AP Calculus AB and BC

Linearization vs Differential

Q: What is the difference between linearization and the differential?

Linearization estimates f(x) itself; the differential estimates the change \Delta y. The linearization equals f(a) plus the differential.

Both use the same tangent line. Linearization estimates the VALUE of the function near a point, giving f of a plus f prime of a times the change in x. The differential estimates the CHANGE in the function, which is just f prime of a times dx. One answers how much, the other how much more.

Linearization

Use when: You are asked to approximate a function value, such as estimating the square root of 4.1.

Differential

Use when: You are asked how much a quantity changes, or for propagated error from a measurement.

Side by side

	Linearization	Differential
Formula	$L(x) = f(a) + f'(a)(x-a)$	$dy = f'(a)\,dx$
Estimates	The value $f(x)$	The change $\Delta y$
Includes $f(a)$	Yes	No
Typical question	Approximate $\sqrt{4.1}$	Estimate the error in the volume
Relationship	$L(x) = f(a) + dy$	The correction term inside $L$

The last row is the cleanest way to hold them together: the linearization IS the starting value plus the differential. Confusing them usually means forgetting to add $f(a)$ back, which produces an answer near zero when it should be near the function value.

Concavity decides the direction of the error

Where the graph is concave up the tangent line lies below it, so both the linearization and the differential UNDERestimate. Concave down overestimates. Stating that is what earns the justification point.

Frequently asked questions

What is the difference between linearization and the differential?

Linearization estimates $f(x)$ itself; the differential estimates the change $\Delta y$ . The linearization equals $f(a)$ plus the differential.

Which do I use for error propagation?

The differential. Propagated error is a change in output caused by a change in input, which is exactly what $dy = f'(a)\,dx$ models.

Is the estimate too big or too small?

Concave up means the tangent lies below the curve, so the estimate is too small. Concave down means too big.

In the CED: Unit 4: Contextual Applications