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# MATH 117: Calculus 1
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## Functions
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A **function** is a rule where each input has exactly one output, which can be determined by the **vertical line test**.
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!!! definition
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- The **domain** is the set of allowable independent values.
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- The **range** is the set of allowable dependent values.
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Functions can be **composed** to apply the result of one function to another.
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$$
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(f\circ g)(x) = f(g(x))
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$$
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!!! warning
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Composition is not commutative: $f\circ g \neq g\circ f$.
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## Inverse functions
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The inverse of a function swaps the domain and range of the original function: $f^{-1}(x)$ is the inverse of $f(x)$.. It can be determined by solving for the other variable:
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$$
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\begin{align*}
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y&=mx+b \\
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y-b&=mx \\
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x&=\frac{y-b}{m}
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\end{align*}
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$$
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Because the domain and range are simply swapped, the inverse function is just the original function reflected across the line $y=x$.
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<img src="https://upload.wikimedia.org/wikipedia/commons/1/11/Inverse_Function_Graph.png" width=300>(Source: Wikimedia Commons, public domain)</img>
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If the inverse of a function is applied to the original function, the original value is returned.
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$$f^{-1}(f(x)) = x$$
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A function is **invertible** only if it is "**one-to-one**": each output must have exactly one input. This can be tested via a horizontal line test of the original function.
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If a function is not invertible, restricting the domain may allow a **partial inverse** to be defined.
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!!! example
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<img src="https://upload.wikimedia.org/wikipedia/commons/7/70/Inverse_square_graph.svg">(Source: Wikimedia Commons, public domain)</img>
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By restricting the domain to $[0,\inf]$, the **multivalued inverse function** $y=\pm\sqrt{x}$ is reduced to just the partial inverse $y=\sqrt{x}$.
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## Symmetry
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An **even function** satisfies the property that $f(x)=f(-x)$, indicating that it is unchanged by a reflection across the y-axis.
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An **odd function** satisfies the property that $-f(x)=f(-x)$, indicating that it is unchanged by a 180° rotation about the origin.
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The following properties are always true for even and odd functions:
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- even × even = even
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- odd × odd = even
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- even × odd = odd
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Functions that are symmetric (that is, both $f(x)$ and $f(-x)$ exist) can be split into an even and odd component. Where $g(x)$ is the even component and $h(x)$ is the odd component:
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$$
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\begin{align*}
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f(x) &= g(x) + h(x) \\
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g(x) &= \frac{1}{2}(f(x) + f(-x)) \\
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h(x) &= \frac{1}{2}(f(x) - f(-x))
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\end{align*}
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$$
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!!! note
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The hyperbolic sine and cosine are the even and odd components of $f(x)=e^x$.
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$$
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\cosh x = \frac{1}{2}(e^x + e^{-x}) \\
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\sinh x = \frac{1}{2}(e^x - e^{-x})
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$$
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## Piecewise functions
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A piecewise function is one that changes formulae at certain intervals. To solve piecewise functions, each of one's intervals should be considered.
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### Absolute value function
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$$
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\begin{align*}
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|x| = \begin{cases}
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x &\text{ if } x\geq 0 \\
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-x &\text{ if } x < 0
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\end{cases}
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\end{align*}
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$$
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### Signum function
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The signum function returns the sign of its argument.
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$$
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\begin{align*}
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\text{sgn}(x)=\begin{cases}
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-1 &\text{ if } x < 0 \\
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0 &\text{ if } x = 0 \\
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1 &\text{ if } x > 0
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\end{cases}
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\end{align*}
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$$
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### Ramp function
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The ramp function makes a ramp through the origin that suddenly flatlines at 0. Where $c$ is a constant:
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$$
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\begin{align*}
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r(t)=\begin{cases}
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0 &\text{ if } x \leq 0 \\
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ct &\text{ if } x > 0
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\end{cases}
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\end{align*}
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$$
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<img src="https://upload.wikimedia.org/wikipedia/commons/c/c9/Ramp_function.svg" width=700>(Source: Wikimedia Commons, public domain)</img>
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### Floor and ceiling functions
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The floor function rounds down.
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$$\lfloor x\rfloor$$
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The ceiling function rounds up.
