1. Find dz

dt : z = x³ - y, x = te^t, y = sin t.

2. Find dz

dx : z = x² + y²; y = x sin x.

3. A vertical line is moving to the right at 2 cm/min., and a horizontal line is moving upward at 3 cm/min. Find the rate of change of the area of the rectangle formed by the coordinate axes and these two lines at the moment when the lines are x = 5 and y = 4.

4. Find the equation of the tangent plane and the normal line to z = x² - 2y² at (2,12).

5. Find the directional derivative in the direction of the given vector. w = xyz² v=i-2j+2k.

6. Find the normal derivative of z=x² - 2y² at (2,12).

7. Find dy

dx : e^xy + y² + 1 = 0.

8. f(x,y) = x/y +ln y Find the differential df.

9. Use differentials to approximate Ã50, ³Ã24

Find all the critical points and test for relative maximum and minimum for each of the following.

10. z = x² + 2xy - y².

11. z = 2y³ + x² - y² - 4x - 4y + 1.