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Consider the curve \(y=ax^2+bx+c\), where \(a \neq 0\).
At a particular point, the tangent and normal to the curve are given by \(t(x)=2x+3\) and \(n(x)=-\dfrac{1}{2}x-2\) respectively.
The curve has a minimum turning point at \(x=-4\).
Find the values of \(a, b\) and \(c\). (4 marks)
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\(a=\dfrac{1}{2}, b=4, c=5\)
| \(y\) | \(=ax^{2}+bx+c\) | |
| \(y^{′}\) | \(=2ax+b\) |
\(\text{SP’s when}\ \ y^{′}=0\ \text{and}\ \ x=-4:\)
\(-8a+b=0\ …\ (1)\)
\(\text{Find point where}\ t(x), n(x)\ \text{and curve intersect:}\)
| \(2x+3\) | \(=-\dfrac{1}{2}x-2\) | |
| \(4x+6\) | \(=-x-4\) | |
| \(5x\) | \(=-10\) | |
| \(x\) | \(=-2\) |
\(\Rightarrow\ \text{Intersection at}\ (-2,-1)\)
\(\text{At}\ \ x=-2,\ m_{\text{tang}} = -4a+b\)
\(-4a+b=2\ …\ (2) \)
\(\text{Subtract}\ (2)-(1):\)
\(4a=2\ \ \Rightarrow\ \ a=\dfrac{1}{2}\)
\(\text{Substitute}\ \ a=\dfrac{1}{2}\ \ \text{into (1):}\)
\(\Rightarrow\ \ b=4\)
\(\text{Since}\ (-2,-1)\ \text{lies on}\ y: \)
\(-1=\dfrac{1}{2}(-2)^{2}+4(-2)+c\)
\(\Rightarrow c=5\)
\(\therefore a=\dfrac{1}{2}, b=4, c=5\)
The gradient of a curve is given by `dy/dx = 6x-2`. The curve passes through the point `(-1, 4)`.
What is the equation of the curve? (2 marks)
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`y = 3x^2-2x-1`
`dy/dx = 6x-2`
| `y` | `= int 6x-2\ dx` |
| `= 3x^2-2x + c` |
`text{Since it passes through}\ (-1,4),`
| `4` | `= 3 (-1)^2-2(-1) + c` |
| `4` | `= 3 + 2 + c` |
| `c` | `= -1` |
`:. y = 3x^2-2x-1`
The derivative of a function `f(x)` is `f^{′}(x) = 4x-3`. The line `y = 5x-7` is tangent to the graph `f(x)`.
Find the function `f(x)`. (3 marks)
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`f(x) = 2x^2-3x + 1`
`text(Solution 1)`
| `f^{′}(x)` | `=4x-3` |
| `f(x)` | `= int 4x-3\ dx` |
| `= 2x^2-3x + c` |
`text(Intersection when)`
`2x^2-3x + c = 5x-7`
`2x^2-8x + (7 + c) = 0`
`text(S)text(ince)\ \ y=5x-7\ \ text(is a tangent)\ => Delta =0`
| `b^2-4ac` | `=0` |
| `(-8)^2-[4xx2xx(7+c)]` | `= 0` |
| `64-56-8c` | `=8` |
| `8c` | `=8` |
| `c` | `=1` |
`:.f(x) = 2x^2-3x + 1`
`text(Solution 2)`
`f^{′}(x)=4x-3`
`y=5x-7\ \ text{(Gradient = 5)}`
| `=>4x-3` | `=5` |
| `x` | `=2` |
`f(x) = 2x^2-3x + 1`
`f(x)\ text{passes through (2, 3)}`
| `f(2)` | `=2xx 2^2-3(2)+c` |
| `3` | `=8-6+c` |
| `c` | `=1` |
`:.f(x) = 2x^2-3x + 1`