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$$\lceil x \rceil$$
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### Fractional part function
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In a nutshell, the fractional part function:
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- returns the part **after the decimal point** if the number is positive
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- returns 1 - **the part after the decimal point** if the number is negative
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$$\text{FRACPT}(x) = x-\lfloor x\rfloor$$
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Because this function is periodic, it can be used to limit angles to the $[0, 2\pi)$ range with:
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$$f(\theta) = 2\pi\cdot\text{FRACPT}\biggr(\frac{\theta}{2\pi}\biggr)$$
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### Heaviside function
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The Heaviside function effectively returns a boolean whether the number is greater than 0.
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$$
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\begin{align*}
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H(x) = \begin{cases}
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0 &\text{ if } t < 0 \\
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1 &\text{ if } t \geq 0
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\end{cases}
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\end{align*}
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$$
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This can be used to construct other piecewise functions by enabling them with $H(x-a)$ as a factor, where $a$ is the interval.
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In a nutshell:
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- $1-H(t-a)$ lets you "turn a function off" at at $t=a$
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- $H(t-a)$ lets you "turn a function on at $t=a$
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- $H(t-a) - H(t-b)$ leaves a function on in the interval $(a, b)$
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!!! example
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TODO: example for converting piecewise to heaviside via collecting heavisides
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and vice versa
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## Periodicity
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The function $f(t)$ is periodic only if there is a repeating pattern, i.e. such that for every $x$, there is an $f(x) = f(x + nT)$, where $T$ is the period and $n$ is any integer.
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### Circular motion
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Please see [SL Physics 1#6.1 - Circular motion](/g11/sph3u7/#61-circular-motion) and its subcategory "Angular thingies" for more information.
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## Partial function decomposition (PFD)
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In order to PFD:
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1. Factor the denominator into *irreducibly* quadratic or linear terms.
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2. For each factor, create a term. Where capital letters below are constants:
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- A linear factor $Bx+C$ has a term $\frac{A}{Bx+C}$.
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- An *irreducibly* quadratic factor $Dx^2+Ex+G$ has a term $\frac{Hx+J}{Dx^2+Ex+G}$.
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- Duplicate factors have terms with denominators with that factor to the power of 1 up to the number of times the factor is present in the original.
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4. Set the two equal to each other such that the denominators can be factored out.
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5. Create systems of equations to solve for each constant.
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!!! example
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To decompose $\frac{x}{(x+1)(x^2+x+1)}$:
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$$
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\begin{align*}
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\frac{x}{(x+1)(x^2+x+1)} &= \frac{A}{x+1} + \frac{Bx+C}{x^2+x+1} \\
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&= \frac{A(x^2+x+1) + (Bx+C)(x+1)}{(x+1)(x^2+x+1)} \\
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x &= A(x^2+x+1) + (Bx+C)(x+1) \\
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0x^2 + x + 0 &= (Ax^2 + Bx^2) + (Ax + Bx + Cx) + (A + C) \\
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\\
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&\begin{cases}
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0 = A + B \\
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1 = A + B + C \\
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0 = A + C
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\end{cases}
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\\
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A &= -1 \\
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B &= 1 \\
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C &= 1 \\
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\\
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∴ \frac{x}{(x+1)(x^2+x+1)} &= -\frac{1}{x+1} + \frac{x + 1}{x^2 + x + 1}
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\end{align*}
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$$
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## Trigonometry
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1 radian represents the angle when the length of the arc of a circle is equal to the radius. Where $s$ is the arc length:
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$$\theta=\frac{s}{r}$$
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The following table indicates the special angles that should be memorised:
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| Angle (rad) | 0 | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | $\pi$ |
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| --- | --- | --- | --- | --- | --- | --- |
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| cos | 1 | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | 0 | -1 |
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| sin | 0 | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ | 1 | 0 |
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| tan | 0 | $\frac{\sqrt{3}}{3}$ | 1 | $\sqrt{3}$ | not allowed | 0 |
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### Identities
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The Pythagorean identity is the one behind right angle triangles:
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$$\cos^2\theta+\sin^2\theta = 1$$
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Cosine and sine can be converted between by an angle shift:
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$$
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\cos\biggr(\theta-\frac{\pi}{2}\biggr) = \sin\theta \\
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\sin\biggr(\theta-\frac{\pi}{2}\biggr) = \cos\theta
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$$
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The **angle sum identities** allow expanding out angles:
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$$
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\cos(a+b)=\cos a\cos b - \sin a\sin b \\
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\sin(a+b)=\sin a\cos b + \cos a\sin b
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$$
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Subtracting angles is equal to the conjugates of the angle sum identities.
