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Biola Home Page My Math Papers Woopy Home Page |
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Biola Home Page My Math Papers Woopy Home Page |
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1. Of course you should know the formula on http://woobiola.net/math/formulas.htm
2. In a triangle ABC, only b, c, A are given. Find the area of DABC.
3. In a triangle ABC, only b, c, a are given. Find the area of DABC.
4. In a triangle ABC, B,C,a are given. Find the length of altitude AD. Find b,c, and area of ABC.
5. Prove that sin(A+B) = sinA cosB + cosA sinB, assuming 0 < {A,B} < 45°.
6. Prove that sin(A-B) = sinA cosB - cosA sinB, assuming 0 < A < B < 90°.
7. Prove that cos(A+B) = cosA cosB - sinA sinB, assuming 0 < {A,B} < 45°.
8. Prove that cos(A-B) = cosA cosB + sinA sinB, assuming 0 < A < B < 90°.
9. Prove that lim q®0 [sinq / q] = 1.
10. Find ò [dx / xÖ(3x2+2x-1) ] .
11. Find the length and area of the cardioid r = a(1+cosq).
12. FInd the surface area and volume of the cherry, formed by rotating the cardioid above around the x-axis.
13. Find the centroid of a semicircular plate of uniform density.
14. Given A(x',y') and B(x",y"), prove that the point P on AB between A,B, with AP : PB = l : (1-l), is ( l'x'+lx", l'y'+ly" ), where l' = 1-l.
15. Given a triangle ABC with A(0,0), B(5,0), C (3,4). Find R, r, O, I, H, G (circumradius, inradius, circumcenter, incenter, orthocenter, centroid). [Hint: use (14).]
16. Given the focus F(a,0) and directrix (x = -a) of a parabola, find its equation. If a light ray comes from infinity parallel to the x-axis, hit the parabola like a mirror, prove that after reflection it must goes through F.
17. Prove a + ar + ar2 + . . . + arn-1 = a(1-rn)/(1-r).
18. Prove the harmonic series diverges.
19. Find 1/(1×2) + 1/(2×3) + 1/(3×4) + . . . + 1/(n(n+1)).
20. Gabriel's trumpet is formed by rotating the curve y = 1/x, x > 1, around the x-axis. Find its volume and surface area.
21. The cycloid is the trajectory of a point on the rim of a bicycle wheel. Derive x and y in terms of the angle of rotation. Find its length after one rotation of the wheel.
