1.

Find the limit of as x ? 0.

2.

Suppose f( x) = ? x ³ + x + 1. Compute f ?(1) and determine the equation of the line tangent to the graph of f( x) at the point (1, f(1)). Plot both f( x) and the tangent line.

3.

Can you repeat the last exercise, but embed it all within Manipulate so that moving a slider changes the point of tangency between the line and the cubic?

4.

Find the closest point on the hyperbola x ² ? y ² = 1 to the point (3, 5).

5.

What is the minimum distance between any point on the circle ( x ? .5) ² + ( y ? 1) ² = .25 and any point on the hyperbola x ² ? y ² = 1?

6.

Plot the circle and hyperbola of the last question and the line segment connecting the two point. (Hint: You could use ParametricPlot to plot the hyperbola and then Epilog to throw in the circle and the line segment. Note that (cosh t, sinh t) parameterizes the hyperbola.)

7.

Find the area of the bounded region trapped between the graphs of y = e ^x and y = 4 ? x ².

8.

Suppose a thin homogenous metal plate has the shape of the...