AP® Calculus AB review sheet from Aim for Five (aimforfive.com/calc-ab/units/2/2-3)
Unit 2 · Topic 2.3
2.3 Estimating Derivatives of a Function at a Point
When all you have is a table or a graph, you can still estimate a derivative. Use the slope between nearby data points, ideally on either side of the point you care about. A graphing calculator can also compute a derivative value numerically.
Key terms
- estimate
- slope between data points
- table of values
- numerical derivative
- graphing calculator
Estimating from a table
A derivative is a limit of secant slopes, so a secant slope over a small interval is a good estimate. With a table, use the data points that surround the x-value you want.
If you want f′(4) and the table has data at x = 2 and x = 5, estimate f′(4) ≈ (f(5) − f(2)) / (5 − 2). If the x-value is in the table itself, like f′(5), you can use the interval on one side, the other side, or one that straddles it. The straddling interval, (f(b) − f(a)) / (b − a) with a < 5 < b, usually gives the best estimate.
Why straddle? A one-sided interval only sees how the function behaves on one side of the point. If the graph is bending, that slope is tilted toward the side you used. An interval that surrounds the point balances the two sides, so its slope is usually closer to the true derivative.
Estimating from a graph
On a graph, estimate f′(a) by sketching the tangent line at x = a and reading its slope: pick two points on that line and compute rise over run. If the graph is a straight line segment around a, its slope is exactly f′(a).
Quick reading guide: f′(a) > 0 where the graph goes up from left to right, f′(a) < 0 where it goes down, and f′(a) = 0 where the tangent line is horizontal, such as at a peak or valley of a smooth graph.
Using a calculator
Graphing calculators can find a numerical derivative at a point (on TI models, the nDeriv or d/dx template). On calculator-active questions, use it whenever the formula is messy and you only need a number. Report answers correct to three decimal places, which is the exam's standard.
The calculator also helps check work you did by hand: compute f′(a) with your formula and compare with the numerical value.
Units and interpretation
An estimated derivative still has units of output per input. If T is temperature in °F and t is in minutes, T′(4) ≈ 3 means that at t = 4 minutes, the temperature is increasing at about 3°F per minute.
Always say “approximately” for estimates. The exam rewards a correct difference quotient with numbers shown, plus a correct interpretation.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Estimating a rate from a table
The temperature T of a cup of soup, in °F, is recorded at times t (minutes): T(0) = 70, T(2) = 76, T(5) = 85, T(9) = 91. Estimate T′(4) and explain its meaning.
Show the solutionHide the solution
- Step 1: t = 4 lies between the data points t = 2 and t = 5, so use them.
- Step 2: T′(4) ≈ (T(5) − T(2)) / (5 − 2) = (85 − 76) / 3 = 3.
- Step 3: Units: °F per minute.
Answer: T′(4) ≈ 3°F per minute: at t = 4 minutes, the soup's temperature is increasing at about 3°F per minute.
- Example 2
Trap: a lopsided estimate at a data point
Using the same table, estimate T′(5).
Show the solutionHide the solution
- Step 1: t = 5 is a data point, so the most balanced estimate uses the points on either side, t = 2 and t = 9.
- Step 2: T′(5) ≈ (T(9) − T(2)) / (9 − 2) = (91 − 76) / 7 = 15/7 ≈ 2.143.
- Step 3: One-sided estimates are also acceptable if a question allows them: (85 − 76)/3 = 3 from the left, or (91 − 85)/4 = 1.5 from the right. They differ a lot, which shows why the straddling interval is the safer choice.
Answer: T′(5) ≈ 15/7 ≈ 2.143°F per minute.
- Example 3Calculator allowed
Numerical derivative with a calculator
Let f(x) = xˣ for x > 0. Use a calculator to find f′(2).
Show the solutionHide the solution
- Step 1: This function has no basic rule you've learned yet, but a calculator's numerical derivative works.
- Step 2: Enter the derivative command for xˣ at x = 2.
- Step 3: The calculator returns about 6.77259.
Answer: f′(2) ≈ 6.773
Common mistakes
- Using data points far from the target when closer ones are available.
- Dividing by the wrong denominator, like dividing a 3-minute change by 1. The bottom is the difference in t-values.
- Rounding to fewer than three decimal places on calculator answers, or rounding too early in the middle of a problem.
On the exam
- This is one of the most common free-response tasks: a table of values and a request to approximate a derivative like R′(5), showing your work. Write the difference quotient with the actual numbers and include units.
- If the table is uneven, pick the two data points that bracket the value you need.
Connected topics
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Check yourself
5 questions on 2.3 Estimating Derivatives of a Function at a Point. Pick an answer to see if you got it, and why.
Let f be the function given by f(x) = e^(sin x) + x². What is the value of f′(1.2)?
| t (minutes) | H(t) (degrees Celsius) |
|---|---|
| 0 | 90 |
| 2 | 82 |
| 5 | 73 |
| 9 | 64 |
| 12 | 59 |
Invented data: temperature of a cup of hot chocolate
A cup of hot chocolate cools on a counter. Its temperature at time t minutes is H(t) degrees Celsius, where H is differentiable. Selected values of H(t) are shown in the table. Based on the table, what is the best estimate of H′(7) ?
Using the same table, H′(7) is estimated to be −2.25. Which of the following is the best interpretation of this estimate?
| t (minutes) | W(t) (liters) |
|---|---|
| 0 | 120 |
| 4 | 138 |
| 6 | 145 |
| 10 | 152 |
| 15 | 160 |
Invented data: water in a tank
Water is being added to a tank. The amount of water in the tank at time t minutes is W(t) liters, where W is a differentiable, increasing function. Selected values of W(t) are shown in the table. Based on the data in the table, what is the best estimate of W′(5)?
Water is being added to a tank. The amount of water in the tank at time t minutes is W(t) liters, where W is a differentiable, increasing function. Selected values of W(t) are shown in the table. Using the data in the table, W′(8) is estimated to be 1.75. Which of the following is the best interpretation of this estimate?
0 of 5 answered