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Differentiation

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  1. Differentiation

  2. After completing this chapter you should be able to: • Find the gradient of a curve whose equation is expressed in a parametric form • Differentiate implicit relationships • Differentiate power functions such as ax • Use the chain rule to connect the rates of change of two variables • Set up simple differential equations from information given in context

  3. 4.1 You can find the gradient of a curve given in parametric coordinates • You differentiate x and y with respect to t • Then you use the chain rule rearranged into the form

  4. 1. Find the gradient of the curve with parametric equations x = t², y = 2t³ 2. Find the gradient of the curve with parametric equations x =, y = this can be tidied up by multiplying by

  5. 3. Find the gradient of the curve with parametric equations x =, y = so Exercise 4A page 38

  6. 4.2 You can differentiate relations which are implicit, such as x² + y² = 8x, and cos(x + y) = siny The basic skill is to differentiate each term in turn using the chain rule and the product rule as appropriate: ① ②

  7. Find in terms of x and y where Differentiating term by term gives us This gives us

  8. Find in terms of x and y where Differentiating term by term From ② From ① This gives us = 0 Collect all the terms

  9. 4.3 You can differentiate the general power function ax, where a is constant 1. Find 2. Find

  10. 3. Find Use the product rule 4. Find Exercise 4C page 41

  11. 4.4 You can relate one rate of change to another If a question involves two or more variables you can connect the rates of change by using the chain rule.

  12. Given that the volume V cm³ of a hemisphere is related to its radius r cm by the formula V = and that the hemisphere expands so that the rate of increase of it’s radius in cm-1 is given by , find the exact value of when r = 3. V = When r = 3 the value of

  13. 4.5 You can set up a differential equation from information given in context In Core 4 we are only going to look at first order differential equations , which only involves the first derivative

  14. A curve C has equation y = f(x) and it’s gradient at each point on the curve is directly proportional to the square of it’s y coordinate at that point. At the point (0, 3) on the curve the gradient is 1. Write down a differential equation, which could be solved to give the equation of the curve. State the value of any constant of proportionality which you use. At (0, 3) the gradient is 1 Equation is

  15. The head of a snowman of radius R cm loses volume by evaporation at a rate proportional to it’s surface area. Assuming that the head is spherical, that the volume of a sphere is cm³ and that the surface is 4πR² cm², write down a differential equation for the rate of change of radius of the snowman’s head. We need to find we know V is the volume, A is the surface area, k is the constant of proportionality, t is time. (head is shrinking so equation is - ) and and A = 4πR²

  16. all done Exercise 4E page 45