Let `g: R -> R,\ \ g(x) = (a - x)^2`, where `a` is a real constant.
The average value of `g` on the interval `[– 1, 1]` is `31/12.`
Find all possible values of `a.` (3 marks)
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Let `g: R -> R,\ \ g(x) = (a - x)^2`, where `a` is a real constant.
The average value of `g` on the interval `[– 1, 1]` is `31/12.`
Find all possible values of `a.` (3 marks)
`+- 3/2`
`text(Solution 1)`
`1/(1 – (– 1)) int_-1^1 (a – x)^2 dx` | `= 31/12` |
`[ax^2 – ax^2 + x^3/3]_-1^1` | `= 31/6` |
`[(a^2 – a+1/3) – (-a^2-a-1/3)]` | `=31/6` |
`2a^2+2/3` | `=31/6` |
`a^2` | `=27/4` |
`:. a` | `=+- 3/2` |
`text(Solution 2)`
`1/(1 – (– 1)) int_-1^1 (a – x)^2 dx` | `= 31/12` |
`1/2 [(a – x)^3/-3]_-1^1` | `= 31/12` |
`[(a – x)^3]_-1^1` | `= – 31/2` |
`(a – 1)^3 – (a + 1)^3` | `= – 31/2` |
`(a^3 – 3a^2 + 3a – 1) – (a^3 + 3a^2 + 3a + 1)=- 31/2`
`-6a^2 – 2` | `= – 31/2` |
`6a^2` | `= 27/2` |
`a^2` | `= 9/4` |
`:. a` | `= +- 3/2` |
Consider the function `f:[−3,2] -> R, \ \ f(x) = 1/2(x^3 + 3x^2 - 4)`.
The rule for `f` can also be expressed as `f(x) = 1/2(x - 1)(x + 2)^2`.
a. `text(Stationary points when)\ \ f´(x)=0,`
`1/2(3x^2 + 6x)` | `= 0` |
`3x(x + 2)` | `= 0` |
`:. x = 0, − 2`
`:.\ text(Coordinates of stationary points:)`
`(0, − 2), (− 2,0)`
b. |
c. | `text(Avg value)` | `= 1/(2 – 0) int_0^2 f(x) dx` |
`= 1/2 int_0^2 1/2(x^3 + 3x^2 – 4)dx` | ||
`= 1/4[1/4x^4 + x^3 – 4x]_0^2` | ||
`= 1/4[(16/4 + 2^3 – 4(2))-0]` | ||
`= 1` |