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Implicit Differentiation: L'Hopital rule, Convergence

1. Express the distance between the point (3, 0) and the point P (x, y) of the parabola y = as a function of x.

2. Find a function f(x) = and a function g such that f(g(x)) = h(x) =

3. Find the trigonometric limit: .

4. Given , use the four step process to find a slope-predictor
function m(x). Then write an equation for the line tangent to the curve at the point x = 0.

5. Find (x) given .

6. A farmer has 480 meters of fencing. He wishes to enclose a
rectangular plot of land and to divide the plot into three equal
rectangles with two parallel lengths of fence down the middle. What
dimensions will maximize the enclosed area? Be sure to verify that
you have found the maximum enclosed area.

7. Use implicit differentiation to find an equation of the line tangent to
the curve at the point (1, 2).

8. What is the maximum possible area of a rectangle inscribed in the
ellipse with the sides of the rectangle parallel to the
coordinate axes?

9. Evaluate

10. Find the area of the surface obtained when the graph of
, , 0 ≤ x ≤ 1, is rotated around the y-axis.

11. Find the volume of the solid that is generated by rotating the region
formed by the graphs of ,y=2 and x = 0 about the y-axis.

12. A 100-ft length of steel chain weighing 15 lb/ft is hanging from the
top of a tall building. How much work is done in pulling all of the
chain to the top of the building?

13. Differentiate the function f(x) = ln(2x + 3).

14. Find .
Apply l'Hopital's rule as many times as necessary,
verifying your results after each application.

15. Evaluate

16. Determine whether converges or diverges.
If it converges, evaluate the integral.

See attached file for full problem description.