Question 3
(5x)3 cos 2x 4. sin- 1(x x ), 0 ≤ x ≤ 1 5. 1cos 2 2 7 x x − + , - 2 < x < 2 6. 1 1 sin 1 sincot 1 sin 1 sin x x x x − + + − + − − , 0 < x < 2 π 7. (log x)log x, x > 1 8. cos (a cos x + b sin x), for some constant a and b. 9. (sin x - cos x ) (sin x - cos x), 3 4 4xπ π< < 10. xx + xa + ax + aa, for some fixed a > 0 and x > 0 11. ( ) 22 3 3 xxx x− + − , for x > 3 12. Find dy dx , if y = 12 (1 - cos t), x = 10 (t - sin t), 2 2tπ π− < < 13. Find dy dx , if y = sin-1 x + sin-1 21 x− , 0 < x < 1 14. If 1 1 0x y y x+ + + = , for , - 1 < x < 1, prove that ( ) 2 1/1 / dy dx x = − + 15. If (x - a)2 + (y - b)2 = c2, for some c > 0, prove that 2 2 2/2 1 dy dx d y dx + is a constant independent of a and b. 16. If cos y = x cos (a + y), with cos a ≠ ± 1, prove that 2cos ( ) sin
| dy | a | y |
|---|---|---|
| dx a += . 17. If x = a (cos t + t sin t) and y = a (sin t - t cos t), find 2/2 d y dx . 18. If f(x) = | x | 3, show that f ″(x) exists for all real x and find it. 19. Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines. 20. Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer. 21. If |
| ( ) ( ) ( )f | x | g |
| --- | --- | --- |
| y | l | m |
| a | b | c |
= , prove that
| ( ) ( ) ( )f | x | g | x | h | x |
|---|---|---|---|---|---|
| dy l m ndx | a | b | c |
′ ′ ′ = 22. If y = 1cosa xe − , - 1 ≤ x ≤ 1, show that ( ) 2 2 21 0d y dyx x a y dxdx − − − = . MATHEMATICS146