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The **double angle identities** simplify the angle sum identity for a specific case.
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$$
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\sin2\theta = 2\sin\theta\cos\theta \\
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$$
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The **half angle formulas** are just random shit.
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$$
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1+\tan^2\theta = \sec^2\theta \\
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\cos^2\theta = \frac{1}{2}(1+\cos2\theta) \\
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\sin^2\theta = \frac{1}{2}(1-\cos2\theta)
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$$
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### Inverse trig functions
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Because extending the domain does not pass the horizontal line test, for engineering purposes, inverse sine is only the inverse of sine so long as the angle is within $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Otherwise, it is equal to that version mod 2 pi.
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$$y=\sin^{-1}x \iff x=\sin y, y\in [-\frac{\pi}{2}, \frac{\pi}{2}]$$
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This means that $x\in[-1, 1]$.
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$$
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\sin(\sin^{-1}x) = x \\
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\sin^{-1}(\sin x) = x \text{ only if } x\in[-\frac{\pi}{2}, \frac{\pi}{2}]
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$$
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Similarly, inverse **cosine** only returns values within $[0,\pi]$.
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Similarly, inverse **tangent** only returns values within $(-\frac{\pi}{2}, \frac{\pi}{2})$. However, $\tan^{-1}$ is defined for all $x\in\mathbb R$.
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Although most of the reciprocal function rules can be derived, secant is only valid in the odd range $[-\pi, -\frac{\pi}{2})\cup [0, \frac{\pi}{2})$, and returns values $(-\infty, -1]\cup [1, \infty)$.
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### Electrical signals
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Waves are commonly presented in the following format, where $A$ is a **positive** amplitude:
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$$g(t)=A\sin(\omega t + \alpha)$$
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In general, if given a sum of a sine and cosine:
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$$a\sin\omega t + b\cos\omega t = \sqrt{a^2 + b^2}\sin(\omega t + \alpha)$$
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The sign of $\alpha$ should be determined via its quadrant via the signs of $a$ (sine) and $b$ (cosine) via the CAST rule.
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!!! example
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Given $y=5\cos 2t - 3\sin 2t$:
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$$
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\begin{align*}
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A\sin (2t+\alpha) &= A\sin 2t\cos\alpha + A\cos 2t\sin\alpha \\
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&= (A\cos\alpha)\sin 2t + (A\sin\alpha)\cos 2t \\
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\\
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\begin{cases}
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A\sin\alpha = 5 \\
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A\cos\alpha = -3
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\end{cases}
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\\
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\\
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A^2\sin^2\alpha + A^2\cos^2\alpha &= 5^2 + (-3)^2 \\
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A^2 &= 34 \\
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A &= \sqrt{34} \\
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\\
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\alpha &= \tan^{-1}\frac{5}{3} \\
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&\text{since sine is positive and cosine is negative, the angle is in Q3} \\
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∴ \alpha &= \tan^{-1}\frac{5}{3} + \pi
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\end{align*}
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$$
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## Limits
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### Limits of sequences
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!!! definition
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- A **sequence** is an infinitely long list of numbers with the **domain** of all natural numbers (may also include 0).
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- A sequence that does not converge is a **diverging** sequence.
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A sequence is typically denoted via braces.
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$$\{a_n\}\text{ or } \{a_n\}^\infty_{n=0}$$
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Sometimes sequences have formulae.
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$$\left\{\frac{5^n}{3^n}\right\}^\infty_{n=0}$$
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The **limit** of a sequence is the number $L$ that the sequence **converges** to as $n$ increases, which can be expressed in either of the two ways below:
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$$
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a_n \to L \text{ as } n\to\infty \\
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\lim_{n\to\infty}a_n=L
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$$
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: > Specifically, a sequence $\{a_n\}$ converges to limit $L$ if, for any positive number $\epsilon$, there exists an integer $N$ such that $n>N \Rightarrow |a_n - L | < \epsilon$.
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Effectively, if there is always a term number that would lead to the distance between the sequence at that term and the limit to be less than any arbitrarily small $\epsilon$, the sequence has the claimed limit.
