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Section Solutions

Domains of Functions

  1. \(\{ (x,y) \in \re^2 : x^2 + y^2 > 1\}\)

  2. \(\{ (x,y) \in \re^2 : y \neq 0 \}\)

  3. \(\{ (x,y) \in \re^2 : |x| \neq |y| \}\)

  4. all of \(\re^3\) except where \(z\) equals the Pisot numbers

  5. \(\{ (x,y) \in \re^2 : |y| \lt 1 \}\)

Graphing Functions

  1. a saddle centered at \((0,0,1)\)

  2. a plane

  3. squished paraboloid

  4. planar saddle with a `point' at \((0,0,0)\)

  5. a three dimensional sine wave

Limits

  1. No, for a chosen value of \(k\text{,}\) the limit along the path \(y=kx^2\) would be \(\frac{1}{1+k}.\) Therefore the limit along different parabolic paths (different values of \(k\)) would yield different results.

Partial Derivatives and Gradients

  1. \(\dsp g_x(x,y) = 3x^2 - \frac{8}{x\ln(7)} + \frac{y}{\sqrt{1-(xy)^2}}\)

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

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

  4. \(g_x(1,1)=g_y(1,1) = -3\)

  5. \(g_y(\pi,2\pi)= 2\pi 5^{\pi^2}\)

  6. \(\dsp \nabla h(x,y,z) = \big( \frac{z}{\sqrt{2xz-5y}} - 3z\cos(x)\cos^2(z\sin(x))\sin(z\sin(x)), \frac{-5}{\sqrt{2xz-5y}},\) \(\dsp \frac{x}{\sqrt{2xz-5y}} - 3\sin(x)\cos^2(z\sin(x))\sin(z\sin(x)) \big)\)

  7. \(h_x(2,3,4) \approx 17,035\)

  8. \(\dsp f_{xx}=\frac{12y}{x^5}, f_{zy} = -\cos(zy)+zy\sin(zy), f_{zxzy}=0\)

Directional Derivatives

Remember: Directions vectors should be unit vectors.

  1. \(\frac{-3}{\sqrt{2}}\)

  2. \(\frac{1}{13}(588 + 31e^3)\)

  3. \((1-\frac{\sqrt{3}}{2})(e^2 + \frac{1}{2}))\)

  4. \(\frac{42}{\sqrt{89}}\)

Derivatives

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

  2. \(Dg(x,y) = \begin{pmatrix}\frac{2x}{y} - 3y \amp - \frac{x^2}{y^2} - 3x \cr \frac{1}{y}\cos(\frac{x}{y}) \amp - \frac{x}{y^2}\cos(\frac{x}{y}) \end{pmatrix}\)

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

  4. \(\dsp Dr(s,t,u) = \begin{pmatrix}t \amp s \amp 0 \cr 2stu \amp s^2u \amp s^2t \cr \frac{tu}{2\sqrt{stu}} \amp \frac{su}{2\sqrt{stu}} \amp \frac{st}{2\sqrt{stu}} \cr \frac{t^2u}{st^2u} \amp \frac{2stu}{st^2u} \amp \frac{st^2}{st^2u} \end{pmatrix}\)

Chain Rule

  1. \((f \circ g)'(t) = \sin(t)(4\cos(t)+7)\)

  2. \((f \circ g)'(-1) \approx 188\)

  3. (a) \(84\)

  4. \(\nabla (f \circ g) = ( (2t^2-8s^3t)\ln(2st)+(2st^2-2s^4t)\frac{1}{s}, (4st-2s^4)\ln(2st) + (2st^2-2s^4t)\frac{1}{t} )\)

  5. (a) \(f_s(s,t)=2\cos(t)-2(s+t)e^{2t}\) and \(f_t(s,t) = -2s \sin(t) - 2(s+t)e^{2t} -2(s+t)^2e^{2t}\)

Tangent Lines and Planes

  1. \(\dsp \frac{\sqrt{14}}{7}\)

  2. \(6x-8y-z=5\)

  3. \(x + y - z = 1\)

  4. \(L(t) = (2,-4,6)t + (1,-1,1)\)

  5. \(3x + ey - z = 2e\)

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