Tangents, rates of change and optimisation

Question 1 of 10

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

Question 2 of 10

The gradient of the normal equals the gradient of the tangent.

Question 3 of 10

The gradient of the normal at a point is

Question 4 of 10

A positive dy/dx means the function is increasing.

Question 5 of 10

A rate of change of volume with time is written

Question 6 of 10

Find the gradient of the curve y = 3x² + 3x at x = 1.

Question 7 of 10

Find the minimum value of y = x² − 2x + 11.

Question 8 of 10

The radius of a circle increases at 2 cm/s. Find the rate of increase of the area when r = 4 cm in terms of π.

Question 9 of 10

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

Question 10 of 10

A rectangle has perimeter 16 cm. Find the dimensions that give the maximum area.