1
Definition of Function
Let A, B be sets. A function f: A –> B is a unique assignment of an element f(x) є B for each x є A.
To see if a graph of points in the plane R2 is the graph of a funtion: R –> R, do the vertical line test.
Set A: domain, set B: codomain, set f(A) = {f(x) є B| x є A}: range of the function f.
The function f is called surjective if f(A) = B, injective (one-to-one) if f(x) ≠ f(y) whenever x ≠ y.
To see if a function f: R –> R is injective, do the horizontal line test on the graph.
If it is bijective (both surjective and injective), it has an inverse f-1 : B –> A defined by f-1(f(x)) = x.
Limit and Derivative of a Function: Let f: (a, b) –> R be a function and c є (a, b).
L = lim f(x), if the value f(x) becomes as close to L as we please for x sufficiently close to c and x ≠ c.
x -> c
For two functions f, g, and r є R we have
1) lim (f(x) + g(x)) = lim f(x) + lim g(x)
x -> c
x -> c
x -> c
2) lim (r f(x)) = r lim f(x)
x -> c
x -> c
3) lim (f(x) g(x)) = lim f(x) lim g(x)
x -> c
x -> c
x -> c
4) lim (f(x)/g(x)) = lim f(x)/lim g(x) if lim g(x) ≠ 0
x -> c
x -> c
5) lim (f(x)) = ( lim f(x))
r
x -> c
r
x -> c
x -> c
x -> c
The function f is called continuous at c, if lim f(x) exists and lim f(x) = f(c).
x -> c
x -> c
The function f is called differentiable at c if the limit of the slopes of the secant lines through the
point (c, f(c)) and the point (x, f(x)), where x ≠ c, exists.
f(x) – f(c)
The derivative f’(c) is defined by f’(c) = lim
.
x -> c
x–c
For two differentiable functions f, g, and r є R we have
1) (f + g)’ = f’ + g’
Sum Rule
4) (f ◦ g)’ = (f’ ◦ g)g’
Chain Rule
2) (r f)’ = r f’
Factor Rule
Inverse Function Rule
3) (f g)’ = f’ g + f g’
Product Rule
5) (f –1)’ ◦ f = 1/f’
f ’
f’ g – f g’
6)
=
g
g2
Quotient Rule
Theorems about Values: Let f: [a, b] –> R be a continuous function.
For an intermediate value r є (f(a), f(b)) there exists a number c є (a, b) such that f(c) = r.
There exist global extreme values, that is there exists a number u є [a, b] such that f(u) is the
maximum, and there exists a number v є [a, b] such that f(u) is the minimum, of all values on [a, b],
and if u, v є (a, b) and the function f is differentiable at u and v, we have f’(u) = f’(v) = 0.
Let the function f be differentiable on (a, b). Then there exists at least one point c є (a, b) satisfying
f(b) – f(a)
.
f’(c) = mean value =
b–a
If f’(x) = 0 for all x є (a, b), we have f(x) = constant.
2
List of Derivatives
d xa
1) dx = axa – 1 for all a є R
d ex
2)
= ex (e is Euler’s Number) 3)
dx
1
d ln(x)
4)
=
for x > 0
5)
x
dx
d sin x
6)
= cos x
7)
dx
1
8) d tan x = 9)
cos2 x
dx
1
d Arcsin x
d Arccos x
10)
=–
=
dx
dx
√1 – x2
1
d Arctan x
11)
= – d Arccot x =
1
+
x2
dx
dx
d ax
= ax ln(a) for a > 0
dx
d loga(x)
1
=
for a > 0 and x > 0
dx
x ln(a)
d cos x
= – sin x
dx
d cot x = – 1
sin2 x
dx
for |x| < 1
Theorems about Derivatives: Let f: (a, b) –> R be a differentiable function and let c є (a, b).
1) If f’(c) > 0, then f goes from smaller than f(c) to larger than f(c) near c.
2) If f’(c) < 0, then f goes from larger than f(c) to smaller than f(c) near c.
3) If f’(x) > 0 for all x є (a, b), then f is increasing on (a, b).
4) If f’(x) < 0 for all x є (a, b), then f is decreasing on (a, b).
5) If f’’(x) > 0 for all x є (a, b), then f is concave upward, which means that the graph of f
lies above all of its tangents and f’ is increasing on (a, b).
6) If f’’(x) < 0 for all x є (a, b), then f is concave downward, which means that the graph of f
lies below all of its tangents and f’ is decreasing on (a, b).
7) If f’(c) = 0 and f’’(c) < 0 (or f’ is continuous and f’ goes from positive to negative near c),
then f has a local maximum at c.
8) If f’(c) = 0 and f’’(c) > 0 (or f’ is continuous and f’ goes from negative to positive near c),
then f has a local minimum at c..
9) If f’’(c) = 0 and f’’’(c) ≠ 0 (or f’’ is continuous and f’’ changes sign at c), then f has an
inflection point at c, which means that f changes from concave upward to concave downward or
vice versa at c and f’ has a local maximum or minimum at c.
Exponential Growth Let e be Euler’s Number, that is e = lim (1 + 1 )n.
n –> ∞
n
f(x) = ae(r/100)x : Exponential Growth of amount a at r %, where x denotes the number of years.
We have f(0) = a, f(1) = ba, f(2) = b2a, f(3) = b3a, etc., where b = er/100.
ln f(x) = ln a + (r/100)x : linear function. Compute f(x) = 2a, that is ln (2a) = ln a + (r/100)x :
We have x = c = 100ln(2)/r ≈ 69.3/r ≈ 70/r and f(c) = 2a, f(2c) = 22a, f(3c) = 23a, etc.