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Differentiation (Integration (Strategy (1 Identify the variables and…
Differentiation
Integration
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Integrating x^n with respect to x is written as
∫x^n dx = x^(n+1)/n+1 + c, n ≠ -1
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Rates of change
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The rate of change of velocity is called acceleration, a = dv/dt
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Strategy
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4 Apply this to the initial question, being mindful of the context and units
1 Read and understand the context, identifying any function or relationship
Turning points
At a turning point, the gradient of the tangent is zero. Therefore, you can work out the coordinate of the turning point by equating the derivative to zero
A turning point is a stationary point, but a stationary point is not necessarily a turning point
At a maximum turning point, d²y/dx² < 0
At a minimum turning point, d²y/dx² > 0
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Tangents and normals
When lines m1 and m2 are perpendicular to each other, m1 X m2 = -1
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The tangent to the curve y = f(x) which touches the curve at the point (x, f(x)), has the same gradient as the curve at that point, giving mT = f '(x), is perpendicular to the tangent at that point, giving mN = -1/mT = -1/f '(x)
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Strategy 2
To work out the area bound between a tangent, a normal and the x-axis/ y-axis
1 Work out the equation of the and, from it, the equation of the normal
2 Work out where each line crosses the required axis. Lines cut the x-axis when y = 0, and the y-axis when x = 0
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4 Use A = 1/2 X base X height, where the base is the length between the intercepts on the x-axis or y-axis and height is the y-coordinate or x-coordinate respectively
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Area under a curve
To find the area under a curve, you perform a series of calculations using a definite integral
A definite integral is denoted by ∫f(x)dx
b is called the upper limit, and a the lower limit
The area under a curve between the x-axis, x = a, x = b and y = f(x), is given by A = ∫f(x)dx = F(b)-F(a)
Strategy
1 Make a sketch of the function, if there isn't one provided
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