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AS and A Level: Core & Pure Mathematics
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Differentiation and intergration
- 1 It is easy to get differentiation and integration the wrong way round. Remember that the power gets smaller when differentiating.
- 2 Differentiation allows you to find the gradient of a tangent at any point on a curve. The first derivative describes the rate of change.
- 3 If a function is increasing then the first derivative is positive, if a function is decreasing, then the first derivative is negative.
- 4 When asked to find the area under a curve, it is asking you to integrate that curve between two points. Even if you don’t know the points, pick two numbers. You’ll get marks for methods.
- 5 When referring to a min/max/stationary point, the gradient equals 0. Differentiate the curve and set this to equal 0. The second derivative tells you whether it is a maximum or minimum. If the second derivative is positive, the point is a minimum, if the second derivative is negative, then the point is a maximum.
Quadratics and circles
- 1 When solving a quadratic inequality, always draw a picture. The inequality is less than 0, where the curve is below the x-axis and bigger than 0 when the curve is above the x-axis.
- 2 Sometimes in part (a) of a question you are asked to find something, for example a radius. In part (b) you might be then asked to use the radius that you found. If you couldn’t do part (a), don’t give up, choose a random radius.
- 1 To find the distance of a straight line, draw the straight line with the co-ordinates. Then make a right angle triangle, find the lengths of the horizontal and vertical lines, then use Pythagoras.
- 2 When a question asks you for a straight line. The first thing to do is to write down the equation of a straight line. Then find out what information you know, and what information you need. Even if you don’t understand the whole question, it is important to start.
- Marked by Teachers essays 3
I repeat this until I get down to increments the size of 0.00001 x F(x) -3.1 8.429 -3.2 6.712 -3.3 4.843 -3.4 2.816 -3.5 0.625 -3.6 -1.736 -3.7 -4.273 -3.8 -6.992 -3.9 -9.899 -4.0 -13 x F(x) -3.51 0.396649 -3.52 0.166592 -3.53 -0.06518 -3.54 -0.29866 -3.55 -0.53387 -3.56 -0.77082 -3.57 -1.00949 -3.58 -1.24991 -3.59 -1.49208 -3.60 -1.736 x F(x) -3.521 0.143492 -3.522 0.120375 -3.523 0.097241 -3.524 0.07409 -3.525 0.050922 -3.526 0.027736 -3.527 0.004534 -3.528 -0.01869 -3.529 -0.04192 -3.530 -0.06518 x F(x) -3.5271 0.002213 -3.5272 -0.00011 -3.5273 -0.00243 -3.5274 -0.00475 -3.5275 -0.00707 -3.5276 -0.0094 -3.5277 -0.01172 -3.5278 -0.01404 -3.5279 -0.01636 -3.5280 -0.01869 x F(x)
- Word count: 3460
This is the formula used for this method: Gradient = Vertical (change in y axis) Horizontal (change in x axis) Therefore the gradient of the curve equals - Gradient = 12-4/4-2 = 8/2=4 This is most likely wrong and most certainly not accurate - it is only estimation. The second method, the increment method is far more accurate. Increment method - This is where you plot two points, represented by the letters P and Q, on the curve, and draw a line to join them.
- Word count: 6489
Estimate a consumption function for the UK economy explaining the economic theory and statistical techniques you have used.3 star(s)
That is...dC/dY is positive and less than unity."(Keynes, 1936,p.96) From this statement can be derived the Keynesian consumption function (Absolute income) which is usually expressed in the following way: Ct=c0+c1Yt Where Ct is consumption, c0 is autonomous consumption, Yt is income and c1 is the marginal propensity to consume. The Keynesian view is that when income rises, consumption will rises as well, but less than income. This indicates that c1, the marginal propensity of consume, is lees than l. Keynes believed that the main influence on consumption is consumers' disposable income. When their disposable income fluctuates, the consumption would follow.
- Word count: 4377