Applications of Derivatives

Tangents, curve sketching with the first and second derivatives, optimization and related rates.

12 questions, free, marked as you go with a worked explanation on every one.

What this covers

For f(x) = x³ - 3x² + 2, find the x-coordinate of the local minimum.
For that same function f(x) = x³ - 3x² + 2, find the x-coordinate of the point of inflection.
Find the slope of the tangent line to the curve y = x² - 4x + 1 at the point where x = 3.
An open-topped box is made from a 12 cm by 12 cm square sheet of card by cutting an equal square of side c from each corner and folding up the sides. What value of c, in centimetres, gives the box its maximum volume?
Air is pumped into a spherical balloon at a rate of 100 cm³ per second. At the instant when the radius is 5 cm, how fast is the radius increasing, in centimetres per second? Round to three decimal places.
A particle moves along a line so that its position after t seconds is s(t) = t³ - 6t² + 9t metres. What is the acceleration of the particle at t = 1 second?
For a function f, what does it mean if f′(x) > 0 and f″(x) < 0 on an interval?
Which of these describes the horizontal asymptote of f(x) = (2x² + 3) / (x² - 1)?
Find the slope of the tangent line to y = x cubed at the point where x = 2.
Find the slope of the tangent line to y = 3x squared minus 5x plus 1 at the point where x = 1.
At a point where the derivative of a function equals zero, what does this tell you about the tangent line there?
Find the slope of the tangent line to y equals the square root of x at the point where x = 4.