Ch 02.pdf - Maths B Yr 12 Ch 02 Page 43 Thursday 7:05 AM...

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2 syllabus ref efer erence ence Rates of change Optimisation In this cha chapter pter 2A Sketching curves 2B Equations of tangents and normals 2C Maximum and minimum problems when the function is known 2D Maximum and minimum problems when the function is unknown 2E Rates of change Applications of differentiation
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44 M a t h s Q u e s t M a t h s B Ye a r 1 2 f o r Q u e e n s l a n d Introduction In the previous chapter, the concept of a function was used to model relationships between variables. Whether the relationship was between volume and side length, weight loss and time, or height and time, the function and its graph proved to be a powerful tool for analysis. In that chapter, the concept of the derivative, the rate of change of one variable with another, was revisited. Now we shall use the derivative to further analyse functions and show how such analysis can be applied to solving practical problems. In particular the derivative is used to assist in sketching curves, find equations of tangents and normals, optimise functions and analyse rates of change. Sketching curves When the graphs of polynomial functions are being sketched, four main characteristics should be featured: 1. the basic shape (whenever possible) 2. the y -intercept 3. the x -intercept 4. the stationary points. Stationary points A stationary point is a point on a graph where the function momentarily stops rising or falling; that is, it is a point where the gradient is zero. The stationary point (or turning point) of a quadratic function can be found by com- pleting a perfect square in the form y = ( x + h ) 2 + k to obtain ( h , k ), but for cubics, quartics or higher-degree polynomials there is no similar procedure. Differentiation enables stationary points to be found for any polynomial function where the rule is known. The gradient of a function f ( x ) is f ( x ). Stationary points occur wherever the gradient is zero. f ( x ) has stationary points when f ( x ) = 0 or y has stationary points when . The solution of f ( x ) = 0 gives the x -value or values where stationary points occur. y x 0 Function stops falling and rises after this point y x 0 Gradient = 0 where function stops rising momentarily, then continues to rise again after this point or d y d x ------ 0 =
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C h a p t e r 2 A p p l i c a t i o n s o f d i f f e r e n t i a t i o n 45 If f ( a ) = 0, a stationary point occurs when x = a and y = f ( a ). So the coordinate of the stationary point is ( a , f ( a )). Types of stationary points There are four types of stationary point. 1. A local minimum turning point at x = a . If x < a , then f ( x ) < 0 (immediately to the left of x = a , the gradient is negative). If x = a , then f ( x ) = 0 (at x = a the gradient is zero). If x > a , then f ( x ) > 0 (immediately to the right of x = a , the gradient is positive). 2. A local maximum turning point at x = a .
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