Undergraduate · Directional derivatives · 28 of 650
Directional derivatives · x² coefficient a=2.5; y² coefficient b=2.5
Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point. Givens: x² coefficient a=2.5; y² coefficient b=2.5.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Let f(x,y)=2.5x²+2.5y², evaluated at the fixed point (1,1). Use a unit direction u=(cos θ,sin θ). Set θ=6π/5 radians. Find the directional derivative and the largest possible unit-direction derivative at this point.
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A starting point
Evaluate the gradient (2a,2b) at the point, then take its dot product with the unit direction. The maximum is the gradient norm.
Work through the reasoning
Step 1
Identify the model and target
Let f(x,y)=2.5x²+2.5y², evaluated at the fixed point (1,1). Use a unit direction u=(cos θ,sin θ). Set θ=6π/5 radians. The governing relation is f=ax²+by²; ∇f(1,1)=(2a,2b); Dᵤf=∇f·u. Evaluate the gradient (2a,2b) at the point, then take its dot product with the unit direction. The maximum is the gradient norm.
Step 2
Substitute and calculate
The gradient at (1,1) is (5,5). Its dot product with (cos(3.769911),sin(3.769911)) is -6.984011. Cauchy–Schwarz bounds all unit-direction rates by √(5²+5²)=7.071068.
Step 3
Check the mathematical meaning
The gradient is (5,5). Cauchy–Schwarz gives |Dᵤf|≤7.071068, consistent with the computed derivative -6.984011. Direction radians: 3.77; Directional derivative: -6.984; Maximum unit-direction rate: 7.071. Decimal values are rounded, so use unrounded intermediate values.
The answer
The gradient at (1,1) is (5,5). Its dot product with (cos(3.769911),sin(3.769911)) is -6.984011. Cauchy–Schwarz bounds all unit-direction rates by √(5²+5²)=7.071068. The gradient is (5,5). Cauchy–Schwarz gives |Dᵤf|≤7.071068, consistent with the computed derivative -6.984011. Animation check: Direction radians: 3.77; Directional derivative: -6.984; Maximum unit-direction rate: 7.071. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
The directional derivative is a dot product with a unit direction, so it depends on orientation as well as gradient length. Rotating a unit vector through all angles reveals a maximum equal to the gradient norm and a minimum equal to its negative. Perpendicular directions have zero first-order change. Let f(x,y)=2.5x²+2.5y², evaluated at the fixed point (1,1). Use a unit direction u=(cos θ,sin θ). Set θ=6π/5 radians. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
f=ax²+by²; ∇f(1,1)=(2a,2b); Dᵤf=∇f·u
03 · Reflect and transfer
Explain what changes and why.
What changes if the direction vector has length two rather than one, and why must direction vectors be normalized here?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.