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Section Practice

These are practice problems for the tests. Solutions are in the next section. While we will not present these, I am happy to answer questions about them in class.

Domains of Functions

  1. Find the domain of \(f(x,y)= \ln(x^2+y^2 - 1).\)

  2. Find the domain of \(\dsp g(x, y) = \invtan \Big({x \over y} \Big).\)

  3. Find the domain of \(\dsp h(x, y) = {4 \over {|x| - |y|}}.\)

  4. Find the domain of \(\dsp k(x,y,z)= {{2xy} \over{z^2+z -1}}.\)

  5. Find the domain of \(\dsp \ell(x, y, z)={{xz} \over {\sqrt{1-y^2}}}.\)

  6. Determine the domain over which \(f(x, y) = \ln(x^3y^4)\) is continuous.

  7. Determine the domain over which \(\dsp g(x, y) = {{x} \over {|x||y|}}\) is continuous.

  8. Determine the domain over which \(\dsp h(x, y, z) = {{x^2 - 1} \over {z \sqrt{y^2 - 1}}}\) is continuous.

Graphing Functions

  1. Graph \(f(x, y)= 1-x^2 +y^2.\)

  2. Graph \(g(x,y)= x+ y.\)

  3. Graph \(\dsp h(x,y)= {{x^2} \over 9} + {{y^2} \over 4}.\)

  4. Graph \(k(x, y)=|x|-|y|.\)

  5. Graph \(r(x,y)= \sin(x).\)

Limits

  1. Consider the function defined by \(f(x,y)=\left\{ \begin{array}{ll} \dsp{\frac{x^2}{x^2+y}} \amp x^2+y \neq 0\\ 0 \amp (x,y)=(0,0) \end{array} \right.\) Does the limit exist as \((x,y) \to (0,0)\) along \(y=kx\) exist for every \(k \in \re?\) Does the limit exist as \((x,y) \to (0,0)\) along \(y=kx^2\) exist for every \(k \in \re?\) Is the function \(f\) continuous at \((x,y)=(0,0)\text{?}\)

Partial Derivatives and Gradients

  1. Let \(g(x,y)=x^3 - 4 \log _7 x^2 + \invsin(xy)\) and compute \(g_x.\)

  2. Let \(g(x,y)= \dsp {{x^2} \over {\sin(xy)}}\) and compute \(g_y.\)

  3. Let \(g(x,y)= e^{xy^2}\) and compute \(\nabla g.\)

  4. Let \(g(x,y)=x \ln(y)-4xy+x\) and compute \(g_x(1,1)\) and \(g_y(1,1).\)

  5. Let \(g(x,y)= 5^{x^2}y \sin(x - y)\) and compute \(g_y(\pi,2\pi).\)

  6. Let \(h(x,y,z)=\sqrt{2xz-5y}+\cos^{3}(z\sin(x))\) and compute \(\nabla h.\)

  7. Let \(h(x, y,z) = z^{xy}\) and compute \(h_x(2,3,4)\text{.}\)

  8. Find \(f_{xx}\text{,}\) \(f_{zy}\text{,}\) and \(f_{zxzy}\) for \(\dsp f(x,y,z)= \frac{y}{x^{3}}-\sin(zy)-3z^{3}\text{.}\)

Directional Derivatives

Remember: Directions vectors should be unit vectors.

  1. Using the definition of directional derivative, compute the derivative of \(f(x,y) = x-y^2\) at \((1,2)\) in the direction, \((1,1).\)

  2. Find \(D_{\oa u}f(p)\) for \(f(x,y)=e^{xy}+2x^{2}y^{3}\text{;}\) \(p=(3,1)\text{;}\) \(\oa u\) is a unit vector parallel to \(\oa v= (-5, 12)\text{.}\)

  3. Find \(D_{\oa u}f(p)\) for \(f(x,y)=e^{xy} + \ln(xy)\text{;}\) \(p=(1,2)\text{;}\) \(\oa u\) is a unit vector which makes an angle \(\dsp {-{\pi} \over 3}\) from the positive \(x\) axis.

  4. Find \(D_{\oa u}f(p)\) for \(f(x,y,z)=x^2y-4y^2z+xyz^2\text{;}\) \(p=(-2,1, -1)\text{;}\) \(\oa u\) is a unit vector parallel to the vector \(\overrightarrow{AB}\) where \(A=(2,1,-5)\) and \(B=(-2,4,3)\text{.}\)

Derivatives

For each of the following functions, compute the indicated “derivative” of the function. Because of the differing domains of the functions, the derivative could be a function (a partial derivative), a vector of functions (a gradient), or a matrix of functions (the `total' derivative)!

