For f(x,y)=x²+3y², find fₓ at (2,1).
Answer and explanation
4
Differentiate with respect to x to obtain 2x, then substitute x=2.
Subjects
Multivariable calculus
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Sample exercises
Write the tangent plane to z=x²+y² at (1,2,5).
Answer and explanation
z=5+2(x−1)+4(y−2), equivalently z=2x+4y−5.
Credit the correct base height and both gradient coefficients.
For f(x,y)=x²+3y², find the directional derivative at (1,1) in direction (3/5,4/5).
Answer and explanation
6
The gradient is (2,6). Its dot product with (3/5,4/5) is 6/5+24/5=6.
What kind of stationary point is the origin for f=x²−y²?
- A strict local maximum
- A point with no partial derivatives
- A saddle point
- A strict local minimum
Answer and explanation
A saddle point
The function increases along the x-axis and decreases along the y-axis.
Find the determinant of the Hessian of f=2x²+3xy+5y².
Answer and explanation
31
The Hessian is [[4,3],[3,10]]; its determinant is 40−9=31.
Why should the direction vector be normalized before calculating a directional derivative?
Answer and explanation
A directional derivative is conventionally the rate per unit distance in that direction. Without normalization, the dot product is scaled by the vector length.
Credit the unit-distance interpretation and the scaling effect.
Study notes
Gradients, tangent planes and extrema
The gradient collects first partial derivatives and determines directional derivatives through a dot product.
UniversityGradients, tangent planes and extrema · revision notes
The gradient collects first partial derivatives and determines directional derivatives through a dot product.
UniversityMultiple integrals and Jacobians
Draw or describe the region before changing integration order.
UniversityMultiple integrals and Jacobians · revision notes
Draw or describe the region before changing integration order.
UniversityEvaluate a partial derivative
Hold the other independent variable fixed while differentiating.
UniversityEvaluate a partial derivative · formula reference
Hold the other independent variable fixed while differentiating.
UniversityDirectional derivative along a unit vector
Normalize a direction vector before the dot product. Here (3/5,4/5) is already a unit vector.
UniversityDirectional derivative along a unit vector · formula reference
Normalize a direction vector before the dot product. Here (3/5,4/5) is already a unit vector.
UniversityHessian determinant of a quadratic
A positive determinant alone does not distinguish a local minimum from a maximum; inspect fₓₓ as well.
UniversityHessian determinant of a quadratic · formula reference
A positive determinant alone does not distinguish a local minimum from a maximum; inspect fₓₓ as well.
UniversityA separable double integral
The rectangular region allows two independent single-variable integrations.
UniversityA separable double integral · formula reference
The rectangular region allows two independent single-variable integrations.
UniversityJacobian of a planar transformation
An integral change of variables uses the absolute determinant; this question asks for its signed value.
UniversityJacobian of a planar transformation · formula reference
An integral change of variables uses the absolute determinant; this question asks for its signed value.
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