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
- Maximize a directional derivative under a unit-length constraint.
- Justify the conclusion "The maximum is five at u=(4/5,3/5)" using the stated assumptions.
Before you start
Gradients, dot products, and Cauchy–Schwarz.
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
For f(x,y)=x²+3y at (2,1), find the maximum directional derivative.
Why this math matters
Maximize a directional derivative under a unit-length constraint. 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:
- Directions have unit Euclidean length.
- The question concerns local first-order change.
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.
Find the direction of fastest local increase
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For f(x,y)=x²+3y at (2,1), find the maximum directional derivative.
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
∇f(2,1)=(4,3)
The gradient collects the two local sensitivities.
Work through the mathematics
Dᵤf=(4,3)·u≤|(4,3)|·|u|=5
Cauchy–Schwarz bounds the rate for unit directions.
Check the conclusion
The maximum is five at u=(4/5,3/5)
Equality occurs when the unit direction aligns with the gradient.
The result
The maximum is five at u=(4/5,3/5)
Equality occurs when the unit direction aligns with the gradient.
Common mistakes to catch
- An unnormalized direction changes the rate scale.
- A zero directional derivative does not imply the function is globally constant along that line.
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 steepest descent rate?
Show a hint
Reverse the maximizing direction.
Reveal answer and explanation
−5 at u=(−4/5,−3/5)
Negating the unit direction negates the dot product.
Practice 2
What is the derivative in a direction perpendicular to the gradient?
Show a hint
Use orthogonality.
Reveal answer and explanation
Zero
First-order change vanishes along that tangent direction.
Take the idea with you
Choose a local motion direction that increases a measured field most rapidly.
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: Classify a critical point through second-order geometry
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