The Gradient, and Optimization of Multivariable Functions

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?