Mathematics 1

Regole di derivazione

D(xⁿ) = nx^n-1

D(x) = 1

D(k) = 0

D(ln x) = ¹/_x, x > 0

D(log_a x) = ¹/_{x log a}

D(e^x) = e^x

D(a^x) = a^x ln a

D(sen x) = cos x

D(cos x) = -sen x

D(tan x) = ¹/_{1 + tan² x}

D(cot x) = -¹/_{(1 + cot² x)}

D(arcsin x) = ¹/_{√(1 - x²)}

D(arccos x) = -¹/_{√(1 - x²)}

D(arctan x) = ¹/_{1 + x²}

D|f(x)| = ^f'(x)f(x)/_|f(x)|

f(x) si deriva come f(x)

Regole di derivazione delle funzioni

• D[f(x) + g(x)] = f'(x) + g'(x)
• D[cf(x)] = c f'(x)
• D[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
• D(f(x)/g(x)) = ^{f'(x)g(x) - g'(x)f(x)}/_(g(x))²

D(x²) = 2x

D(1_x) = log_e x, x > 0

D(e^x) = e^x, x₀

D(¹/_x²) = -¹/_x³, x

D(x) = 1

(-csec N {0}, x ∈ R, csc ∈ R \ {0}) > 0

sec N {0}, x ∈ R, x ≠ 0, 0 N 0

D(arc csc x) = -1/√(x² - 1)

D(log_e x) = 1/x, x > 0

D(x) = 2*x

D(^{2_ex x)}

D(tan x) = sec² x

D(arcsec x) = 1/x√(x² - 1)

D(arc csc x) = -1/x√(x² - 1)

Function, domain, codomain

A function of a real variable x, with domain D, is one that assigns a unique real number to each real number x in D. And x values over the whole domain, the set of all possible resulting values f(x) is called the range of f.

Types of functions

• Linear
• Quadratic
• Rational
• Irrational
• Exponential
• Logarithmic
• Others

Is it a function?

Is f(x) a function?

Is y = x² a function?

A parabola is a function

Is x² + y² = 1 a function?

NOT a function

Studio di funzioni

DOMINIO:

• log > 0
• √ ≥ 0
• dett ≠ 0

Esempi di domini

f(x) = 2x - 7

D is ℝ → D(-∞,∞)

f(x) = x² + 3x - 5

D(∞)

f(x) = 2x³ - 5x² + 7x - 3

D(∞)

f(x) = 5 / (x - 2)

x - 2 ≠ 0, x ≠ 2, x can be anything except 2

D(-∞,2) ∪ (2,+∞)

f(x) = (3x - 8) / (x² - 3x + 20)

x² - 3x + 20 ≠ 0, ∆ = 81 - 805

x ≠ 4, x ≠ 5

D(-∞,4) ∪ (4,5) ∪ (5,+∞)

f(x) = (2x - 3) / (x² + 4)

x² + 4 ≠ 0

x ≠ 4 ⇒ the denominator will never be 0

All ℝ numbers

D(-∞,+∞)

f(x) = √x - ux - u ≥ 0, x ≥ 4

D(4,+∞)

f(x) = √x² + 3x - 28

∆ = 9 + 12 = 121

x₁,₂ = (-3 ± 11) / 2, -7, 4

x ≥ 4, x ≥ -7

D(-∞,-7) ∪ [4,+∞)

f(x) = (2x - 7) / (x + 3)

x + 3 ≠ 0

D(-3,+∞)

R(x) - Range of functions

R(x) = ^{x - 4}/_{x² - 25}

x ≠ ±5

D (4,5) ∪ (5,+∞)

R(x) = ^{√(x + 3)}/_{√(x² - 16)}

x ≥ 3

x² - 16 > 0

x > 4

D (4,+∞)

Range examples

• y = x + 3 ℝ (-∞,+∞) for any linear function
• y = x² ℝ [0,+∞)
• y = x² - 3 ℝ [−3,+∞)
• y = 4 - x² ℝ [−∞,4]
• y = x² - 4x + 5

Value of the vertex

-b / 2a = ±4 / 2 = 2 → y = 2² - 4(2) + 5 = 1

V (2,1) ℝ [1,+∞)

• y = x³ ℝ (-∞;+∞)
• y = x³ + 5x² - 8x
• y = x⁵ + 6x³ - x⁸ + ∞

Continuous functions

• → ℝ (-∞;+∞)

Graph examples

• y = |x| ℝ [0,∞)
• y = |x-2|-3 ℝ [-3,∞)
• y = |x| + 3 ℝ [3,∞)
• y = 2Θ|x-3| ℝ (-∞,+2]
• y = √x ℝ [0,+∞)
• y = √Θx ℝ [0,+∞)
• y = 1/√x+x ℝ (-∞,0]
• y = - √-x-x ℝ (-∞,0]
• y = - √x-3 +4 ℝ (-∞,4]

Composition of functions

f(x) = 3x - 4

f(g(x)) = 3(x² - 3) - 4 = 3x² - 9 - 4 = 3x² - 13

g[f(x)] = (3x - 4)² - 3 = 9x + 16 - 24x - 3 = 9x² - 24x + 13

f(x) = 5x + 2

g(x) = x³ - 4

g(2) = 8 - 4 = 4