The Gradient, and Optimization of Multivariable Functions

Trivium

Given an equation involving the multiple variables, be able to compute the partial derivative of any of them with respect to one of the others, both in the case that those symbols represent independent variables and in the case that they represent functions of other independent variables. If possible, be able to solve this implicit partial derivative to express the partial derivative explicitly.

Compute the gradient vector of a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) and the directional derivative of \(f\) in the direction of a vector \(\bm{v}.\)

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

Determine the extreme values of a multivariable function \(f\colon \mathbf{R}^2 \to \mathbf{R}\) 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

  1. If \(x = \sin(2t)\) and \(y = \cos(t)\) and \(z = x^2y+3xy^4,\) calculate the value of \(\frac{\mathrm{d}z}{\mathrm{d}t}\) when \(t = 0.\)
  2. If \(x = st^2\) and \(y = s^2t\) and \(z = \mathrm{e}^x\sin(y),\) determine formulas for \(\frac{\partial z}{\partial s}\) and \(\frac{\partial z}{\partial t}.\)
  3. If \(x = rs\mathrm{e}^t\) and \(y = rs^2\mathrm{e}^{-t}\) and \(z = r^2s\sin(t)\) and \(\omega = x^4y + y^2z^3,\) calculate the value of \(\frac{\partial \omega}{\partial s}\) when \(r = 2\) and \(s = 1\) and \(t = 0.\)
  4. If \(x^3+y^3 = 6xy,\) calculate a formula for \(\frac{\mathrm{d}y}{\mathrm{d}x}.\)
  5. For the function \(f\) defined by the formula \(f(x,y) = x^3-3xy+4y^2\) and the unit vector \(\bm{u}\) in the direction given by an angle of \(\theta = \frac{\pi}{6}\) measured from the positive \(x\)-axis, determine a formula for \(\operatorname{D}_{\bm{u}}f(x,y)\) and compute \(\operatorname{D}_{\bm{u}}f(1,2).\)
  6. Compute \(\nabla f(x,y)\) for \(f(x,y) = \sin(x)+\mathrm{e}^{xy}.\)
  7. For the function \(f\) defined by the formula \(f(x,y) = x^2y^3-4y,\) compute the directional derivative of \(f\) at the point \((2,-1)\) in the direction \(\bm{v} = \langle 2,5 \rangle.\)
  8. For the function \(f\) defined by the formula \(f(x,y,z) = x\sin(yz),\) determine the gradient of \(f,\) and compute the directional derivative of \(f\) at the point \((1,3,0)\) in the direction \(\bm{v} = \langle 1,2,-1 \rangle.\)
  9. For the function \(f\) defined by the formula \(f(x,y) = x\mathrm{e}^y,\) compute the rate at which \(f\) is changing at the point \((2,0)\) in the direction of the point \((1/2, 2).\) At the point \((2,0),\) in what direction is the rate of change of \(f\) maximal?
  10. Suppose the temperature at a point \((x,y,z)\) in space is given by the function \(T(x,y,z) = 80/\bigl(1+x^2+2y^2+3z^2\bigr),\) the coordinates measured in meters and \(T\) measured in °C. At the point \((1,1,-2)\) in what direction is the temperature increasing the fastest, and what is this maximal rate of increase?
  11. Determine any local minimum or maximum values of the function \(f(x,y) = x^2+y^2-2x-6y+14.\)
  12. Determine any local minimum or maximum values of the function \(f(x,y) = y^2-x^2.\)
  13. Determine any local minimum or maximum values or saddle points of the function \(f(x,y) = x^4+y^4-4xy+1.\)
  14. Determine any local minimum or maximum values or saddle points of the function \(f(x,y) = 10x^2y-5x^2-4y^2-x^4-2y^4.\)
  15. Determine the absolute minimum and maximum values of the function \(f(x,y) = x^2-2xy+2y\) whose domain is the closed rectangle \(\bigl\{(x,y) \mid 0\leq x \leq 3, 0\leq y \leq 2\bigr\}.\)
  16. Determine the absolute minimum and maximum values of the function \(f(x,y) = x^2+2y^2\) whose domain is the circle \(x^2+y^2=1.\)
  17. Determine the point on the sphere \(x^2+y^2+z^2=4\) that is closest to the point \((3,1,-1).\)
  18. Determine the absolute minimum and maximum values of the function \(f(x,y) = x^2+2y^2\) whose domain is the disk \(\bigl\{(x,y) \mid x^2+y^2\leq 1\bigr\}.\)
  19. Determine the absolute minimum and maximum values of the function \(f(x,y,z) = x+2y+3z\) whose domain is curve that is the intersection of the plane \(x-y+z=1\) and the cylinder \(x^2+y^2=1.\)

Problems & Challenges

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

    1. Calculate the gradient of \(f,\) and list all points such that the gradient is \(\bm{0}.\)
    2. Calculate the directional derivative of \(f\) at the point \(p\) in the direction \(\langle 4,-3 \rangle.\)
    3. Write down an equation for the plane tangent to the graph of \(f\) at \(p.\)
    4. Does there exist a direction \(\bm{u}\) in which the directional derivative of \(f\) at \(p\) in the direction \(\bm{u}\) is \(6\)?
    5. Explain why there must exist a direction \(\bm{u}\) in which the directional derivative of \(f\) at \(p\) in the direction \(\bm{u}\) is zero, and calculate a unit vector in that direction.
  2. James Stewart If the ellipse \(x^2/a^2 + y^2/b^2 = 1\) is to enclose the circle \(x^2 + y^2 = 2y,\) what values of \(a\) and \(b\) minimize the area of the ellipse?