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

By Christine Tootill

Who else are looking to be triumphant in arithmetic? Calculus frequently factors panic in scholars, yet with this publication, that quickly could be a factor of the prior. jam-packed with transparent factors and written via a hugely skilled and sympathetic instructor with a long time of expertise in getting ready scholars for complex arithmetic examinations, it is a convinced must-have booklet for all scholars learning arithmetic.

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Extra resources for Calculus: How Calculus Works (Studymates in Focus) (In-Focus)

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In this Chapter we see how to apply differentiation to problems such as: ● minimising the length of fencing needed for an enclosure ● minimising the amount of material needed to enclose a fixed volume Areas, perimeters and volumes Before continuing with this Chapter, read again through the solution to Practical Assignment (7). This cardboard box exercise gives an example of the use of differentiation to find the maximum volume of a container, and is a simple illustration of how calculus can be used very effectively in solving simple design problems.

3. 125). 4. (i) At which two points on the curve y = 1/x is the line y = x a normal to the curve? (ii) What are the equations of the two corresponding tangents? 5. (i) Write down the co-ordinates of the points where the curve y = x(x − 1)(x + 1) cuts the x-axis. (ii) Find the gradient function of this curve, and the value of the gradient at the origin. (iii) Use the results of (i) and (ii) to sketch the curve. (iv) Deduce the equation of the normal to the curve at the origin. (v) Find the co-ordinates of the points P and R where the normal meets the curve again.

2 y = (2x + 3)½ In this case, we can only differentiate this function by using the chain rule, because expanding the bracket would give an infinite series. 1) Differentiate these functions using the chain rule: (a) y = (x − 2)6 (b) y = (x2 + 1)2 2 −1 (c) y = (x + 2x) (d) y = (x2 − 3x)4 ½ 3 (e) y = (1 − x ) (f) y = (x + 1/x)2 (g) y = (1 + 2x)½ (h) y = (x 3 − x)−3 2 3 (i) y = (½ x + 1) (j) y = (2x − 3)4 Check your answers before you move on. We can write the chain rule like this: If y is a function of x, then the derivative of y n is ny n–1 × dy/dx We can also use the chain rule in this form, if a function is written in the form yn = f(x).

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