Unit 3 · Lesson 1 of 7

A tangent is a local prediction tool.

A small measurement change can be translated into an approximate output change using sensitivity at a reference point.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its signed linear-estimate-minus-exact error.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

What will change if you increase change h from x=4? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

Read the model

f(x)=\sqrt x,\quad L(x)=2+\frac14(x-4)

Linearization near $x=4.$ All controls keep x positive.

ExactLinearization · dashed

Change h from x=40.5

signed linear-estimate-minus-exact error

L(4+0.5)=2.125,\quad L-f=0.00367966

The tangent approximation is local. Concavity of square root makes the tangent an overestimate away from the base point.

Assumptions, units, and limits

Linearization near $x=4.$ All controls keep x positive.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

From the situation to an equation

Linearization preserves the function value and slope at the base point. Approximate output change by derivative times input change. Concavity predicts whether the tangent estimate lies above or below the exact curve.

f(x)=\sqrt x,\quad L(x)=2+\frac14(x-4)

Keep the reasoning visible

f(a+h)\approx f(a)+f'(a)h

The tangent approximation is local. Concavity of square root makes the tangent an overestimate away from the base point.

What makes this model useful?

Linearization near $x=4.$ All controls keep x positive.

Local does not mean exact away from the reference point. Check the size and sign of the approximation error.

A worked example

Calculate the starting scenario

Use change h from x=4 = 0.5 .
$L(4+0.5)=2.125,\quad L-f=0.00367966$
The tangent approximation is local. Concavity of square root makes the tangent an overestimate away from the base point. The checked quantity is signed linear-estimate-minus-exact error.

Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the signed linear-estimate-minus-exact error using change h from $x=4$ 0.5 . State the meaning of its sign and units.

Checked quantity: signed linear-estimate-minus-exact error · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

f(x)=\sqrt x,\quad L(x)=2+\frac14(x-4)

Linearization near $x=4.$ All controls keep x positive.

ExactLinearization · dashed

Change h from x=40.5

signed linear-estimate-minus-exact error

L(4+0.5)=2.125,\quad L-f=0.00367966

The tangent approximation is local. Concavity of square root makes the tangent an overestimate away from the base point.

Assumptions, units, and limits

Linearization near $x=4.$ All controls keep x positive.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

f(x)=\sqrt x,\quad L(x)=2+\frac14(x-4)

Calculate and explain

Apply the idea to a new scenario

Use $h=1$ to estimate sqrt(5) from the tangent at $x=4$ ; find signed estimate-minus-exact error.

Checked quantity: signed linear-estimate-minus-exact error · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

f(x)=\sqrt x,\quad L(x)=2+\frac14(x-4)

Linearization near $x=4.$ All controls keep x positive.

ExactLinearization · dashed

Change h from x=41

signed linear-estimate-minus-exact error

L(4+1)=2.25,\quad L-f=0.013932

The tangent approximation is local. Concavity of square root makes the tangent an overestimate away from the base point.

Assumptions, units, and limits

Linearization near $x=4.$ All controls keep x positive.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

f(x)=\sqrt x,\quad L(x)=2+\frac14(x-4)

Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual

Read the model

f(x)=\sqrt x,\quad L(x)=2+\frac14(x-4)

Linearization near $x=4.$ All controls keep x positive.

ExactLinearization · dashed

Change h from x=40.5

signed linear-estimate-minus-exact error

L(4+0.5)=2.125,\quad L-f=0.00367966

The tangent approximation is local. Concavity of square root makes the tangent an overestimate away from the base point.

Assumptions, units, and limits

Linearization near $x=4.$ All controls keep x positive.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Linearization preserves the function value and slope at the base point. Approximate output change by derivative times input change. Concavity predicts whether the tangent estimate lies above or below the exact curve.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

Completion is saved on this browser. This is separate from account practice progress.