II Calculus Fun Problems
1 1. Show that 4 tan,1 1 , tan,1 239 = . 5 4
2. Notice that 2 2 = 2 and 2 4 = 4 . Show analytically that 2 x 6= x for any other real x value. 3. Which one is larger e or e ? You must prove your answer analytically! Z1 1 4. Suppose f is a 1-1 continuous function with f 0 = 0, f 1 = 1, and f x dx = 3 . 0 Z1 ,1 Find the value of the integral f y dy . 5. Suppose f 00 is a continuous function on 0; 1 and f 1 = f 01 = 0 . Show that Z1 Z 2 f 00 x dx = 2 1 f x dx . x
0 0 0
p
p
p
dx 2 6. Prove that 0 1 + tan xa = 4 for any real number a di erent from zero.
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7. Suppose f is an even continuous function. Show that if tlim f x dx = L, then !1 ,t Z1 f x dx = L . 8. Suppose l : + ,! is a function with l0 = 0, l 0 x and continuous l 0 x 1 for all x. Prove that l is an arclength function. That is, lx is the arclength of a curve y = f t from t = 0 to t = x. 9. If the circular wheel x = cos t, y = 1 + sin t, 0 t 2, is rolled along the x-axis, its top touches the line y = 2 continuously. Find another wheel which will do the same. 10. A ball is dropped from a height of 10 feet and bounces. Each bounce is 43 of the height of the bounce before. This ball will boun...