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Chapter 11. Exponential Functions, Logarithmic Functions, Trigonometric Functions

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1. Exponential Functions and Logarithmic Functions

2. Trigonometric 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 Angle notation

① Degree measure: A notation for angles that defines one full rotation as 360°

② Radian measure(radian measure, circular measure): θ = ℓ / r

⑵ Trigonometric Functions Definition

① Example

Let $c:[0,1)\to\mathbb{R}^2$ be given by $c(t)=(\sqrt{1-t^2},t)$, and let $x(t)=\sqrt{1-t^2}$ and $y(t)=t$. If $\theta(t)$ denotes the length of the curve $c([0,t])$, show that

Using $6\theta(1/2)=\pi$, show that the length of a semicircle is $\pi$. Also, regarding $x$ and $y$ as differentiable functions of $\theta$, show that $x’(\theta)=-y(\theta)$, $y’(\theta)=x(\theta)$, $x(0)=1$, and $y(0)=0$.

○ Solution

Assume $0<t<1$ and consider the length of the curve $c([0,t])$. Since $c’(\tau)=(-\tau/\sqrt{1-\tau^2},1)$, we have $\lVert c’(\tau)\rVert=1/\sqrt{1-\tau^2}$. Hence, by the arc-length formula,

When $t=1/2$, the points $c(0)=(1,0)$ and $c(1/2)=(\sqrt{3}/2,1/2)$ lie on the unit circle, and the corresponding central angle is $\pi/6$. Therefore, $\theta(1/2)=\pi/6$, so $6\theta(1/2)=\pi$. Thus, the length of a semicircle is $\pi$. By the Fundamental Theorem of Calculus, $\theta’(t)=1/\sqrt{1-t^2}$ for $0\le t<1$. Since $\theta$ is strictly increasing, $t$ may be regarded as a differentiable function of $\theta$, and $dt/d\theta=\sqrt{1-t^2}$. Using the chain rule, $x’(\theta)=(dx/dt)(dt/d\theta)$. Since $dx/dt=-t/\sqrt{1-t^2}$, we obtain $x’(\theta)=-t=-y(\theta)$. Similarly, since $dy/dt=1$, we have $y’(\theta)=\sqrt{1-t^2}=x(\theta)$. Finally, $\theta=0$ corresponds to $t=0$, so $x(0)=1$ and $y(0)=0$. Hence $x’(\theta)=-y(\theta)$, $y’(\theta)=x(\theta)$, $x(0)=1$, and $y(0)=0$.

② Definition: Extend the functions $x,y:[0,\pi/2)\to\mathbb{R}$ to functions $x,y:\mathbb{R}\to\mathbb{R}$ by the following relations, and write $x(\theta)=\cos\theta$ and $y(\theta)=\sin\theta$:

③ Example: Show that $\sin\theta:[-\pi/2,\pi/2]\to[-1,1]$ is a bijection and hence has an inverse function $\varphi:[-1,1]\to[-\pi/2,\pi/2]$. Also show that, for $-1<y<1$,

○ Solution

First, consider the graphs of $\sin\theta$ and $\cos\theta$ on $0\le\theta<\pi/2$. Both functions are continuous, and $\sin\theta$ is strictly increasing on this interval. Also, $\sin(\pi/2)=1$ and $\cos(0)=1$. For $-\pi/2\le\theta<0$, the relation $\sin\theta=-\cos(\theta+\pi/2)$ shows that $\sin\theta$ is also strictly increasing. Therefore, $\sin\theta:[-\pi/2,\pi/2]\to[-1,1]$ is continuous and strictly increasing, and hence it is a bijection. Let $f(\theta)=\sin\theta$, and denote its inverse by $\varphi:[-1,1]\to[-\pi/2,\pi/2]$. For $

y

<1$, let $\varphi(y)=\theta$, where $-\pi/2<\theta<\pi/2$. By the inverse function theorem,

For $0<\theta<\pi/2$, we have $f’(\theta)=\cos\theta$. On the other hand, for $-\pi/2<\theta<0$, using $\sin\theta=-\cos(\theta+\pi/2)$ gives $f’(\theta)=\sin(\theta+\pi/2)=\cos\theta$. The same formula also holds at $\theta=0$. Since $y=\sin\theta$ and $\cos\theta>0$ for $-\pi/2<\theta<\pi/2$, the identity $\cos^2\theta+\sin^2\theta=1$ gives $\cos\theta=\sqrt{1-y^2}$. Therefore,

④ Types of 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

⑤ 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

⑶ Basic properties of trigonometric functions

① The addition formulas of trigonometric functions can be understood through rotation transformations, which are linear mappings

② $\tan(A+B+C) = \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1 - \tan A \tan B - \tan B \tan C - \tan C \tan A}$ (Advanced)

⑷ 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

⑸ 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