The Gradient, and Optimization of Multivariable Functions

Trivium

Compute the gradient vector of a multivariable function f ⁣:R2→Rf\colon \mathbf{R}^2 \to \mathbf{R} and the directional derivative of ff in the direction of a vector v.\bm{v}.

Determine an equation of the plane tangent to a point on the graph z=f(x,y)z = f(x,y) of a multivariable function f ⁣:R2→R.f\colon \mathbf{R}^2 \to \mathbf{R}.

Determine the local extreme values of a multivariable function f ⁣:R2→Rf\colon \mathbf{R}^2 \to \mathbf{R} on an open subset of the interior of its domain by analyzing its partial derivatives, gradient, and Hessian.

Determine the local extreme values of a multivariable function f ⁣:R2→Rf\colon \mathbf{R}^2 \to \mathbf{R} on a closed subset of the boundary of its domain by either directly analyzing a parameterization of that boundary or by the method of Lagrange multipliers.

Exercises

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    y=mx+b\displaystyle y = mx+b
    y=mx+b\displaystyle y = mx+b

Problems & Challenges

  1. Given the function f ⁣:R2→Rf\colon \mathbf{R}^2 \to \mathbf{R} defined by the formula f(x,y)=1−xexy ,f(x,y) = 1-x\mathrm{e}^{xy}\,, consider the graph z=f(x,y)z = f(x,y) and let pp be the point (1,1)(1,1) in the domain of f.f.

    1. Calculate the gradient of f,f, and list all points such that the gradient is 0.\bm{0}.
    2. Calculate the directional derivative of ff at the point pp in the direction ⟨4,−3⟩.\langle 4,-3 \rangle.
    3. Write down an equation for the plane tangent to the graph of ff at p.p.
    4. Does there exist a direction u\bm{u} in which the directional derivative of ff at pp in the direction u\bm{u} is 66?
    5. Explain why there must exist a direction u\bm{u} in which the directional derivative of ff at pp in the direction u\bm{u} is zero, and calculate a unit vector in that direction.
  2. James Stewart Suppose ff is a differentiable function of one variable. Show that all tangent planes to the surface z=xf(y/x)z = xf(y/x) intersect in a common point.
  3. James Stewart Among all planes that are tangent to the surface xy2z2=1,xy^2z^2 = 1, find the ones that are farthest from the origin.
  4. James Stewart If the ellipse x2/a2+y2/b2=1x^2/a^2 + y^2/b^2 = 1 is to enclose the circle x2+y2=2y,x^2 + y^2 = 2y, what values of aa and bb minimize the area of the ellipse?