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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