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!!! example
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A limit can be proved to exist with the above definition. To prove $\left\{\frac{1}{\sqrt{n}}\right\}\to0$ as $n\to\infty$:
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$$
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\begin{align*}
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\text{Proof:} \\
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n > N &\Rightarrow \left|\frac{1}{\sqrt{n}} - 0\right| < \epsilon \\
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&\Rightarrow \frac{1}{\epsilon^2} < n
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\end{align*} \\
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\ce{Let \epsilon\ be any positive number{.} If n > \frac{1}{\epsilon^2}, then \frac{1}{\sqrt{n}}-> 0 as n -> \infty{.}}
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$$
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Please see [SL Math - Analysis and Approaches 1#Limits](/g11/mhf4u7/#limits) for more information.
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The **squeeze theorem** states that if a sequence lies between two other converging sequences with the same limit, it also converges to this limit. That is, if $a_n\to L$ and $c_n\to L$ as $n\to\infty$, and $a_n\leq b_n\leq c_n$ is **always true**, $b_n\to L$.
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!!! example
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$\left\{\frac{\sin n}{n}\right\}$: since $-1\leq\sin n\leq 1$, $\frac{-1}{n}\leq\frac{\sin n}{n}\leq \frac{1}{n}$. Since both other functions converge at 0, and sin(n) is always between the two, sin(n) thus also converges at 0 as n approaches infinity.
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If function $f$ is continuous and $\lim_{n\to\infty}a_n$ exists:
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$$\lim_{n\to\infty}f(a_n)=f\left(\lim_{n\to\infty}a_n\right)$$
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On a side note:
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$$\lim_{n\to\infty}\tan^{-1} n = \frac{\pi}{2}$$
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### Limits of functions
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The definition is largely the same as for the limit of a sequence:
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: > A function $f(x)\to L$ as $x\to a$ if, for any positive $\epsilon$, there exists a number $\delta$ such that $0<|x-a|<\delta\Rightarrow|f(x)-L|<\epsilon$.
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Again, for the limit to be true, there must be a value $x$ that makes the distance between the function and the limit less than any arbitrarily small $\epsilon$.
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The extra $0 <$ is because the behaviour for when $x=a$, which may or may not be defined, is irrelevant.
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!!! example
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To prove $3x-2\to 4$ as $x\to 2$:
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$$
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\ce{for any \epsilon\ > 0, there is a \delta\ > 0\ such that:}
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$$
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$$
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\begin{align*}
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|x-2| < \delta &\Rightarrow|(3x-2) - 4| &< \epsilon \\
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&\Leftarrow |(3x-2) -4| &< \epsilon \\
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&\Leftarrow |3x-6| &< \epsilon \\
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&\Leftarrow |x-2| &< \frac{\epsilon}{3} \\
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\delta &= \frac{\epsilon}{3}
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\end{align*}
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$$
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$$
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\ce{Let \epsilon\ be any positive number{.} If }|x-2|<\frac{\epsilon}{3}, \\
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\text{then }|(3x-2)-4|<\epsilon\text{. Therefore }3x-2\to 4\text{ as }x\to 2.
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$$
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!!! warning
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When solving for limits, negatives have to be considered if the limit approaches a negative number:
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$$\lim_{x\to -\infty}\frac{x}{\sqrt{4x^2-3}} = \frac{1}{-\frac{1}{\sqrt{x}^2}\sqrt{4x^2-3}}$$
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As the angle in **radians** of an arc approaches 0, it is nearly equal to the sine (vertical component).
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$$
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\lim_{\theta\to 0}\frac{\sin\theta}{\theta} = 1
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$$
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This function is commonly used in engineering and is known as the sinc function.
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$$
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\text{sinc}(x) = \begin{cases}
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\frac{\sin x}{x}&\text{ if }x\neq 0 \\
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0&\text{ if }x=0
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\end{cases}
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$$
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## Continuity
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Please see [SL Math - Analysis and Approaches 1#Limits and continuity](/g11/mhf4u7/#limits-and-continuity) for more information.
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Most common functions can be assumed to be continuous (e.g., $\sin x,\cos x, x, \sqrt{x}, \frac{1}{x}, e^x, \ln x$, etc.).
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: > $f(x)$ is continuous in an interval if for any $x$ and $y$ in the interval and any positive number $\epsilon$, there exists a number $\delta$ such that $|x-y|<\delta\Rightarrow |f(x)-f(y)| < \epsilon$.
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Effectively, if $f(x)$ can be made infinitely close to $f(y)$ by making $x$ closer to $y$, the function is continuous.
