Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compute curvature of a graph at a specified point.
- Justify the conclusion "κ(0)=2; κ(1)=2/(5√5)" using the stated assumptions.
Before you start
First and second derivatives.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Find the curvature of y=x² at x=0 and x=1.
Why this math matters
Compute curvature of a graph at a specified point. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The graph is twice differentiable.
- Curvature is unsigned in this lesson.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Measure how quickly a curve changes direction
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the curvature of y=x² at x=0 and x=1.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
κ=|y″|/(1+(y′)²)^(3/2)
For a regular graph, curvature normalizes bending by the rate of travel along the curve.
Work through the mathematics
y′=2x, y″=2, so κ(x)=2/(1+4x²)^(3/2)
The denominator distinguishes slope from actual directional bending.
Check the conclusion
κ(0)=2; κ(1)=2/(5√5)
The parabola bends most sharply near its vertex despite being steeper farther away.
The result
κ(0)=2; κ(1)=2/(5√5)
The parabola bends most sharply near its vertex despite being steeper farther away.
Common mistakes to catch
- A large slope alone does not mean large curvature.
- Use radians when interpreting turning angles.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the radius of curvature at zero?
Show a hint
Use R=1/κ.
Reveal answer and explanation
1/2
The osculating circle has reciprocal-curvature radius.
Practice 2
What is the curvature of y=3x+2?
Show a hint
Its second derivative is zero.
Reveal answer and explanation
Zero
A straight line does not change tangent direction.
Take the idea with you
Explain how a road can be steep while locally almost straight.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Separate spatial twisting from bending
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.