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Estimating Derivatives
at a Point

Learn how to approximate the instantaneous rate of change of a function using tables, graphs, and the limit definition of \( f'(a) \).

AP Calculus BC

What Is a Derivative?

The derivative of a function \( f \) at a point \( x = a \) measures the instantaneous rate of change: how fast \( f(x) \) is changing at that exact moment. Geometrically, it equals the slope of the tangent line to the curve at \( (a, f(a)) \).

When we cannot compute this limit exactly, we estimate \( f'(a) \) by choosing small values of \( h \) and computing the difference quotient.

Average vs Instantaneous Rate

Average Rate of Change
Slope of the secant line between two points: \(\displaystyle \frac{f(b) - f(a)}{b - a}\)
Instantaneous Rate of Change
Slope of the tangent line at one point: \(\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\)

From Secant to Tangent

x y a b secant b' secant tangent

The Difference Quotient

The core tool for estimating derivatives is the difference quotient. There are three forms, depending on how you choose points around \( a \).

Forward
\(\displaystyle \frac{f(a+h) - f(a)}{h}\)
Backward
\(\displaystyle \frac{f(a) - f(a-h)}{h}\)
Symmetric
\(\displaystyle \frac{f(a+h) - f(a-h)}{2h}\)

Comparing Estimation Methods

Forward Backward Symmetric
Formula \(\frac{f(a+h)-f(a)}{h}\) \(\frac{f(a)-f(a-h)}{h}\) \(\frac{f(a+h)-f(a-h)}{2h}\)
Points used \(a\) and \(a+h\) \(a-h\) and \(a\) \(a-h\) and \(a+h\)
Accuracy Good Good Best
Error order \(O(h)\) \(O(h)\) \(O(h^2)\)

Estimating from a Table

When given a table of values for \( f(x) \), pick the two data points closest to and on either side of \( a \), then apply the symmetric quotient.

\(x\) 1.0 1.5 2.0 2.5 3.0
\(f(x)\) 3.0 4.2 5.1 5.7 5.9

Estimating from a Graph

x y 0 1 2 3 4 2 4 (3, 4.33) tangent run rise

How to Estimate \( f'(a) \)

1
Identify the point. Determine the value of \( a \) where you need \( f'(a) \).
2
Find nearby data. Locate values of \( f \) at points close to \( a \): ideally one on each side.
3
Choose a quotient. Use the symmetric difference quotient if you have data on both sides: otherwise, use forward or backward.
4
Compute and interpret. Calculate the quotient. State the units and meaning: "At \( x = a \), \( f \) is changing at approximately ... per unit."

Example 1: Table Estimate

The table gives values of a differentiable function \( g(t) \). Estimate \( g'(4) \).

\(t\) 2 3 4 5 6
\(g(t)\) 10 14 17 19 20

Example 2: Shrinking \( h \)

Let \( f(x) = x^3 \). Estimate \( f'(2) \) using forward differences with \( h = 1 \), \( h = 0.1 \), and \( h = 0.01 \). The exact value is \( f'(2) = 12 \).

\(h\) Difference Quotient Estimate Error
\(1\) \(\frac{27 - 8}{1}\) 19 7
\(0.1\) \(\frac{9.261 - 8}{0.1}\) 12.61 0.61
\(0.01\) \(\frac{8.120601 - 8}{0.01}\) 12.0601 0.06

Example 3: Graph Estimation

x y 0 1 2 3 4 5 1 2 3 4 x = 1 tangent

What If \( f(a) \) Is Not Given?

Sometimes you need \( f'(a) \) but the table does not include \( x = a \) directly. In this case, use the two closest values that straddle \( a \).

\(x\) 1 3 5 7
\(f(x)\) 2 10 26 50

Units and Interpretation

The derivative \( f'(a) \) always has units of "output units per input unit." On the AP exam, you must state the meaning of your answer in context.

If \( f(t) \) = distance (m)
\( f'(t) \) has units m/s: the instantaneous velocity at time \( t \).
If \( P(t) \) = population
\( P'(t) \) has units people/year: the rate of population change.
If \( C(x) \) = cost ($)
\( C'(x) \) has units $/unit: marginal cost per additional item.

Common Mistakes

Wrong Denominator
Using \( h \) instead of \( 2h \) in the symmetric quotient. If points are at \( x = 1 \) and \( x = 5 \), the denominator is \( 5 - 1 = 4 \), not \( 2 \).
Using Distant Points
Choosing points far from \( a \) when closer data is available. Always pick the nearest neighbors to minimize error.
Forgetting Units
Giving a numerical answer without stating units or interpretation. The AP exam requires a verbal explanation of the derivative's meaning.

Summary of Methods

Source Method Key Tip
Table of values Symmetric difference quotient Use nearest data on both sides
Graph Sketch tangent, estimate slope Pick two clear points on tangent
Formula Small \( h \) in limit definition Smaller \( h \) = better accuracy
Any source State units and meaning Always interpret in context

Question 1 of 5

The table shows selected values of a differentiable function \( f \). What is the best estimate of \( f'(3) \)?

\(x\) 1 2 3 4 5
\(f(x)\) 5 8 12 18 26

Question 2 of 5

Which difference quotient gives the most accurate estimate of \( f'(a) \)?

Question 3 of 5

A particle's position \( s(t) \) in meters is recorded in the table. The best estimate of the velocity at \( t = 6 \) seconds is:

\(t\) (s) 4 6 8
\(s(t)\) (m) 20 35 44

Question 4 of 5

If \( f(2) = 7 \) and \( f(2.01) = 7.05 \), the forward difference quotient gives \( f'(2) \approx \):

Question 5 of 5

The cost of producing \( x \) items is \( C(x) \) dollars. If \( C(100) = 5000 \) and \( C(101) = 5034 \), which statement best interprets \( C'(100) \approx 34 \)?