions

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. $f(1) = -2$, $f(3) = 4$, $f(5) = 1$, $f(7) = 6$.

Key Takeaways — Part 4

1. Look for the target value between consecutive $f$-values
2. The value must be between $f(a)$ and $f(b)$ to apply IVT

Working with Tables & Data

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

Reading $f'$ from a Table of $f$

If $f$ values go from 3 to 7, $f$ is increasing ($f' > 0$).

If $f$ values change rapidly, $f'$ is large.

Second Derivative from Tables

$f''$ tells us about concavity. If $f'$ is increasing, $f'' > 0$.

Interpreting Data 🎯

Key Takeaways — Part 5

1. Increasing differences → concave up
2. Decreasing differences → concave down

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 🎯

$g$ is twice-differentiable. $g(0) = 1$, $g(2) = 5$, $g(4) = 4$, $g(6) = 10$.

Tables & Data — Complete! ✅