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Differentiation and Integration

Differentiation and Integration. Differentiation and Integration together are called Calculus. Differentiation. How do you know when to differentiate?. Differentiate if the question says: Derive Find the derivative Find the rate of change Find or f ’ (x) Maximise or minimise.

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Differentiation and Integration

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  1. Differentiation and Integration Differentiation and Integration together are called Calculus. Maths revision course by Miriam Hanks

  2. Differentiation How do you know when to differentiate? • Differentiate if the question says: • Derive • Find the derivative • Find the rate of change • Find or f ’(x) • Maximise or minimise Maths revision course by Miriam Hanks

  3. Differentiation How do you prepare for differentiation? • Change to • Get rid of fractions with x on the bottom: • eg • Multiply out brackets Maths revision course by Miriam Hanks

  4. Differentiation How do you differentiate? Bring the power to the front (to multiply) Subtract 1 from the power Maths revision course by Miriam Hanks

  5. Differentiation in real life Why was differentiation developed originally? In Isaac Newton's day, shipwrecks occurred regularly because the ship was not where the captain thought it was using only the stars for guidance. Calculus was developed to improve navigation techniques. Maths revision course by Miriam Hanks

  6. Differentiation in real life Differentiation is used in economics and business to find optimum solutions. eg to use the minimum amount of material to make the packaging, or to maximise efficiency of products or to minimise their costs. Maths revision course by Miriam Hanks

  7. Differentiation and Integration Click for calculus video Maths revision course by Miriam Hanks

  8. Graphs of the derivative If you are asked to draw the graph of the derivative, keep in mind that this is a graph of the gradient. • Start by marking points on the axes where there are turning points on the original graph. • Now look at the gradient before and after each of these points: Is it up (in which case, draw above the x- axis) or down (draw below x-axis)? • Make sure you curve is smooth Maths revision course by Miriam Hanks

  9. Integration How do you know when to integrate? • Integrate if the question says: • Find the integral • Find the area under a curve • Find the area between 2 curves • Find • Find y when you are given Maths revision course by Miriam Hanks

  10. Integration How do you prepare for integration? • Change to • Get rid of fractions with x on the bottom: • eg • Multiply out brackets Maths revision course by Miriam Hanks

  11. Integration How do you integrate? Divide by the new power Add 1 to the power + c Maths revision course by Miriam Hanks

  12. Integration in real life Integration is used to analyse crash test data, to improve safety of cars. It is also used by engineers to find the force of water from the surface area of a reservoir, to design dams. Maths revision course by Miriam Hanks

  13. Further Calculus To differentiate or integrate a trig function, use the formula sheet: • sin ax a cos ax • cos ax -a sin ax • sin ax - cos ax • cos ax sin ax differentiates to integrates to Maths revision course by Miriam Hanks

  14. Further Calculus To differentiate or integrate a bracket with a power: 1. Think of the whole bracket as an “x” and differentiate or integrate it as usual 2. Multiply by the derivative of the bracket (if differentiating) orDivide by the derivative of the bracket (if integrating) Maths revision course by Miriam Hanks

  15. Integration Click for an integration joke Maths revision course by Miriam Hanks

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