The graph of the function `f(x)=2xe^((1-x^(2)))`, where `0 <= x <= 3`, is shown below.
- Find the slope of the tangent to `f` at `x=1`. (1 mark)
- Find the obtuse angle that the tangent to `f` at `x = 1` makes with the positive direction of the horizontal axis. Give your answer correct to the nearest degree. (1 mark)
- Find the slope of the tangent to `f` at a point `x =p`. Give your answer in terms of `p`. (1 mark)
- i. Find the value of `p` for which the tangent to `f` at `x=1` and the tangent to `f` at `x=p` are perpendicular to each other. Give your answer correct to three decimal places. (2 marks)
- ii. Hence, find the coordinates of the point where the tangents to the graph of `f` at `x=1` and `x=p` intersect when they are perpendicular. Give your answer correct to two decimal places. (3 marks)
Two line segments connect the points `(0, f(0))` and `(3, f(3))` to a single point `Q(n, f(n))`, where `1 < n < 3`, as shown in the graph below.
- i. The first line segment connects the point `(0, f(0))` and the point `Q(n, f(n))`, where `1 < n < 3`.
- Find the equation of this line segment in terms of `n`. (1 mark)
- ii. The second line segment connects the point `Q(n, f(n))` and the point `(3, f(3))`, where `1 < n < 3`.
- Find the equation of this line segment in terms of `n`. (1 mark)
- iii. Find the value of `n`, where `1 < n < 3`, if there are equal areas between the function `f` and each line segment.
- Give your answer correct to three decimal places. (3 marks)