Topic 4.6 Approximating Values of a Function Using Local Linearity and Linearization: use the tangent line to approximate function values near a point.

What this topic is asking

The College Board (Topic 4.6) uses the idea of local linearity - that a differentiable function looks like a straight line when you zoom in - to approximate function values near a known point. The approximating line is the tangent line at that point, and using it as a stand-in for the function is called linearization or linear approximation.

The linearization formula

This is just the point-slope equation of the tangent line, with the function value $f(a)$ as the base point and the derivative $f'(a)$ as the slope.

Why the tangent line approximates

A reliable procedure

Build the tangent line at a convenient nearby point where $f$ and $f'$ are easy, then evaluate it at the target input.

Over- or under-estimate from concavity

A signature AP question asks whether your linear approximation is too big or too small, and the answer comes from the second derivative. The tangent line is straight while the curve bends, so the curve sits on one side of its tangent. If $f''> 0$ near $a$ (concave up, the curve cups upward), the tangent line lies below the curve, and the approximation under-estimates the true value. If $f'' < 0$ (concave down), the tangent line lies above the curve, and the approximation over-estimates. To justify on the exam, compute $f''$, state its sign on the relevant interval, name the concavity, and conclude the direction of the error. This concavity argument is worth a scored point and is the part students most often leave incomplete by stating the conclusion without the $f''$ evidence.

Using differentials and reading the error

The same idea appears in differential language: writing $dy = f'(a)\,dx$, the differential $dy$ approximates the actual change $\Delta y = f(a + \Delta x) - f(a)$ for a small step $dx = \Delta x$. This is the marginal-change idea from Topic 4.1 in disguise, and it is why estimating "the next unit" of cost and approximating $\sqrt{9.2}$ are the same mathematics. The approximation is most accurate when (a) the target is close to the base point and (b) the curve is nearly straight there (small $|f''|$). Choosing a base point where the function is easy to evaluate exactly - a perfect square, a perfect cube, a multiple of $\pi$ - is the practical art of these problems. On the no-calculator section, linearization is the standard way to produce a decimal estimate for an awkward root or trig value without a calculator, so recognizing "approximate this value near a nice point" as a tangent-line problem is the key cue.