Differentiation Rates of change KUS objectives BAT Use

Differentiation: Rates of change • KUS objectives BAT Use the chain rule to connect rates of change and solve problems Starter differentiate :

WB 40 Given that the area of a circle A, is related to its radius r by the formula A = πr 2, and the rate of change of its radius in cm is given by dr/dt = 5, find d. A/dt when r = 3 You are told a formula linking A and r You are told the radius is increasing by 5 at that moment in time You are asked to find how much the area is increasing when the radius is 3 This is quite logical – if the radius is increasing over time, the area must also be increasing, but at a different rate… We can set up a Chain rule like this WORKING OUT SPACE You need to find a derivative of A with respect to r

You need to use the information to set up a chain of derivatives that will leave Rate of change of surface area over time We are told d. V/ dt in the question so we will use it We know a formula linking V and r so can work out d. V/dr d. S/ dt We know a formula linking S and r so can work out d. S/dr This isn’t all correct though, as multiplying these will not leave d. S/dt However, if we flip the middle derivative, the sequence will work!

WB 41 b The total surface area is given by the formula: Now work out d. V/dr and DS/dr using the given formulas WORKING OUT SPACE

WB 42 expanding cube The length of a cube is increasing at a constant rate of 1. 5 mms-1 At the moment when the length of the edge is 30 mm find the rate of increase of the surface area Chain rule surface area, differentiated • from the Question Substitute in

WB 43 circle A circular stain is increasing so that its radius is increasing at a constant rate of 0. 8 mms-1 Find the rate at which the area is increasing when the radius is 19 mm Chain rule area, differentiated • From the Question Substitute in

WB 44 water tank A water tank has a rectangular base measuring 2. 6 m by 6. 8 m The sides are vertical. Water is pumped in to the tank at a rate of 32 m 3 per minute At what rate is the depth of water increasing? Volume, differentiated, let x = depth Chain rule • From the Question Substitute in

WB 45 expanding sphere The radius of a sphere is increasing at a rate of 0. 2 ms-1 Find the rate of increase of a) The volume b) The surface area Of the balloon at the instant when r = 1. 6 m Chain rule Volume of sphere, differentiated Chain rule S’ Area of sphere, differentiated • Substitute in

WB 46 decreasing sphere Air is being lost from a spherical air balloon at a constant rate of 3. 6 m 3 per minute Find the rate at which the radius is decreasing at the instant when the radius is 14. 5 m Start with Volume, differentiated • Read Question again Substitute in Chain rule

• KUS objectives BAT Use the chain rule to connect rates of change and solve problems Write one thing you have learned Write one thing you need to improve

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