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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. Functions. Consider the function y = ±√x. This equation gives two values of y for every value of x. This is called a one-to-many rule. .

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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. Functions Consider the function y = ±√x This equation gives two values of y for every value of x. This is called a one-to-many rule. x is known as the independent variable and y is known as the dependent variable. A graph of the equation y = ±√x can be drawn by constructing a table of values. A function is defined as a rule that gives one value of y for each value of x. x 0 1 2 3 4 y 0 ±√1 ±√2 ±√3 ±√4

  3. Functions This equation has two values of x for each value of y (except 0). This is called a many-to-one rule. This is a function. Consider the function f(x) = x2 Range A function is a rule that returns a unique value ‘f(x)’ to each value ‘x’ in a given domain. The set of values given by ‘f(x)’ for all values of ‘x’ is called the range of the function. Domain

  4. Functions Is this graph described by a many-to-one or a one-to-many equation? This is a many-to-one equation

  5. Functions Exponential Functions are always of the form Graphs of exponential functions with different bases are shown below. Where a is called the base of the exponential function. The natural exponent is defined by or and may be defined by these conditions or

  6. Functions The hyperbolic functions are defined by A graph of hyperbolic functions is shown below. y = cosh(x) Properties y = tanh(x) y = sinh(x)

  7. Functions log10(1000) = y What is y? 103 = 1000 Hence y = 3 Logarithms are the inverse of Exponents. The Natural Logarithm is commonly used and it is the inverse of the natural exponent. It is written as Logarithms help solve the following problem Where x and a are known but y is not. which is spoken as ‘log to the base e’ The solution to this problem is written as Properties Loga(ax) ≡ x Loga(x.y) ≡Logax + Logay Loga(x/y) ≡Logax - Logay Loga(xp) ≡p.Loga(x) Loga(1) ≡ 0 And spoken as ‘log to the base a of x’

  8. Coordinates in a Plane We can deduce that The general formula for a straight line is given by A common formula describing a straight line is given by Where a, b and c are constants. y Determine m and c in terms of x1, x2, y1, and y2 x2, y2 x, y x1, y1 x

  9. Graphs Find the radius and centre of the circle described by the following formula The general formula for a circle is given by y (a,b) r x If a curve is given by f(x,y) then f(x-a, y-b) represents the same curve translated by a distance ‘a’ on the x axis and ‘b’ on the y axis. Therefore the centre is at (1/2, -1) and the radius is 2.

  10. Graphs The general formula for an ellipse is given by The general formula for a hyperbola is given by a > 0, b > 0 a > 0, b > 0 y y y=bx/a y=-bx/a b a -a x x -b

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

  12. 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

  13. 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

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

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

  16. 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

  17. 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

  18. L’Hopital’s Rule Question What is the solution to the following equation? A systematic way of finding the limits to difficult functions is provided by L’Hopital’s rule. We can prove that if Hence Then Note: zero divided by zero does not equal zero. The answer does not exist.

  19. Hence Limits 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

  20. Graph Sketching • It easy to sketch a graph of a function using the following guidelines. We will learn how to sketch graphs by following an example. • i) Zeros - obtain values of x where y = 0 • ii) Poles - obtain values of x that make the denominator equal to zero (called poles). iii) Asymptotes - these are straight lines that the curve approaches as either x → ±∞ or y → ±∞. Determine if the curve approaches the asymptote from above or below. iv) Special points - calculate values of f(x) explicitly at selected points to give further insight. e.g. at x = 0

  21. Graph Sketching x=-1, y=∞ y (-1/2,2) (0,1) (1/2,0) x (2,-1) y=-2, x=∞

  22. Graph Sketching x=-1, y=∞ y y=1, x=∞ (2,1/3) (1,0) x (0,-1) • Sketch the function (-1/2,3)

  23. 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.

  24. 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

  25. The Second Derivative y y x x Minima Maxima

  26. The Second Derivative Inflection Point y x

  27. The Second Derivative y Inflection Point x

  28. 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 lectures • 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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