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

By Christine Tootill

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

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Example text

1 So the length of the enclosure will be 50, the width is 25, and the area is 50 × 25 = 1250. This time a square enclosure is not the best option. Why is this? ). 33 metres of wall, but the best solution uses 50 metres of wall. 1) 1. Two walls at right angles can be used to form part of a rectangular enclosure. One of the walls is 30 metres long, and the other is 100 metres long. How should the 100 metres of fencing be arranged to maximise the enclosed area? What will the maximum area be? 2. A box without a lid is to be made from a square piece of cardboard measuring 10 cm by 10 cm, by cutting squares from each corner and folding the sides up.

Look back at Practice Question (1). We observed from the quadratic curve that the gradient is negative when x is negative, zero at the origin, and large when x is large. Our gradient function dy/dx = 2x confirms these observations. 2) 1. Compare the curve y = 2x2 with the curve y = x2. How does doubling the y co-ordinates affect the gradient? Deduce the gradient function of the curve y = 2x2. 2. a) What is the gradient function of the line y = 3x? b) What is the gradient function of the line y = −x?

2 Find the equation of the normal to the curve y = −x 2 + 2x + 3 at the point (2,3). First, we find the gradient of the curve at the point (2,3): dy/dx = 2x + 2 When x = 2, dy/dx = −2. The normal is perpendicular to the tangent, and we know from Co-ordinate Geometry that if gradients m1 and m2 are perpendicular, then m1 × m2 = –1 Using m1 for the tangent and m2 for the normal, we have m1 = −2 and m2 = ½ We have x1 = 2 and y1 = 3, so using the straight line formula: y – y1 = m(x – x1) we get y – 3 = ½(x – 2) which simplifies to y = ½x + 2 This is the equation of the normal to the curve y = −x 2 + 2x + 3 at the point (2,3).