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The function with rule `f(x)` has derivative `f^{prime}(x) = cos\ 3x`.
If `f(pi/6) = 1,` find `f(x).` (3 marks)
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`f(x)= 1/3 sin\ 3x + 2/3`
| `int f(x)\ dx` | `=int cos\ 3x\ dx` |
| `= 1/3 sin\ 3x + c` | |
| `f(pi/6)` | `= 1/3\ [sin\ (3 xx pi/6) ]+ c` |
| `1` | `= 1/3\ sin\ pi/2+c` |
| `c` | `= 2/3` |
`:.f(x)= 1/3 sin\ 3x + 2/3`
The function with rule `g(x)` has derivative `g^{prime} (x) = sin (2 pi x).`
Given that `g(1) = 1/pi`, find `g(x).` (2 marks)
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`(3-cos (2 pi x))/(2 pi)`
| `g^{prime} (x)` | `= int sin (2 pi x) dx` |
| `= (-cos (2 pi x))/(2 pi) + c` |
`text(Substitute)\ \ (1, 1/pi)\ \ text(into)\ g prime(x):`
| `1/pi` | `= (-cos (2 pi))/(2 pi) + c` |
| `1/pi` | `= -1/(2 pi) + c` |
| `:. c` | `= 3/(2 pi)` |
`:. g(x) = (3-cos (2 pi x))/(2 pi)`
If `f^{prime}(x) = 2cos(x)-sin(2x)` and `f(pi/2) = 1/2`, find `f(x)`. (3 marks)
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`2sinx + 1/2cos(2x)-1`
| `f(x)` | `= int(2cosx-sin2x)dx` |
| `= 2sinx + 1/2cos(2x) + c` |
`text(Substitute)\ \ f(pi/2) = 1/2:`
| `1/2` | `= 2sin(pi/2) + 1/2cos(pi) + c` |
| `1/2` | `= 2-1/2 + c` |
| `c` | `=-1` |
| `:. f(x)` | `= 2sinx + 1/2cos(2x)-1` |