8. (a) Sketch the curve with equation
$$y = a ^ { - x } + 4$$
where \(a\) is a constant and \(a > 1\)
On your sketch show
- the coordinates of the point of intersection of the curve with the \(y\)-axis
- the equation of any asymptotes to the curve.
(3)
(b) Use the trapezium rule with 5 trapeziums to find an approximate value for
$$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$
giving your answer to two significant figures.
(3)
(c) Using the answer to part (b), find an approximate value for
- \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x\)
- \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x\)