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Published byPamela Feemster Modified about 1 year ago

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Differential calculus is concerned with the rate at which things change. For example, the speed of a car is the rate at which the distance it travels changes with time. First we shall review the gradient of a straight line graph, which represents a rate of change.

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Gradient of a straight line graph The gradient of the line between two points (x 1, y 1 ) and (x 2, y 2 ) is where m is a fixed number called a constant. A gradient can be thought of as the rate of change of y with respect to x.

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Gradient of a curve A curve does not have a constant gradient. Its direction is continuously changing, so its gradient will continuously change too. The gradient of a curve at any point on the curve is defined as being the gradient of the tangent to the curve at this point. y = f(x)

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A tangent is a straight line, which touches, but does not cut, the curve. x y O A Tangent to the curve at A., as we know only one point on the tangent and we require two points to calculate the gradient of a line. We cannot calculate the gradient of a tangent directly

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Using geometry to approximate to a gradient x O y A Tangent to the curve at A. B1B1 B2B2 B3B3 Look at this curve. Look at the chords AB 1, AB 2, AB 3,... For points B 1, B 2, B 3,... that are closer and closer to A the sequence of chords AB 1, AB 2, AB 3,... move closer to becoming the tangent at A. The gradients of the chords AB 1, AB 2, AB 3,... move closer to becoming the gradient of the tangent at A.

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A numerical approach to rates of change Here is how the idea can be applied to a real example. Look at the section of the graph of y = x 2 for 2 > x > 3. We want to find the gradient of the curve at A(2, 4). A (2, 4) B 1 (3, 9) B 2 (2.5, 6.25) B 3 (2.1, 4.41) B 4 (2.001, 4.004001) The gradient of the chord AB 1 is Chord x changes from y changes from gradient AB 1 AB 2 AB 3 AB 4 AB 5 2 to 3 4 to 9 = 5 2 to 2.5 4 to 6.25 = 4.5 2 to 2.1 4 to 4.41 = 4.1 2 to 2.001 4 to 4.004001 = 4.001 2 to 2.00001 4 to 4.0000400001 4.00001 Complete the table

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As the points B 1, B 2, B 3,... get closer and closer to A the gradient is getting closer to 4. This suggests that the gradient of the curve y = x 2 at the point (2, 4) is 4. x y 2 4 y = x 2

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Find the gradient of the chord joining the two points with x-coordinates 1 and 1.001 on the graph of y = x 2. Make a guess about the gradient of the tangent at the point x = 1. Example (1) (1, ) (1.001, ) 1.001 2 1) = 2.001 The gradient of the chord is I’d guess 2.

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Find the gradient of the chord joining the two points with x-coordinates 8 and 8.0001 on the graph of y = x 2. Make a guess about the gradient of the tangent at the point x = 8. Example (2) (8, ) (8.0001, ) The gradient of the chord is 8.0001 2 64 = 16.0001 I’d guess 16.

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Let’s make a table of the results so far: x-coordinategradient 12 24 3 4 5 6 7 816 You’re probably noticing a pattern here. But can we prove it mathematically?

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(2, 4) (2 + h, (2 + h) 2 ) I need to consider what happens when I increase x by a general increment. I will call it h. h I will call it ∆x.

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Let y = x 2 and let A be the point (2, 4) x y A(2, 4) y = x 2 O B(2 + h, (2 + h) 2 ) Let B be the point (2 + h, (2 + h) 2 ) Here we have increased x by a very small amount h. In the early days of calculus h was referred to as an infinitesimal. Draw the chord AB. Gradient of AB = 4 + h As h approaches zero, 4 + h approaches 4. So the gradient of the curve at the point (2, 4) is 4. Use a similar method to find the gradient of y = x 2 at the points (i) (3, 9) (ii) (4, 16) If h ≠ 0 we can cancel the h’s.

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We can now add to our table: x-coordinategradient 12 24 3 4 5 6 7 816 It looks like the gradient is simply 2x. 6868

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Let’s check this result. y = x 2

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Let’s check this result. y = x 2 Gradient at (3, 9) = 6

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Let’s check this result. y = x 2 Gradient at (2, 4) = 4

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Let’s check this result. y = x 2 Gradient at (1, 1) = 2

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Let’s check this result. y = x 2 Gradient at (0, 0) = 0

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Let’s check this result. y = x 2 Gradient at (–1, 1) = –2

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Let’s check this result. y = x 2 Gradient at (–2, 4) = –4

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Let’s check this result. y = x 2 Gradient at (–3, 9) = –6

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Another way of seeing what the gradient is at the point (2, 4) is to plot an accurate graph and ‘zoom in’. ZOOM IN y = x 2

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When we zoom in the curve starts to look like a straight line which makes it easy to estimate the gradient. 0.2 0.8

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