mat9004复习总结(2)

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

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.

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) = {1over 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) = {1over 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 = cint_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]