Linear Interpolation Only

6. $$f ( x ) = \tan \left( \frac { x } { 2 } \right) + 3 x - 6 , \quad - \pi < x < \pi$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 1,2 ]\).
2. Use linear interpolation once on the interval \([ 1,2 ]\) to find an approximation to \(\alpha\). Give your answer to 2 decimal places.

2. $$\mathrm { f } ( x ) = \cos \left( x ^ { 2 } \right) - x + 3 , \quad 0 < x < \pi$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 2.5,3 ]\).
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2. Use linear interpolation once on the interval [2.5,3] to find an approximation for \(\alpha\), giving your answer to 2 decimal places.

4. $$f ( x ) = 2 ^ { x } - 6 x$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [4,5]. Using the end points of this interval find, by linear interpolation, an approximation to \(\alpha\).

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1. Show that the equation $$x ^ { 3 } + 2 x - 2 = 0$$ has a root between 0.5 and 1 .
2. Use linear interpolation once to find an estimate of this root. Give your answer to two decimal places.

$$f(x) = 2 \sin 2x + x - 2.$$ The root \(\alpha\) of the equation \(f(x) = 0\) lies in the interval \([2, \pi]\). Using the end points of this interval find, by linear interpolation, an approximation to \(\alpha\). [4]

$$f(x) = 2^x + x - 4.$$ The equation \(f(x) = 0\) has a root \(\alpha\) in the interval \([1, 2]\). Use linear interpolation on the values at the end points of this interval to find an approximation to \(\alpha\). [2]

The temperature \(\theta\) °C of a room \(t\) hours after a heating system has been turned on is given by $$\theta = t + 26 - 20e^{-0.5t}, \quad t \geq 0.$$ The heating system switches off when \(\theta = 20\). The time \(t = \alpha\), when the heating system switches off, is the solution of the equation \(\theta - 20 = 0\), where \(\alpha\) lies in the interval \([1.8, 2]\).

1. Using the end points of the interval \([1.8, 2]\), find, by linear interpolation, an approximation to \(\alpha\). Give your answer to 2 decimal places. [4]
2. Use your answer to part (a) to estimate, giving your answer to the nearest minute, the time for which the heating system was on. [1]

$$f(x) = 3x^2 + x - \tan \left( \frac{x}{2} \right) - 2, \quad -\pi < x < \pi.$$ The equation \(f(x) = 0\) has a root \(\alpha\) in the interval \([0.7, 0.8]\). Use linear interpolation, on the values at the end points of this interval, to obtain an approximation to \(\alpha\). Give your answer to 3 decimal places. [4]