Calculus Rates of Change

Remember: The rate of change, at a point is often called the instantaneous rate of change

(b) What does h’(t) represent? A cup of coffee can be made by pouring boiling water onto coffee grounds placed into a special funnel lined with a filter. The level of coffee in the cup rises as the liquid soaks through the grounds and filter. The depth of coffee h cm in the cup t seconds after the water is poured onto the grounds is given by the equation h = t – 0.025t2 (0 ≤ t ≤ 20) (a) Find h’(t) (b) What does h’(t) represent? (c) At what rate is the depth changing when (i) t = 5 (ii) t = 19

There are two main points to remember from this section of work (1) if asked for a rate of change between two given values, find the average rate of change between these points. (2) if asked for the rate of change at one given value, find the instantaneous rate of change at the point. This involves differentiation

A horticulturalist knows that the height (h mm) of a new breed of tomatoes closely follows the equation h = 8t – 0.2t2 + 6 Where t is the number of days since planting out. Find the rate of growth for the tomatoes over the first four days (b) Find the rate at which the tomatoes are growing at the end of the forth day.

The reserves in oil (V million barrels) in an oil field in the North Sea, x years after 2005, are expected to be given by V = 1.80 x 105 – 10.5 (x4 – 40x3 + 450x2) Find an expression for the rate at which V is decreasing. At what rate is V decreasing (i) when x = 5 (at the end of 2010) (ii) when x = 10 (at the end of 2015) (c) What is the average rate of decrease between the end of 2010 and 2015?

Kinematics Distance, Velocity and Acceleration Definitions: Velocity is a measure of a rate of change of distance compared with time Acceleration is a measure of the rate of change of velocity, compared with time. s for distance s(t) for distance at time t v for velocity v(t) for velocity at time t a for acceleration

Distance, Velocity and Acceleration Positive acceleration indicates that an object is speeding up Negative acceleration indicates that an object is slowing down If the acceleration, a=0, the object is travelling at constant speed Note that s, v and a are all functions of time s differentiate v differentiate a

Distance, Velocity and Acceleration Average velocity = change in distance change in time = s(t2) – s(t1) t2 - t1 (From time t1 to t2) Velocity at a point = ds dt

Distance, Velocity and Acceleration Average acceleration = change in velocity change in time = v(t2) – v(t1) t2 - t1 (From time t1 to t2) Velocity at a point = dv dt

Distance, Velocity and Acceleration If the distance (in metres) of a body at time t (in seconds) is given by s = t3 – 5 t2 + 2 t + 6 2 Find an equation for the velocity, v m/s, of the body at time t seconds. (b) What is the velocity after 3 seconds? (c) What is the average velocity during the first 3 seconds? (d) Find an equation for the acceleration, a m/s2, of the body at time t seconds. (e) What is the acceleration after 3 seconds?