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to ible use the sign of d 2y/dx 2 to test which we’ve got. 8.E Some uses for differentiation 339 Here is a summary of the above results, so that we can use them to find out how particular curves will behave. Finding and classifying turning points and points of inflection For a point to be a local maximum, dy/dx must be equal to zero. Then use either Test (1): the gradients of the tangents move through the point in the sequence + 0 –, so test the value of dy/dx either side of this point, or Test (2): if the value of d 2y/dx 2 is negative at this point, then it is a local maximum, but if d 2y/dx 2 = 0 then Test (1) must be used. For a point to be a local minimum, dy/dx must be equal to zero. Then use either Test (1): the gradients of the tangents move through the point in the sequence – 0 +, so test the value of dy/dx either side of this point, or Test (2): if the value of d 2y/dx 2 is positive at this point, then it is a local minimum, but if d 2y/dx 2 = 0 then Test (1) must be used. For a point of inflection, (1) the value of dy/dx does not change sign as it moves through the point (it may or may not be equal to zero at the point itself), and (2) the value of d 2y/dx 2 at the point must be equal to zero. 8.E.(c) General rules for sketching curves The tests outlined in this previous section give us useful extra information which we can use for sketching graphs. I have already listed informally the other questions which we need to answer in order to draw a graph sketch Section in 3.B.(i) where we sketched y = (x + 3)/(x – 2). You should look back at how we built up this sketch before going on. Now that we can include finding the turning points, I can give you a complete summary of the questions which you need to answer in order to sketch a curve. For convenience, I will call this curve y = f (x ) but, of course, other letters can be used. 340 Differentiation Questions to answer in order to draw a graph sketch (1) Does the curve cut the y-axis? If so, where? (Try putting x = 0.) (2) Does the curve cut the x-axis? If so where? (This is the same as asking if the equation f (x ) = 0 has any roots on the x-axis.) (3) Are there any values of x which have to be excluded because they would mean trying to divide by zero? If so, what are they? (Such values of x will give you vertical asymptotes. An asymptote is a line which the curve of the graph of the function becomes closer and closer to.) What happens to the values of f (x ) for values of x just either side of the forbidden values? (4) What happens to the values of f (x ) when x takes very large positive or negative values? (If it gets closer and closer to some fixed limit then this will give you a horizontal asymptote.) (5) Are there any turning points? (That is, are there any values of x for which f (x ) or dy/dx = 0?) If so, what are they? You will need to find the value of f (x ) (the stationary value ) for each of these values of x. Test each turning point to find whether it is a l... View Full Document

Olive - Maths - A Student's Survival Guide 2e HQ

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