  1. \(f(x,y) = x^2y^3\)

  2. \(\dsp g(x,y) = \big( \frac{x^2}{y} - 3xy , \sin(\frac{x}{y}) \big)\)

  3. \(h(x,y) = x^2- e^{xy\sqrt{z}} + \sinh(yz)\)

  4. \(r(s,t,u) = \big( st, s^2tu, \sqrt{stu}, \ln(st^2u) \big)\)

Chain Rule

  1. Let \(f(x,y)=2x^{2}-7y\) and \(\oa g(t) = (\sin(t), \cos(t)).\) Compute the derivative of \(f \circ g\) in two ways. First compose \(f\) and \(g\) and take the derivative. Second, apply the chain rule. Verify that your solutions are the same.

  2. Let \(\dsp f(x,y)=4x^{3}y+e^{3y}+\frac{2}{x},\) \(x(t)=t^{2}\text{,}\) \(y(t)=4t-3,\) and \(g(t)=(x(t),y(t)).\) Compute \((f \circ g)'(-1).\)

  3. For each of the following problems, find \(\dsp \frac{dg}{dt}\) and evaluate at the given value of \(t\text{.}\)

    1. \(g(x,y)=3xy+e^{x}y^{2}\) where \(x(t)=4t^{2}+t\text{,}\) \(y(t)=6+5t\text{,}\) and \(t=0\)

    2. \(\dsp g(x,y,z)=x^{3}y+xz+\frac{x}{y-z}\) where \(x(t)=t^{2}+3\text{,}\) \(y(t)=4t-t^2\text{,}\) \(z(t)= \cos(t-3 \pi)\text{,}\) and \(t=0\)

  4. Let \(f(x,y)=xy \ln(x)\) and \(\oa g(s,t)=(2st, t-s^{3}).\) State the domain \(f,\) \(g,\) and \(f \circ g.\) Compute the gradient of \((f \circ g).\)

  5. For each of the following problems, find \(f_s\) and \(f_t\) (i.e. \(\dsp {{\partial f} \over {\partial s}}\) and \(\dsp {{\partial f} \over {\partial t}}\)).

    1. \(f(x,y)=2x-y^2\text{,}\) \(x(s, t)=s \cos(t)\text{,}\) and \(y(s, t)=(s+t)e^t\)

    2. \(f(x, y, z) = (x + 2y + 3z)^4\) and \(x(s, t) = s + t\text{,}\) \(y(s, t) = s-t\text{,}\) \(z(s, t)=st\)

  6. For each of the following problems, find \(\dsp {{\partial g} \over {\partial u}},\) \(\dsp {{\partial g} \over {\partial v}},\) and \(\dsp {{\partial g} \over {\partial w}}\) (i.e. \(g_u\text{,}\) \(g_v\text{,}\) and \(g_w\)).

    1. \(g(x, y) = (x+y) \ln(xy)\text{,}\) \(x(u, v, w) = u+v-3w\text{,}\) and \(y(u, v, w) = uv+3w\)

    2. \(g(x, y, z) = yz+xz+xy\text{,}\) \(x(u, v, w)=u+v-3vw\text{,}\) \(y(u, v, w)=v+w+4uw\text{,}\) and \(z(u, v, w)=u+w-5uv\)

  7. For each of the following three problems, find \(\dsp {{\partial z} \over {\partial x}}\) and \(\dsp {{\partial z} \over {\partial y}}\) at the indicated point.

    1. \(z^3 -xy+2yz+y^3-3=0\text{;}\) \((1, 1,1)\)

    2. \(\dsp {1 \over {x^2}}+{1 \over {y^2}} + {1 \over {z^2}} = {{49} \over {36}}\text{;}\) \((-1, 2, 3)\)

    3. \(ye^{xyz} \cos(2xz) = 1\text{;}\) \((\pi, 1, 4)\)

Tangent Lines and Planes

  1. Find the (shortest) distance from the point \((0, 1, 0)\) to the plane \(x + 2y+ 3z=4\text{.}\)

  2. Find the equation of the tangent plane to the function \(z=x^{2}-4y^{2}\) at \((3,1,5).\)

  3. Find the equation of the tangent plane to \(z+1=xe^y \cos(z)\) at the point \((1,0,0).\)

  4. Find the equation of the line normal (perpendicular) to \(x^2+2y^2 + 3z^2 = 6\) and passing through \((1,-1,1).\)

  5. Find the tangent plane approximation of \(h(x,y)=x+x \ln(xy)\) when \(x=e\) and \(y=1\text{.}\)

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