Academic Skills Advice Rate of Change Summary If you are asked to find the rate of something you need to do: Change in something π(πππππππππ) ππ‘ For example: Change in time Find the rate of increase of the area (A) = π(π΄) ππ‘ Find the rate of increase of the height (h) = π(β) ππ‘ π(π) Find the rate of increase of the radius (r) = ππ‘ etc Fractions Rates of change can be manipulated the same as normal fractions to find the one you want. Examples: Find Find π π π π π π π π , given , given πβ ππ ππ ππ ππ and and ππ‘ ππ‘ ππ multiplying gives: dividing gives: πβ ππ ππ ππ × ππ ÷ ππ‘ ππ‘ ππ = π π = ππ π π ππ × ππ ππ‘ = π π π π Example: The area of the surface of a sphere is given by π΄ = 4ππ 2 , where π is the radius. If the radius of a sphere is increasing at a rate of 3cm/s, find the rate of increase of the area when π = 5cm. Weβre given the rate of increase of the radius, i.e. ππ ππ‘ We want the rate of increase of (=3) the area, i.e. ππ΄ ππ‘ We also know that π΄ = 4ππ 2 Therefore: ππ΄ ππ = 8ππ We can use the fact that ππ΄ ππ‘ = ππ΄ ππ × ππ ππ‘ , therefore ππ΄ ππ‘ = 8ππ × 3 = 24ππ When π = 5, the rate of increase of the area = 120π H Jackson 2010 / Academic Skills 1 Except where otherwise noted, this work is licensed under http://creativecommons.org/licenses/by-nc-sa/3.0/
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