Math · Calculus I · Concept
Linear approximation and differentials
Near a point a, a differentiable function is close to its tangent line, so L(x) = f(a) + f′(a)(x − a) approximates f(x) for x near a. The differential dy = f′(x) dx uses the same idea to estimate how much the output changes when the input changes by a small amount dx.
The tangent line as an approximation
Zoom in far enough on a smooth graph and it looks straight. The tangent line at x = a has the function’s value and slope there, so for x close to a it gives a good estimate. It is called the linearization of f at a.
Choose a convenient point
Pick a close to the x you need, at a point where f(a) and f′(a) are easy to find exactly. To estimate √4.1, use a = 4, because √4 = 2.
Differentials
If x changes by a small amount dx, the tangent line predicts a change in y of dy = f′(x) dx. The true change, Δy, is close to dy when dx is small.
Propagating a measurement error
Differentials estimate how an error in a measurement affects a result calculated from it. For a sphere, V = (4/3)πr³ gives dV = 4πr² dr, so the relative error in the volume is about three times the relative error in the radius.
Over or under?
Concavity decides the direction of the error. Where a graph is concave down it bends below its tangent line, so the linear approximation overestimates; where it is concave up, it underestimates. √x is concave down, so L(4.1) = 2.025 is slightly above √4.1.
Common mistakes
- Using a point a far from x: the approximation is only good near a.
- Mixing up dx and dy: dx is the change in the input, dy the predicted change in the output.
- Using degrees with a trigonometric function: the derivative of sin x is cos x only in radians.
- Reporting the estimate as if it were exact.
Key terms
- Linearization
- The tangent-line function L(x) = f(a) + f′(a)(x − a), which approximates f(x) for x near a.
- Differential
- For y = f(x), the differential dy = f′(x) dx is the change in y predicted by the tangent line when x changes by dx. For small dx it approximates the true change Δy.
- Tangent line
- The line that matches a curve’s direction at a point, with slope equal to the derivative there. It can cross the curve; in trigonometry, “tangent” also names tan θ = sin θ/cos θ.
- Derivative
- The instantaneous rate of change of a function: the limit of the average rate of change as the step shrinks to zero, when that limit exists. On a graph it is the slope of the tangent line.
Work through an example
Use a linear approximation to estimate √4.1.
Estimate a square root with a tangent line →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Graph √x and its tangent line in Graph Check each step in Math Open worked example on a board Linearization in Math ReferenceYour existing work stays on this device. Examples open as editable copies.