Questions in Section Calculus III: page 3

Use Lagrange multipliers to find the maximum and minimum values of `f(x)=x+y+z` along the intersection of the constraints `x^2-y^2=z` and `x^2+z^2=4`.

Find the partial derivative with respect to `x` for the expression `x^2(y+1)^3=(y^2)/(x-1)`.

Consider the function below. By straightforward inspection (no differentiation is necessary), determine the coordinates `(x,y)` of the point of minimum.

Further, by using the standart procedure involving differentiation, determine the coordinates `(x,y)` of all critical points (you are NOT required to classify which point is minimum or maximum or saddle). Is the above mentioned point of minimum among them?

Find the partial derivatives `(partialf)/(partialt)` and `(partialf)/(partials)` of the function `f(s,t)=e^(s/t)ln(s^2+t^2)`.

Evaluate the double integral `iint(2y)/(1+x^2)dydx` over the region shown in the Figure.

Find the equation for the tangent plane to the surface given by `z=arctan(x+y^2)+tan(x/y)` at the point with the coordinates `x=0`, `y=1`, `z=pi/4`.

Note: `arctan` means inverse tangent (same as `tan^-1`) and `(arctanx)'=1/(1+x^2)`.

Calculate the integral `I=int_A(x^2-y^2)dA`, where `A` is the region between the circles of radius 1 and radius 2, axis `x` and the line `y=x` in the first quadrant. Make a sketch and shade the integration region. For your information: `cos^2 theta-sin^2theta=cos(2theta)`.

Find a directional derivative of the function `f(x,y)=(x^2-sqrt(y))ln[cos(x/y)]-sinx+sqrt(3)e^(y-1)` in the direction of the vector `a=5/2i+(5sqrt(3))/2j`.

A particle moves along the line that starts at `(1,1)` and ends at `(2,2)`, and is acted upon by the force `F=3x^2i+3y^2j`.

  1. Calculate the work `W` done by the force on the particle using the definition of the work `W=intF*dr`.
  2. Find the potential of the force `F`.
  3. Calculate the work using the potential and compare to your answer in a).

Find the equation for the tangent plane to the surface given by `z=e^-x+sin(y)` at the point with the coordinates `x=0,y=0`.

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