Find the coordinates of the point `P` on the curve `y = 2e^x + 3x` at which the tangent to the curve is parallel to the line `y = 5x - 3`. (3 marks)
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Find the coordinates of the point `P` on the curve `y = 2e^x + 3x` at which the tangent to the curve is parallel to the line `y = 5x - 3`. (3 marks)
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`(0, 2)`
`text(Gradient of)\ \ y = 5x − 3\ text(is)\ 5.`
`y` | `= 2e^x + 3x` |
`(dy)/(dx)` | `= 2e^x + 3` |
`text(Find)\ \ x\ \ text(when)\ \ (dy)/(dx) = 5`
`5` | `= 2e^x + 3` |
`2e^x` | `= 2` |
`e^x` | `= 1` |
`x` | `= 0` |
`text(When)\ \ x = 0`
`y` | `= 2e^0 + (3 xx 0)` |
`= 2` |
`:.P\ \ text{has coordinates (0, 2)}`
The line `y = mx` is a tangent to the curve `y = e^(2x)` at a point `P`.
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i. |
ii. | `y` | `= e^(2x)` |
`dy/dx` | `= 2e^(2x)` |
`text(Gradient of)\ \ y = mx\ \ text(is)\ \ m`
`text(Gradients equal when)`
`2e^(2x)` | `= m` |
`e^(2x)` | `= m/2` |
`ln e^(2x)` | `= ln (m/2)` |
`2x` | `= ln (m/2)` |
`x` | `= 1/2 ln (m/2)` |
`text(When)\ \ x = 1/2 ln (m/2)`
`y` | `= e^(2 xx 1/2 ln (m/2))` |
`= e^(ln(m/2))` | |
`= m/2` |
`:.\ P (1/2 ln (m/2), m/2)`
iii. `y=mx\ \ text(passes through)\ \ (0,0)\ text(and)\ (1/2 ln (m/2), m/2)`
`text(Equating gradients:)`
`(m/2 – 0)/(1/2 ln (m/2) – 0)` | `=m` |
`m/2` | `=m xx 1/2 ln(m/2)` |
`ln (m/2)` | `= 1` |
`m/2` | `= e^1` |
`m` | `= 2e` |