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Introduction

Introduction. We will cover 4 topics today 1. Gradient of a Slope 2. Rates of Change 3. Important Derivatives 4. Higher Order Derivatives. The Gradient of a Slope. y. y. q. q. q. dy. p. p. dx. tangent. x. x.

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Introduction

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  1. Introduction We will cover 4 topics today 1. Gradient of a Slope 2. Rates of Change 3. Important Derivatives 4. Higher Order Derivatives

  2. The Gradient of a Slope y y q q q dy p p dx tangent x x As the length of PQ decreases, the gradient of PQ moves closer to the gradient of the tangent through p. As dx → 0 ; q → p

  3. The Gradient of a Slope dy = (y at q) – (y at p) Obtain an expression for dy q dy Hence p dx Find the gradient of the curve at Thus

  4. Rates of Change Find the rate of change of the area of a circle with respect to its radius. Call the radius ‘r’ and the area ‘A’. The rate of change of A with respect to r is Therefore Now let dr → 0 or

  5. Definitions If y is a constant i.e. y = c then Also If y = xn then

  6. Proof Substituting More generally Hence If f(x) = xn then Thus

  7. Questions If y = x3 Obtain the general expression for dy/dx What is the gradient of the curve at x = 2 The angle of the tangent to the curve that passes through x = 2 The equation of this tangent The curve passes through the point (2,8), hence

  8. Questions A car travels along a straight road with varying velocity for one hour. After t hours, its displacement (x) from the starting point is given by Find an expression for the velocity The velocity is the rate of change of displacement with time Find an expression for the acceleration The acceleration is the rate of change of velocity with time

  9. Important Derivatives Question What is the solution to the following equation? Let y = ex Thus By taking successively smaller values of ε we find that and Hence Conversely

  10. Hence Important Derivatives Let y = sin(x) If x is an angle then x and dx are in radians. Recall the identity Thus Also Let C = x + dx and D = x

  11. Higher Order Derivatives By differentiating again we get We may differentiate a function and then differentiate the result. E.g. This is called the second derivative of the function. In general, we use the notation Let y = x4 Then If y = f(x) then we use the following notation This is called the first derivative of the function.

  12. Higher Order Derivatives Let n and r be any integers with The general case is Prove that n! is defined as If y = xn then we know that And successively Thus

  13. The Second Derivative y y x x Minima Maxima

  14. Conclusion Today we have looked at 1. Gradient of a Slope 2. Rates of Change 3. Important Derivatives 4. Higher Order Derivatives • Essential reading for the next two weeks • HELM Workbook 11.1 Introducing Differentiation • HELM Workbook 11.2 Using a Table of Derivatives • HELM Workbook 11.3 Higher Derivatives • HELM Workbook 11.4 Differentiating Products and Quotients • HELM Workbook 11.5 The Chain Rule • HELM Workbook 11.6 Parametric Differentiation • HELM Workbook 11.7 Implicit Differentiation

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