Math 1220-1
Review 3
1. Expand x4 − 3x3 + 2x2 + x − 2 in a Taylor polynomial of order 4 based at 1.
2. Find the third-order Maclaurin polynomial for (1 + x)3/2 .
Z 1
cos(sin x) dx.
3. Use the trapezoidal rule with n = 10 to approximate
4. Use Simpon’s rule with n = 6 to approximate
Z
0
π
sin x dx.
0
5. Approximate the real root of 7x3 + x − 5 = 0 using Newton’s Method to five decimal places.
6. Approximate the real root of 2 cos x − e−x = 0 to two decimal places by using the Bisection Method
on [1, 2].
7. Find the standard equation of the parabola with vertex at the origin and directrix y = 27 .
8. Find the equation of the ellipse with a focus at (0,3) and minor diameter 8.
9. Find the equation of the hyperbola with focus at (5,0) and vertex at (3,0).
10. Sketch a design for a reflecting telescope that uses a parabola and an ellipse.
11. If a planet orbits a star in an elliptical path, and the star is located at (50,0), and the planet is measured
to be 150 million miles from the star when the angle from the origin to the star to the planet is 135
degrees, give the equation of the planet’s orbit, assuming the star is at one focus, and the center of the
ellipse is the origin.
12. Find the Cartesian equation of r2 − 8r cos θ + 12r sin θ = 0.
13. Graph r = 2 cos 3θ.
14. Graph r = 2θ for θ ≥ 0.
15. Sketch the following graphs and give their points of intersection:
√
r2 = 4 cos 2θ, r = 2 2 sin θ
.
16. Sketch one leaf of the rose r = 3 cos 2θ and find the area of the region enclosed by it.
√
17. Find the area of the region outside the circle r = 3 sin θ and inside the cardioid r = 1 + cos θ.