1. Write an equation for the circle that passes through the points:
(1, -1), (-5, 7), and (-6, 0).
2. Express the polar equation in rectangular form.
3. Find the total area enclosed by the graph of the polar equation r = 1 + cos 2θ.
4. Write the equation of the line tangent to the parametric curve x = tcos t, y = tsin t at the point corresponding to t = 2π.
Then calculate to determine whether the curve is concave upward or concave downward at this point.
5. Find the arc length of the curve given by x = sin t - cos t,
6. Find an equation of the ellipse with center (0, 0), vertical major axis 14, and minor axis 10.
7. Find an equation for the hyperbola with focus (11, 2), and asymptotes 4x - 3y = 18 and 4x + 3y = 30.
8. Determine whether or not the sequence converges and find its limit if it does converge.
9. A ball has bounce coefficient r <1 if, when it is dropped from a height h, it bounces back to a height of rh. A ball with a bounce coefficient 0.81 is dropped from an initial height of a = 4 ft. Use a geometric series to compute the total time required for it to complete its infinitely many bounces. The time required for a ball to drop h feet (from rest) is
10. Write the Taylor polynomial with center zero and of degree 4 for the function
11. Determine the values of p for which the series converges.
12. Calculate the sum of the first ten terms of the series , then estimate the error made in using this partial sum to approximate the sum of the series.
13. Determine whether the series converges absolutely, converges conditionally, or diverges.
14. Find the interval of convergence of the power series
15. Calculate sin 87° accurate to five decimal places using Taylor's formula for an appropriate function centered at .
16. First derive a recurrence relation giving cn for n ≥ 2 in terms of or c1 (or both). Then apply the initial conditions to find the values of and c1. Next determine cn in terms of n, (as in the text) and, finally, identify the particular solution in terms of familiar elementary functions. y'' - 9y = 0; y(0) = 0, y'(0) = 3.
Algebra and trigonometry are solved. The solution is detailed and well presented.