Chapter 11. Transcendental Functions(Exponential Functions, Logarithmic Functions, Trigonometric Functions)
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1. Exponential Functions and Logarithmic Functions
1. Exponential Functions and Logarithmic Functions
⑴ Natural Constant(natural constant): Also called Napier’s constant
① Existence of the infinite series sequence xn
○ Binomial theorem for $x_n$
○Theorem 1. $x_n$ is a monotonically increasing function
○ Theorem 2. $x_n$ is bounded
○ Conclusion: By the completeness axiom, xn converges
② Definition of the natural constant e
○ Definition in the range of natural numbers
○ Definition in the range of positive real numbers
○ Definition in the range of negative real numbers
⑵ Exponential Function
① Exponential function(exponential function)
○ For a ∈ ℝ, define a0 = 1
○ For a ∈ ℝ, define an for every integer n greater than or equal to 0 by an+1 = an × a
○ For a ∈ ℝ, n ∈ ℕ
○ For a ∈ ℝ, n ∈ ℕ, m ∈ ℤ
○ When the base is a ＞ 0 and the exponent is x ∈ ℝ, define the exponential function as follows
② Natural exponential function(natural exponential function): Exponential function with base e
○ Derivative coefficient of $e^x$ at the origin
○ Derivative coefficient of $e^x$: The derivative is equal to itself
○ Maclaurin series of $e^x$ (Maclaurin series)
③ Properties of exponential functions
⑶ Logarithmic Function: Inverse function of an exponential function
① Definition: When the base is a, the antilogarithm is b, and the exponent is y
② Properties
③ Natural logarithms: Logarithms with base e
④ Common logarithms: Logarithms with base 10
⑷ Analytic definition
① Natural logarithm function: Define the function ℓ: (0, ∞) → ℝ as follows. Also denoted by ln x
○ Theorem 1. The logarithmic function is a bijective function
○ Theorem 2. For each a, b ＞ 0, ℓ(ab) = ℓ(a) + ℓ(b)
② Natural exponential function: Refers to the inverse function k: ℝ → (0, ∞) of ℓ(x). Also denoted by $e^x$
○ Theorem 1. k’(x) = k(x)
○ Theorem 2. For x, y ∈ ℝ, k(x+y) = k(x) × k(y)
③ Exponential function
○ For a ＞ 0, x ∈ ℝ
○ If a ＞ 0, a ≠ 1, and x ＞ 0
○ Theorem 1. For a ＞0, x, y ∈ ℝ, $a^{x+y} = a^{x}a^{y}$
④ Logarithmic function
○ Theorem 1. When a ＞0, a ≠ 1, x ＞ 0, y ∈ ℝ
○ Theorem 2. When a ＞ 0, a ≠ 1, x ＞ 0, y ∈ ℝ, logax = y and x = ay are equivalent
⑸ Function with exponential order: A function f for which there exist M, γ ＞ 0 such that the following is satisfied on the interval [0, ∞)
2. Trigonometric Functions
⑴ Definition
① Angle notation 1. Degree measure: A notation for angles that defines one full rotation as 360°
② Angle notation 2. Radian measure(radian measure, circular measure): θ = ℓ / r
③ Definition of trigonometric functions
○ Sine function(sine function): For a counterclockwise angle θ expressed in radians, the y-coordinate is denoted by sin θ
○ Cosine function(cosine function): For a counterclockwise angle θ expressed in radians, the x-coordinate is denoted by cos θ
○ Tangent function(tangent function): tan θ ≡ sin θ / cos θ
○ Cosecant function(cosecant function): csc θ ≡ 1 / sin θ
○ Secant function(secant function): sec θ ≡ 1 / cos θ
○ Cotangent function(cotangent function): cot θ ≡ cos θ / sin θ
④ Definition of inverse functions of trigonometric functions
○ Arcsine function(arcsine function): arcsin θ ≡ sin-1 θ
○ Arccosine function(arccosine function): arccos θ ≡ cos-1 θ
○ Arctangent function(arctangent function): arctan θ ≡ tan-1 θ
⑵ Basic properties of trigonometric functions
① The addition formulas of trigonometric functions can be understood through rotation transformations, which are linear mappings
⑶ Transformations of trigonometric functions
① Converting products into sums and differences
② Converting sums or differences into products
③ Double-angle formulas and half-angle formulas
⑷ Laws related to trigonometric functions
① Law of Sines: When the radius of the circumcircle of ΔABC is R and the lengths of the sides corresponding to the angles are a, b, c, the following holds
② Area of a Triangle and the Law of Sines: When the lengths of the sides corresponding to the angles of ΔABC are a, b, c, the following holds
③ First Law of Cosines: When the lengths of the sides corresponding to the angles of ΔABC are a, b, c, the following holds
④ Second Law of Cosines: When the lengths of the sides corresponding to the angles of ΔABC are a, b, c, the following holds
⑤ Trigonometry: In △ABC
○ cos2A + cos2B + cos2C = 1 - 2 cosA cosB cosC
○ cos2A + cos2B + cos2C = 1 - 4 cosA cosB cosC
⑥ Heron’s Formula: If the lengths of the three sides of a triangle are a, b, c, respectively, then the area is as follows
⑥ Useful Memorization Formula: Can be proved by integration by parts
⑦ Sum of a Sequence of Cosines
○ Proof 1. Geometric sequence of complex exponential functions
○ Method 2. Mathematical induction
Input: 2020.03.19 09:47
Modified: 2023.08.03 23:27