Mathematical Scratchpad

252 final review

  1. Use implicit differentiation to find the curvature of an ellipse MATH at $(\pm 4,0)$ and $(0,\pm 2).$

  2. If $f(x,y)=x^{2}-y$, sketch the level curves for $f(x,y)=-1,$ $f(x,y)=0,$ and $f(x,y)=1$.

  3. If MATH

    1. Prove that MATH does not exist.

    2. Find $f_{x}(x,y)$ and $f_{y}(x,y).$

  4. Consider MATH

    1. find $f_{y}(x,y),$

    2. find $f_{x}(x,y).$

  5. Use the limits to write down the definition for the partial derivative of $f_{xy}(a+1,b)$. (b) Interpret $f_{x}(a,b)$ and $f_{y}(a,b)$ geometrically.

  6. Find the tangent plane at the given point for the following surfaces.

    1. MATH

    2. $x^{3}+y^{3}=3xyz,$ $(1,2,\frac{3}{2})$

    3. $xy^{2}+2z^{2}=12$, $(1,2,2)$

    4. $z=axy,$ $(1,\frac{1}{a},1)$

    5. MATH $(4,-2,-10).$

  7. Review sheet for 'Practices on Gradients'.

  8. Review sheet for 'Max/Min'.

  9. Use the Midpoint Rule with $m=n=2$ to estimate the value of the integral MATH where MATH

  10. Use the Chain Rule to find MATH if MATHplus related problems.

  11. Practices on 'Double Integrals'.

  12. Know when to use $dxdy$ or $dydx;$

    1. Know how to transform from rectangular coordinate to polar coordinate.

    2. Know when to use $drd\theta $ and when to use $d\theta dr.$

  13. The center of mass.

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