AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/4/4-6)
Unit 4 · Topic 4.6
4.6 Approximating Values of a Function Using Local Linearity and Linearization
Zoom in far enough on a smooth curve and it looks like a straight line: its tangent line. So near x = a, the tangent line L(x) = f(a) + f′(a)(x − a) gives good approximations of f(x). Concavity tells you whether the estimate is too high or too low.
Key terms
- local linearity
- linearization
- tangent line approximation
- underestimate
- overestimate
Local linearity
A differentiable function is locally linear: near the point of tangency, the graph and its tangent line are almost the same. That's why a straight-line formula can estimate values of a complicated function, as long as you stay close to the point.
The linearization
The tangent line at x = a is L(x) = f(a) + f′(a)(x − a). This is called the linearization of f at a, or the tangent line approximation. To estimate f(b) for b near a, compute L(b).
You only need two numbers: f(a) and f′(a). That's why tables often give exactly those values.
Overestimate or underestimate?
Concavity decides which side the tangent line is on:
- Picture y = x² (concave up). Every tangent line sits under the curve, so tangent line values are too low.
- Picture y = √x (concave down). Tangent lines sit above the curve, so tangent line values are too high.
- To justify, state the sign of f″ on the interval between a and the estimated point, not just at a.
| Near x = a | Graph vs. tangent line | Tangent line estimate is |
|---|---|---|
| Concave up (f″ > 0) | Graph lies above the tangent line | An underestimate |
| Concave down (f″ < 0) | Graph lies below the tangent line | An overestimate |
Linearization in context
Approximation questions often come with a story. If H(t) is the depth of water in feet and you know H(2) = 4.0 and H′(2) = 0.3 feet per hour, then the depth at t = 2.1 hours is about H(2) + H′(2)(0.1) = 4.0 + 0.03 = 4.03 feet.
This is the same as saying “the change in output is about the rate times the change in input,” which is a quick way to think about any local linear estimate.
When the approximation gets worse
The farther you move from a, the more the curve bends away from the line, and the worse the estimate gets. A sharply curved graph (large |f″|) pulls away faster. In Unit 7 (Euler's method, BC) and Unit 10 (Taylor polynomials, BC), this same idea gets extended to better approximations.
Words to know
The exam might say “use the line tangent to the graph of f at x = 2 to approximate f(2.1)” or “use local linearity.” Both mean the same thing: find L(x) and evaluate it.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Approximating a square root
Use the tangent line to f(x) = √x at x = 25 to approximate √26. Is your estimate too high or too low?
Show the solutionHide the solution
- Step 1: f(25) = 5. f′(x) = 1/(2√x), so f′(25) = 1/10.
- Step 2: L(x) = 5 + (1/10)(x − 25).
- Step 3: L(26) = 5 + 1/10 = 5.1.
- Step 4: f″(x) = −1/(4x^(3/2)) < 0 for all x > 0, so f is concave down on [25, 26]. The tangent line lies above the graph.
Answer: √26 ≈ 5.1, an overestimate (the actual value is about 5.0990).
- Example 2
Linearization from table values
f(2) = 5, f′(2) = −3, and f″(x) > 0 for all x. Estimate f(2.1) using the tangent line at x = 2, and say whether it is an overestimate or underestimate.
Show the solutionHide the solution
- Step 1: L(x) = 5 − 3(x − 2).
- Step 2: L(2.1) = 5 − 3(0.1) = 4.7.
- Step 3: f″ > 0, so the graph is concave up and lies above its tangent line.
Answer: f(2.1) ≈ 4.7, an underestimate because f is concave up.
- Example 3
Trap: using the wrong point in the formula
A student approximating √26 writes L(x) = f(26) + f′(26)(x − 25). What's the problem?
Show the solutionHide the solution
- Step 1: The tangent line has to be built at a point where you know the values exactly. You don't know f(26) = √26; that's what you're estimating.
- Step 2: Build the line at a = 25: use f(25) = 5 and f′(25) = 1/10.
- Step 3: Then evaluate at x = 26: L(26) = 5.1.
Answer: The line must use f(a) and f′(a) at the known point a = 25, not at 26. Correct estimate: 5.1.
Common mistakes
- Mixing up which concavity gives an overestimate. Concave down means the line is above the curve, so the estimate is too high.
- Justifying over or under with the sign of f′ instead of f″.
- Forgetting the (x − a) factor and writing L(x) = f(a) + f′(a)x.
On the exam
- Free-response questions often ask you to write a tangent line, use it to approximate a value, and then explain whether the approximation is too big or too small. The last part needs concavity as the reason.
- If you only know f″ from a graph or table, check its sign over the whole interval from a to the estimated point.
Connected topics
Videos
Check yourself
4 questions on 4.6 Approximating Values of a Function Using Local Linearity and Linearization. Pick an answer to see if you got it, and why.
A function f satisfies f(3) = 5 and f′(x) = √(x + 1) for x > −1. The line tangent to the graph of f at x = 3 is used to approximate f(3.2). What is the approximation, and is it an overestimate or an underestimate?
Let f(x) = e^(sin x) + x. What is the approximation of f(1.2) found using the line tangent to the graph of f at x = 1?
The line tangent to the graph of f(x) = ∛x at x = 8 is used to approximate ∛8.12. What is the approximation, and is it an overestimate or an underestimate?
A twice-differentiable function f has f(2) = 3, f′(2) = −1 and f″(x) > 0 for all x. The line tangent to the graph of f at x = 2 is used to approximate f(1.8). What is the approximation, and is it an overestimate or an underestimate?
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