The Gradient; Optimization of Multivariable Functions

Exercises

  1. If \(x\) and \(y\) are functions of \(t\) defined as \(x = \sin(2t)\) 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. Consider the function defined by the formula \(f(x,y) = \frac{1}{2}ax^2 + bxy + \frac{1}{2}cy^2,\) for real coefficients \(a\) and \(b\) and \(c,\) and note that \(f\) has a critical point at \((0,0).\) Classify all combinations of \(a\) and \(b\) and \(c\) that will result in \(f\) having a local minimum versus a local maximum versus a saddle point.

  12. For each of the following functions \(f \colon \mathbf{R}^2 \to \mathbf{R}\) determine its local minimum and maximum values and the locations of its saddle points. Are any of the local extrema the function’s global minimum or maximum?

    \(\displaystyle f(x,y) = x^2+y^2-2x-6y+14\)
    \(\displaystyle f(x,y) = y^2-x^2\)
    \(\displaystyle f(x,y) = x^4+y^4-4xy+1\)
    \(\displaystyle f(x,y) = x^2+xy+y^2+y \)
    \(\displaystyle f(x,y) = x^3+3xy+y^3 \)
    \(\displaystyle f(x,y) = xy^2-x^3\)
    \(\displaystyle f(x,y) = xy^3-x^3y \)
    \(\displaystyle f(x,y) = (y-x)(1-xy) \)
    \(\displaystyle f(x,y) = x\bigl(\mathrm{e}^y-1\bigr) \)
    \(\displaystyle f(x,y) = \mathrm{e}^y\cos(x) \)
    \(\displaystyle f(x,y) = \bigl(x^2+y^2\bigr)\mathrm{e}^{-y} \)
    Mostly from Stewart’s Calculus 9E
    \(\displaystyle f(x,y) = 48+3xy-x^2-xy^2 \)
    \(\displaystyle f(x,y) = \cos(x)+\sin(y) \)
    \(\displaystyle f(x,y) = \frac{1}{2}x^2-9x-3xy+3y^3+9y^2+9y \)
    \(\displaystyle f(x,y) = y\mathrm{e}^{-x^2-2y^2} \)
    From Active Calculus §10.7 (CC BY-NA-SA)
    \(\displaystyle f(x,y) = 2-10x-3^2-20y+8xy+3y^2 \)
    \(\displaystyle f(x,y) = 5-2x-2x^2-2xy+2y-y^2 \)
    \(\displaystyle f(x,y) = x^4+2y^4+3x^2y^2+4x^2+5y^2 \)
    \(\displaystyle f(x,y) = x^3+x^2+4xy-y^2+9 \)
    \(\displaystyle f(x,y) = 2x^3+y^2-6xy \)
    \(\displaystyle f(x,y) = 4x^3+y^3+(1-x-y)^3 \)
    From Multivariable Calculus §4 (CC BY)
    \(\displaystyle f(x,y) = x^3-y^3-2xy+6 \)
    \(\displaystyle f(x,y) = x\bigl(x^2+xy+y^2-9\bigr) \)
    \(\displaystyle f(x,y) = x^2+y^2+x^2y+4 \)
    \(\displaystyle f(x,y) = x^3+x^2-2xy+y^2-x \)
    \(\displaystyle f(x,y) = x^3+xy^2-3x^2-4y^2+4 \)
    From CLP-3 Multivariable Calculus §2.9 (CC BY-NA-SA)

  13. For each of the following functions \(f \colon D \to \mathbf{R}\) defined on some bounded domain \(D \subset \mathbf{R}^2,\) determine its global minimum and maximum values.

    1. The function \(f(x,y) = x^2-2xy+2y\) whose domain is the closed rectangle \(\bigl\{(x,y) \mid 0\leq x \leq 3, 0\leq y \leq 2\bigr\}.\)
    2. The function \(f(x,y) = x^2+2y^2\) whose domain is the circle \(x^2+y^2=1.\)
    3. The function \(f(x,y) = x^2+2y^2\) whose domain is the closed unit disk \(\bigl\{(x,y) \mid x^2+y^2\leq 1\bigr\}.\)
    4. Active Calculus The function \( f(x,y) = 2+2x+2y-x^2-y^2 \) whose domain is the closed triangular region in the first quadrant bounded by the lines \(x=0\) and \(y=0\) and \(y=9-x.\)
    5. 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 \(x^2+y^2=1.\)
    6. The function \( f(x,y) = x^2+y^2 \) whose domain is the curve where \(xy=1.\)
    7. The function \( f(x,y) = y^2-x^2 \) whose domain is the ellipse \(x^2+4y^2=4.\)
    8. The function \( f(x,y) = \mathrm{e}^{xy} \) whose domain is the curve \(x^3+y^3 = 25.\)
    9. The function \( f(x,y,z) = xyz \) whose domain is the spheroid defined by the equation \(x^2+2y^2+3z^2=6.\)
    10. The function \( f(x,y,z) = x+2y \) whose domain is the intersection of the plane \(x+y+z=1\) and the cylinder \(y^2+z^2=4.\)
    11. The function \( f(x,y,z) = x^2+y^2+z^2 \) whose domain is the intersection of the plane \(x=1+y\) and the surface \(y^2-z^2=1.\)
  14. Determine the point on the radius four sphere \(x^2+y^2+z^2=4\) that is closest to the point \((3,1,-1).\)

Problems & Challenges

  1. 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?