Parametric differentiation

In this question you must show detailed reasoning.

A curve has parametric equations x = t³ + t², y = t² + 2t for all real values of t.

The curve passes through the point P with coordinates (2, 3).

Show that the equation of the tangent to the curve at the point P can be written as 5 y = 4 x + 7

Determine the coordinates of the point where the tangent to the curve at P meets the curve

again.

Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, B, and the number of wildflower plants, F, at time t weeks after the start of the study.

In the model B = 20 + 2t + cos 3t and F = 50e^0.1t.

The model assumes that B and F can be treated as continuous variables.

Suggest a reason why this model may not be valid for values of t greater than 12.

Show that the equation of the tangent at P can be written as y =-2p²x + 4p.

The tangent to this curve at P crosses the x-axis at the point A and the normal to this curve at P crosses the x-axis at the point B.

Show that the ratio PA: PB is 1:2p².

A particle moves in the x - y plane so that at time t seconds, where t ⩾ 0, its coordinates are given by

x = e^2t - 4e^t + 3, y = 2e^-3t.

Explain why the path of the particle never crosses the x-axis.

Determine the exact values of t when the path of the particle intersects the y-axis.

Hence find the coordinates of the particle when its path is parallel to the y-axis.

The diagram shows the curve C with parametric equations

The maximum point on C is denoted by P.

Determine the exact coordinates of P.

The shaded region R is enclosed by the curve, the x-axis and the lines x = 1 and x = 6.

Show that the area of R is given by

where a and b are constants to be determined.

Hence determine the exact area of R.