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Section 12.6 Assignment 6

  1. Find the equations of the tangent planes to the function

    \begin{equation*} f(x,y) = x^2 + 4y^2 \end{equation*}

    at the point \((4,1,20)\text{.}\) (4)

  2. Find the equations of the tangent planes to the function

    \begin{equation*} f(x,y) = \frac{x+y}{x-y} \end{equation*}

    at the point \((2,4,-3)\text{.}\) (4)

  3. Consider the function \(f(x,y) = \frac{3}{x^2 + y^2 + 1}\text{.}\) Where does the graph of this function intersect with the sphere or radius \(3\text{?}\) Is the tangent plane at any of those points also tangent to the sphere? (The normal of the sphere has the same direction as the vector defining a point on the sphere.) (8)

  4. For the following function, calculate \(\nabla f\text{,}\) draw a contour diagram, and draw several of the gradient directions. (6)

    \begin{equation*} f(x,y) = \frac{x^2 + y^2}{12} \end{equation*}

  5. For the following function, calculate \(\nabla f\text{,}\) draw a contour diagram, and draw several of the gradient directions. (6)

    \begin{equation*} f(x,y) = \frac{(x-y)x^3}{5} \end{equation*}

  6. Consider a comet heading towards the sun. The heat energy of the function could be modelled by a function

    \begin{equation*} H(x,y) = \frac{A}{x^2 + y^2 + 1} \end{equation*}

    and the path of the coment could be given as

    \begin{equation*} \gamma(t) = (B e^{-t} \cos t, B e^{-t} \sin t) \end{equation*}

    for positive \(t\text{.}\) How fast is the comet heatin gup as it approaches the sun? Solve the problem first by using the chain rule, and then by using a direct substitution and ordinary single derivative in \(t\text{.}\) (10)

  7. Find and classify the critical point and extrema of the following function. (8)

    \begin{equation*} f(x,y) = 2x^2 + 3y^2 - 2xy \end{equation*}

  8. Find and classify the critical point and extrema of the following function. (8)

    \begin{equation*} f(x,y) = \frac{1}{xy} + \frac{1}{x^2 y^2} \end{equation*}

  9. Consider the function \(f(x,y) = x^2 y^2 - yx^2\text{.}\) (8)

    1. Find the critical points of \(f\) and try to classify them.

    2. The classification using the Hessian determinant doesn't really work here. Instead, there is a line of critical points. Look at the values of the function on paths perpendicular to the line of critical points and use those values to describe the nature of the critical points. (There will be ranges along th eline where the critical points have different behaviours.)

    3. What is it like to walk tthis graph along the \(x\) axis or along the \(y\) axis? What is different about each walk?