Tangents, rates of change and optimisation

Question 1 of 10

The radius of a circle increases at 2 cm/s. Find the rate of increase of the area when r = 3 cm in terms of π.

Question 2 of 10

Find the gradient of the curve y = 3x² + 5x at x = 3.

Question 3 of 10

The gradient of y = 1x² at x = 1 is ___

Question 4 of 10

The gradient of the normal at a point is

Question 5 of 10

Find the equation of the tangent to y = x³ + 4 at the point where x = 1.

Question 6 of 10

Find the gradient of the curve y = 1x² + 4x at x = 1.

Question 7 of 10

Find the stationary points of y = x³ − 3x² − 9x + 5 and determine their nature.

Question 8 of 10

A particle moves so that s = t³ − 6t² + 9t. Find its velocity and acceleration at t = 2 and when it is momentarily at rest.

Question 9 of 10

A farmer has 100 m of fencing to enclose a rectangular plot against a wall (no fencing along the wall). Find the dimensions that maximise the area.

Question 10 of 10

The gradient of y = 1x² at x = 5 is ___