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# Differentiation and integration

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1. Express the distance between the point (3, 0) and the point P(x, y) of the parabola y = x^2 as a function of x.

2. Find a function f(x) = x^k and a function g such that f(g(x)) = h(x) = sqrt(3x +x^2).

3. Find the trigonometric limit: lim(x->0) (x - tan2x)/sin2x

4. Given f(x) = 2/(x -1), 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 f'(x) given f(x) = (5x^3 - 4x^2 +3x - 2)/x^2.

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 x^3 + 2xy + y^3 = 13 at the point (1, 2).

8. What is the maximum possible area of a rectangle inscribed in the ellipse x^2 + 4y^2 = 4 with the sides of the rectangle parallel to the coordinates axes?

9. Evaluate the definite integral

10. Find the area of the surface obtained when the graph of y = ^2, 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 = x^2, 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 the limit. Apply l'Hopital's rule as many times as necessary. verifying your results after each application.

15. evaluate the integral xsinh(x)dx.

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