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year 1 calculas
chapter 12/13 - differentiation and integration
| Term | Definition |
|---|---|
| the derivative of a function | used to get the gradient of a function at any given point via a tangent of the function at that point |
| derivative from first principles | f’(x) = limit(h->0) [(f(x+h)-f(x))/h] |
| power rule of derivatives | if f(x) = kx^n f’(x) = knx^(n-1) |
| linearity rule of derivatives | each term of a function can be differentiated separately. the derivative of the function is the sum of the new terms |
| normals to the function at a point | are perpendicular to the tangent at that point (and could be thought as -dx/dy) |
| increasing functions | when the derivative is <=0 at a range of points |
| decreasing functions | when the derivative is >= 0 at a range of points |
| stationary points | points where the derivative of the function is 0 |
| maxima / minima | points where the derivative of a function is 0 and the second derivative is negative (convex) or positive (concave) respectively, can be local or global |
| points of inflection | points where the second derivative is 0 |
| sketching graphs of derivatives | increasing function = positive decreasing function = negative stationary point = 0 point of inflection = 0 (touches the x-axis) asymptotes are shown as normal (horizontal ones are always at the x-axis though) |
| modelling with derivatives | useful when one the rate at which one variable changes is linked to another variable (normally with time or space) |
| points of inflection | points where there is a change of sign with the second derivative of a function |
| integration | anti-differentiation, looks at the Riemann sum of a function as the width of the rectangles approaches 0 |
| power rule of integration | int(ax^n) = a*(1/(n+1))*x^(n+1) + c, where c is a constant and n =/=1 |
| linearity rule of integration | every individual term can be integrated individually. the final result is the sum of the integrated results |
| indefinite integration | integration without bounds |
| bounded integration | the SIGNED area of a graph between two points |
| definite integrals | the fundamental theorem of calculas: int(f(x)) from a,b = F(a) - F(b) where F(x) is the anti-differential of f(x) |
| total area between curves and the x-axis | the sum of the absolute value of the definite integral - summing the definite integrals between roots and any other needed points |
| area of curves and lines | often the area of the triangle formed by the line subtracted from the area of the curve between the intersections of the curve and line, context dependent |
| integral of x^-1 | ln(x), where ln = log base e |
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That cool NAMe
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