Let `f(x) = (e^x)/(cos(x))`.
Evaluate `f′(pi)`. (2 marks)
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Let `f(x) = (e^x)/(cos(x))`.
Evaluate `f′(pi)`. (2 marks)
`text(See Worked Solutions)`
`f′(x) = (e^x)/(cos(x))`
`u` | `= e^x` | `v` | `= cos(x)` |
`u′` | `= e^x` | `v′` | `= −sin(x)` |
`f′(x)` | `= (u′v – uv′)/(v^2)` |
`= (e^x · cos(x) + e^x sin(x))/(cos^2(x))` |
`f′(pi)` | `= (e^pi · cospi + e^pi sinpi)/(cos^2 pi)` |
`= (e^pi(−1) + e^pi · 0)/((−1)^2)` | |
`= −e^pi` |
For `y = e^(2x) cos (3x)` the rate of change of `y` with respect to `x` when `x = 0` is
`B`
`y` | `= e^(2x) cos (3x)` |
`dy/dx` | `=e^(2x) xx -3sin(3x) + 2e^(2x) xx cos (3x)` |
`=e^(2x)(-3sin(3x) + 2cos(3x))` |
`text(When)\ \ x = 0,`
`dy/dx= 2`
`=> B`
Let `g(x) = log_e(tan(x))`. Evaluate `g′(pi/4)`. (2 marks)
`g′(pi/4) = 2`
`g(x) = log_e (tan(x))`
`text(Using Chain Rule:)`
`g′(x) = (sec^2(x))/(tan(x))`
`text(When)\ \ x = pi/4,`
`g′(pi/4)` | `= (sec^2(pi/4))/(tan (pi/4))` |
`=1/(1/sqrt2)^2` | |
`=1/(1/2)` | |
`=2` |