Partial derivatives, multiple integrals, and vector calculus terms from a multivariable calculus course.
40 cards · basic cards · AI-written, checked twice. Edit anything.
- What does the partial derivative ∂f/∂x measure?
- The rate of change of f with respect to x while treating all other variables as constants.
- What is the notation for the partial derivative of f with respect to x?
- ∂f/∂x or f_x
- What are mixed partial derivatives?
- Partial derivatives taken with respect to different variables in sequence, such as ∂²f/∂x∂y.
- What does Clairaut's theorem state about mixed partial derivatives?
- If f_xy and f_yx are continuous, then f_xy = f_yx (the order of differentiation does not matter).
- What is the second partial derivative test used for?
- Determining whether a critical point is a local maximum, local minimum, or saddle point.
- State the chain rule for partial derivatives when z = f(x,y) and x and y are functions of t.
- dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)
- What is the equation of the tangent plane to a surface z = f(x,y) at point (x0, y0, z0)?
- z - z0 = f_x(x0, y0)(x - x0) + f_y(x0, y0)(y - y0)
- What is the linear approximation formula for f(x,y) near (a,b)?
- f(x,y) ≈ f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)
- What is the directional derivative of f at point P in the direction of unit vector u?
- D_u f = ∇f · u, the dot product of the gradient with the unit vector.
- What is the gradient vector ∇f for a function f(x,y,z)?
- ∇f = (f_x, f_y, f_z), a vector of all partial derivatives.
- In which direction does the gradient vector point?
- The direction of greatest increase of the function.
- What is a critical point of a multivariable function?
- A point where all first partial derivatives are zero or undefined.
- What is a saddle point on a surface?
- A critical point that is a local maximum in one direction and a local minimum in another direction.
- How is a double integral defined as a limit?
- ∬_R f(x,y) dA = lim (sum of f(x_ij*, y_ij*) ΔA_ij) as the partition size approaches zero.
- What is Fubini's theorem for double integrals?
- ∬_R f(x,y) dA = ∫∫ f(x,y) dy dx = ∫∫ f(x,y) dx dy (under appropriate conditions).