Calculus: How Calculus Works (Studymates in Focus) by Christine Tootill

By Christine Tootill

Who else are looking to succeed in arithmetic? Calculus frequently explanations panic in scholars, yet with this ebook, that quickly could be a factor of the previous. packed with transparent motives and written through a hugely skilled and sympathetic instructor with a long time of expertise in getting ready scholars for complex arithmetic examinations, this can be a certain must-have e-book for all scholars learning arithmetic.

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E. we can obtain the gradient function of the gradient function! 1 y = x 3 dy/dx = 3x 2 The function dy/dx = 3x 2 can be differentiated. But first we need a “label” for the new function. The notation for this is: d2y/dx 2 (This is pronounced as “d 2 y by d x squared” or “d squared y by d x squared”) In this case, d2y/dx2 = 6x. indb 29 1/22/2008 1:01:00 PM Calculus Terminology A function which is obtained from another function by differentiation is called a derived function or a derivative. These terms are more general than gradient function, and are used because in many applications of calculus, a function is differentiated for reasons other than finding a gradient.

To solve this problem, we need to find an expression for the amount of aluminium used – this is directly related to the surface area of the can. But there are two variables – the height and the diameter. How are these two variables related? The key is the volume of the can, which is directly related to the capacity. One millilitre (ml) occupies 1 cm3, so the volume of the can is 250 cm3. We know that the volume of a cylinder is given by the formula: V = πr 2 h where r is the radius and h is the height.

We have these results: x 1 2 –3 gradient dy/dx 3 12 27 We need to establish a relationship between x and dy/dx for the function y = x3, and we see from the table that it is not a straight line. A moment’s observation shows that dy/dx is divisible by 3 in each case, and we can rewrite the table as: x 1 2 –3 gradient dy/dx 3x1 3x4 3x9 and we see that dy/dx = 3x2. This is our second important result: The gradient function of the cubic curve is a quadratic function. indb 21 1/22/2008 1:00:58 PM Calculus Tutorial Progress question (4) This question summarises the results of this chapter.

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