Calculus

I can’t stand my poor math anymore, and start to re-learn math from calculus.

This notes will record some formulas and anything intersting associates with calculus.

Trigonometry

ASTC method

ASTC method

Trig Identities

$$ \begin{array}{l} cos^2(x) + sin^2(x) =1 \\
1 + tan^2(x) = sec^2(x) \end{array} $$

$$ \begin{array}{ll} sin(A+B) & = & sin(A)cos(B) + cos(A)sin(B) \\
cos(A+B) & = & cos(A)cos(B) - sin(A)sin(B) \\
sin(2x) & = & 2 sin(x) cos(x) \\
cos(2x) & = & 2 cos^2(x) - 1 = 1 - 2 sin^2(x) \end{array} $$

Tangent half-angle formula

又称为“万能公式”。

$$ \begin{array}{ll} sin \alpha & = & \dfrac{2tan\dfrac{\alpha}{2}}{1+tan^2\dfrac{\alpha}{2}} \\
cos \alpha & = & \dfrac{1 - tan^2\dfrac{\alpha}{2}}{1+tan^2\dfrac{\alpha}{2}} \\
tan \alpha & = & \dfrac{2tan\dfrac{\alpha}{2}}{1-tan^2\dfrac{\alpha}{2}} \\
\end{array} $$

Limits

Definition

“\(\displaystyle\lim_{x \to a} f(x) = L \)” means that for any choice of \(\epsilon \gt 0\), there is always a \(\delta \gt 0\), such that:
\(|f(x)-L| \lt \epsilon\) for all \(x\) satisfying \(0 \lt |x-a| \lt \delta\).

DNE: Do not exist.

New limits form old ones

The limit of sum is the sum of limits. $$ \lim_{x\to a}(f(x)+g(x)) = \lim_{x\to a}f(x) + \lim_{x\to a}g(x) $$

Proof tip:\(\displaystyle\frac\epsilon 2\)

The limit of product is the product of the limits $$ \lim_{x\to a}f(x)g(x) = \lim_{x\to a}f(x) \times \lim_{x\to a}g(x) $$

Proof tip:

  1. \(|f(x)g(x) - Lg(x) + Lg(x) - LM|\)
  2. \(g(x) - M < 1 \Rightarrow |g(x)| < |M| + 1\)

The limit of quotient is the quotient of the limits. $$ \lim_{x\to a}\frac{f(x)}{g(x)} = \frac{\displaystyle\lim_{x\to a}f(x)}{\displaystyle\lim_{x\to a}g(x)} $$

Proof tip:

  1. \(\displaystyle|\frac{f(x)}{g(x)} - \frac{M}{g(x)} + \frac{M}{g(x)} - \frac LM|\)
  2. \(g(x) > M - \epsilon \Rightarrow |g(x)| > \displaystyle\frac{|M|}2 \)

Sandwich principle

Also called as squeeze principle.

If \(g(x) < f(x) < h(x)\) for all x near a, and \(\displaystyle \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L, \text{then} \displaystyle \lim_{x \to a} f(x) = L\)

Proof tip:

\(L-\epsilon < g(x) \le f(x) \le h(x) < L + \epsilon \Rightarrow |f(x) - L| < \epsilon \)

Limit problems involving polynomials

  1. Substitute (直接代入)
  2. Factor (因式分解)
    • \(\displaystyle\lim_{x\to 2} \frac{x^2-3x+2}{x-2}\)
    • \(a^3-b^3=(a-b)(a^2+ab+b^2)\)
  3. Using (a-b)(a+b)
    • \(\displaystyle\lim_{x\to 5} \frac{\sqrt{x^2-9}-4}{x-5}\)
    • conjugate (共轭) expression: \(\sqrt{x^2-9}-4\)
  4. Focus on leading term
    • \(\displaystyle\lim_{x\to \infty} \frac{C}{x^n} = 0, \ n > 0 \)
  5. Be careful of positive and negative.
    • \(\displaystyle\lim_{x\to \infty} \frac{(x^2-9)(9-x^3)}{5x^5}\)
    • \(\displaystyle\lim_{x\to -\infty} \frac{\sqrt{x^2-9}}{x} = -1\)
    • \(\displaystyle\lim_{x\to 0^-} \frac{|x|}{x} = -1\)

Continuity and Differentiability

Continuity

Definition: A function \(f\) is continuous at \(x=a\) if \(\displaystyle\lim_{x\to a}f(x)=f(a)\)

The intermediate value theorem

介值定理。

If \(f\) is continuous on \([a, b]\), \(f(a)<0 \text{ and } f(b)>0\), then there is at least one number \(c\) in the interval \((a,b)\) such that \(f(c)=0\). The same is true if instead \(f(a)>0 \text{ and } f(b)<0\).

Max-Min Theorem

最大最小值定理。

If \(f\) is continuous on \([a, b]\), then \(f\) has at least one maximum and one minimum on \([a,b]\).

Derivative function

$$ f’(x) = \lim_{\Delta{x}\to 0}\frac{f(x+\Delta{x})-f(x)}{\Delta{x}}=\lim_{\Delta{x}\to 0}\frac{\Delta{y}}{\Delta{x}}=\frac{dy}{dx} $$

  • \(\Delta{x}\) means change in \(x\)
  • \(dx\) means really tiny change in \(x\)

$$ f^″(x) = f^{(2)}(x) = \frac{d^2y}{dx^2} $$

Differentiability and continuity

If a function \(f\) is differentiable at \(x\), then it’s continuous at \(x\).

Proof tip.

$$ \lim_{\Delta{x}\to 0}\Delta{x}\cdot f’(x) = \lim_{\Delta{x}\to 0}\frac{f(x+\Delta{x})-f(x)}{\Delta{x}}\cdot \lim_{\Delta{x}\to 0}\Delta{x} $$

  • Differentiable functions are automatically continuous.
  • Continuous functions aren’t always differentiable.

Differentiating with basic operations

  • constant multiples of functions
  • sums and differences of functions
  • product rule
  • quotient rule
  • chain rule

Constant multiplication

$$ \frac{d}{dx}(cy) = c\frac{dy}{dx} $$

Proof tip.

Substitute into the formula of \(f’(x)\) directly.

Sum and difference

$$

$$

Proof tip.

Substitute into the formula directly.