MAT9004复习总结(2)

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MAT9004复习总结(2)

这部分记录一些很简单的函数相关值知识，基本高中范围.

Concave Function

A function is concave if, for any two points in its plot, the straight line between both points is entirely below (or touching) the plot of the function. convex function

Convex Function

A function is convex if, for any two points in its plot, the straight line between both points is entirely above (or touching) the plot of the function. convex function

Bijection

A function \(f : X → Y\) is called :

  • injective (or one-to-one) if for all distinct \(x1, x2 ∈ X\) and \(f(x1) != f(x2)\)

  • surjective if \(Y = f (X)\), that is if for every \(y ∈ Y\) there is an \(x ∈ X\) with \(f (x) = y\)

  • bijective if it is both injective and surjective

Log-Log Plot

The log-log plot of a data set \((x_1, y_1), . . . ,(x_n, y_n)\) is the plot of the data \((ln(x_1), ln(y_1)), . . .(ln(x_n), ln(y_n))\)

If \((x_1, y_1), . . . ,(x_n, y_n)\) are points of the graph of a power-law function, then \((ln(x_1), ln(y_1)), . . . ,(ln(x_n), ln(y_n))\) are points of the graph of a linear function with :

  • slope equal to the exponent of the power-law function

  • y-intercept equal to ln(b), if the original function was \(f (x) = bx^{−a}\)

Derivertive Rules

  • If \(f (x) = f_1(f_2(x))\) then \(f'(x) = f'_2 (x)f'_1(f'_2(x)\)

  • If \(g(x) = g_1(x)g_2(x)\) then \(g'(x) = g'_1(x)g_2(x) + g_1(x)g'_2 (x)\)

  • If \(h_1(x) = x^b\) then \(h'_1 (x) = bx^{b−1}\)

  • If \(h_2(x) = a^x\) then \(h'_2 (x) = ln(a)a^x\)

  • If \(h3(x) = log_a(x)\) then \(h'_3(x) = {1 \over ln(a)x}\)

Increasing/Decreasing of Function

For an arbitrary function f :

  • On any interval where \(f'(x)\) is positive, \(f\) increases

  • On any interval where \(f'(x)\) is negative, \(f\) decreases

Local Maxima/Minima

  • A local maximum of f is a stationary point where f 0 changes from positive to negative (as x moves left to right)

  • A local minimum of f is a stationary point where f 0 changes from negative to positive

Sufficient Conditions for Judging

If a is a stationary point of \(f\) and \(f''(a)\) exists then :

  • \(f\) has a local minimum at \(x = a\) if \(f''(a) > 0\)

  • \(f\) has a local maximum at \(x = a\) if \(f''(a) < 0\)

  • \(f''(a) = 0\) gives no conclusion

Antiderivatives

A function \(F\) is an antiderivative of f if \(F' = f\)

Some basic antiderivatives

  • If \(f (x) = x^a\) where \(a \neq −1\); \(F(x) = {1\over a+1} x^{a+1}\)

  • If \(f (x) = x^{−1}\) ; \(F(x) = ln(x)\)

  • If \(f (x) = e^{ax}\) where \(a \neq 0\); \(F(x) = {1\over a} e^{ax}\)

Calculus

If \(F\) is an antiderivative of \(f\) then : \[\int_a^b f(x) dx = F(b) - F(a)\]

And \(F(x)+c\) is the indefinite integral of \(f\)

Linearity

  1. If \(f\) and \(g\) are functions then : \[\int_a^b f(x)+g(x) dx = \int_a^b f(x)dx + \int_a^b g(x)dx\]

  2. If \(f\) is a function and \(c\) is a constant then : \[\int_a^b cf(x)dx = c\int_a^b f(x) dx\]

Another Property

If \(a,b,c\) are real numbers with \(a<b<c\) then : \[\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx\]

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