riaology
Unit 3 · Lesson 3 of 7

An average rate must occur somewhere instantaneously.

A trip with a known average velocity must match that velocity at some instant under the theorem's smoothness conditions.

Change the input. Follow the mathematics.

What you'll be able to do
  • Set up the model and calculate its matching input c.
  • Connect the equation to the changing visual.
  • Explain the assumptions and identify a misleading interpretation.
Predict before exploring

What will change if you increase right endpoint b? Predict the direction, then explain the response.

A model you can changeMove one control at a time.
Motion illustration: position 2.25. Velocity direction shown where applicable.Position 2.25 · arrow: velocity direction
Input x versus x². Function, Endpoint secant, Parallel tangent. Selected matching input c: 1.5 . At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside.-2.2503.3751.593x²Input x
FunctionEndpoint secant · dashedParallel tangent

matching input cb2−0b−0=b=3,2c=b⇒c=1.5\frac{b^2-0}{b-0}=b=3,\quad2c=b\Rightarrow c=1.5

At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside.

Assumptions, units, and limits

The function is continuous on [0,b] and differentiable on (0,b), so the mean value theorem applies.

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

01

From the situation to an equation

Continuity on the closed interval and differentiability inside it allow the mean value theorem. Its secant slope matches a tangent slope at an interior point. Rolle's theorem is the special case of equal endpoint values.

f′(c)=f(b)−f(a)b−a,f=x2, a=0⇒c=b/2f'(c)=\frac{f(b)-f(a)}{b-a},\quad f=x^2,\ a=0\Rightarrow c=b/2
02

Keep the reasoning visible

f(b)−f(a)=f′(c)(b−a)f(b)-f(a)=f'(c)(b-a)

At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside.

03

What makes this model useful?

The function is continuous on [0,b] and differentiable on (0,b), so the mean value theorem applies.

A corner can violate differentiability even when the graph is continuous. The theorem is an existence statement, not a claim that every instant has the average rate.

A worked example

Calculate the starting scenario

  1. Use right endpoint b = 3 .
  2. b2−0b−0=b=3,2c=b⇒c=1.5\frac{b^2-0}{b-0}=b=3,\quad2c=b\Rightarrow c=1.5
  3. At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside. The checked quantity is matching input c.
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 matching input c using right endpoint b 3 . State the meaning of its sign and units.

Checked quantity: matching input c · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Continuity on the closed interval and differentiability inside it allow the mean value theorem. Its secant slope matches a tangent slope at an interior point. Rolle's theorem is the special case of equal endpoint values.

Explore the visual solution
Motion illustration: position 2.25. Velocity direction shown where applicable.Position 2.25 · arrow: velocity direction
Input x versus x². Function, Endpoint secant, Parallel tangent. Selected matching input c: 1.5 . At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside.-2.2503.3751.593x²Input x
FunctionEndpoint secant · dashedParallel tangent

matching input cb2−0b−0=b=3,2c=b⇒c=1.5\frac{b^2-0}{b-0}=b=3,\quad2c=b\Rightarrow c=1.5

At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside.

Assumptions, units, and limits

The function is continuous on [0,b] and differentiable on (0,b), so the mean value theorem applies.

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.

Continuity on the closed interval and differentiability inside it allow the mean value theorem. Its secant slope matches a tangent slope at an interior point. Rolle's theorem is the special case of equal endpoint values.

f′(c)=f(b)−f(a)b−a,f=x2, a=0⇒c=b/2f'(c)=\frac{f(b)-f(a)}{b-a},\quad f=x^2,\ a=0\Rightarrow c=b/2
Calculate and explain

Apply the idea to a new scenario

For x² on [0,4], find the point where instantaneous slope equals average slope.

Checked quantity: matching input c · dimensionless. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Continuity on the closed interval and differentiability inside it allow the mean value theorem. Its secant slope matches a tangent slope at an interior point. Rolle's theorem is the special case of equal endpoint values.

Explore the visual solution
Motion illustration: position 4. Velocity direction shown where applicable.Position 4 · arrow: velocity direction
Input x versus x². Function, Endpoint secant, Parallel tangent. Selected matching input c: 2 . At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside.-4062164x²Input x
FunctionEndpoint secant · dashedParallel tangent

matching input cb2−0b−0=b=4,2c=b⇒c=2\frac{b^2-0}{b-0}=b=4,\quad2c=b\Rightarrow c=2

At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside.

Assumptions, units, and limits

The function is continuous on [0,b] and differentiable on (0,b), so the mean value theorem applies.

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.

Continuity on the closed interval and differentiability inside it allow the mean value theorem. Its secant slope matches a tangent slope at an interior point. Rolle's theorem is the special case of equal endpoint values.

f′(c)=f(b)−f(a)b−a,f=x2, a=0⇒c=b/2f'(c)=\frac{f(b)-f(a)}{b-a},\quad f=x^2,\ a=0\Rightarrow c=b/2
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
Motion illustration: position 2.25. Velocity direction shown where applicable.Position 2.25 · arrow: velocity direction
Input x versus x². Function, Endpoint secant, Parallel tangent. Selected matching input c: 1.5 . At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside.-2.2503.3751.593x²Input x
FunctionEndpoint secant · dashedParallel tangent

matching input cb2−0b−0=b=3,2c=b⇒c=1.5\frac{b^2-0}{b-0}=b=3,\quad2c=b\Rightarrow c=1.5

At c the tangent is parallel to the endpoint secant. The theorem guarantees a matching instantaneous rate somewhere inside.

Assumptions, units, and limits

The function is continuous on [0,b] and differentiable on (0,b), so the mean value theorem applies.

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

The reasoning. Continuity on the closed interval and differentiability inside it allow the mean value theorem. Its secant slope matches a tangent slope at an interior point. Rolle's theorem is the special case of equal endpoint values.

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.

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