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If two functions are continuous:
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- $(f\circ g)(x)$ is continuous
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- $(f\pm g)(x)$ is continuous
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- $(fg)(x)$ is continuous
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- $\frac{1}{f(x)}$ is continuous anywhere $f(x)\neq 0$
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### Intermediate value theorem
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The IVT states that if a function is continuous and there is a point between two other points, its term must also be between those two other points.
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: > If $f(x)$ is continuous, if $f(a)\leq C\leq f(b)$, there must be a number $c\in[a,b]$ where $f(c)=C$.
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The theorem is used to validate using binary search to find roots (guess and check).
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### Extreme value theorem
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The EVT states that any function continuous within a **closed** interval has at least one maximum and minimum.
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: > If $f(x)$ is continuous in the **closed interval** $[a, b]$, there exist numbers $c$ and $d$ in $[a,b]$ such that $f(c)\leq f(x)\leq f(d)$.
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## Derivatives
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Please see [SL Math - Analysis and Approaches 1#Rate of change](/g11/mhf4u7/#rate-of-change) and [SL Math - Analysis and Approaches#Derivatives](/g11/mhf4u7/#derivatives) for more information.
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The derivative of a function $f(x)$ at $a$ is determined by the following limit:
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$$\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
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If the limit does not exist, the function is **not differentiable at $a$**.
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Alternative notations for $f'(x)$ include $\dot f(x)$ and $Df$ (which is equal to $\frac{d}{dx}f(x)$).
|
||
|
||
Please see [SL Math - Analysis and Approaches 1#Finding derivatives using first principles](/g11/mhf4u7/#finding-derivatives-using-first-principles) and [SL Math - Analysis and Approaches 1#Derivative rules](/g11/mhf4u7/#derivative-rules) for more information.
|
||
|
||
Some examples of derivatives of inverse functions:
|
||
|
||
- $\frac{d}{dx}f^{-1}(x) = \frac{1}{\frac{dx}{dy}}$
|
||
- $\frac{d}{dx}\sin^{-1} x = \frac{1}{\sqrt{1-x^2}}$
|
||
- $\frac{d}{dx}\cos^{-1} x = -\frac{1}{\sqrt{1-x^2}}$
|
||
- $\frac{d}{dx}\tan^{-1} x = \frac{1}{1+x^2}$
|
||
- $\frac{d}{dx}\log_a x = \frac{1}{(\ln a) x}$
|
||
- $\frac{d}{dx}a^x = (\ln a)a^x$
|
||
|
||
### Implicit differentiation
|
||
|
||
Please see [SL Math - Analysis and Approaches 1#Implicit differentiation](/g11/mhf4u7/#implicit-differentiation) for more information.
|
||
|
||
### Mean value theorem
|
||
|
||
The MVT states that the average slope between two points will be reached at least once between them if the function is differentiable.
|
||
|
||
: > If $f(x)$ is continuous in $[a, b]$ and differentiable in $(a, b)$, respectively, there must be a $c\in(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$.
|
||
|
||
### L'Hôpital's rule
|
||
|
||
As long as $\frac{f(x)}{g(x)} = \frac{0}{0}\text{ or } \frac{\infty}{\infty}$:
|
||
|
||
$$\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$$
|
||
|
||
: > If $f(x)$ and $g(x)$ are differentiable (except maybe at $a$), and $\lim_{x\to a}f(x) = 0$ and $\lim_{x\to a}g(x) = 0$, the relation is true.
|
||
|
||
### Related rates
|
||
|
||
Please see [SL Math - Analysis and Approaches 1#Related rates](/g11/mhf4u7/#related-rates) for more information.
|
||
|
||
## Differentials
|
||
|
||
$\Delta x$ and $\Delta y$ represent tiny increments of $x$ and $y$. $dx$ and $dy$ are used when those tiny ammounts approach 0.
|
||
|
||
Specifically, by rearranging the definition of the deriative, $df$ is a short form for the **differential** of $f$:
|
||
|
||
$$f'(x)dx=dy=df$$
|
||
|
||
By abusing differentials, the tangent line of a point in a function can be approximated.
|
||
|
||
$$\Delta f\approx f'(x)\Delta x$$
|
||
|
||
!!! example
|
||
If $f(x) = \sqrt{x},x_0=81$, $\sqrt{78}$ can be estimated by:
|
||
|
||
$$
|
||
\begin{align*}
|
||
\Delta x&=dx=78-81=-3 \\
|
||
\frac{df}{dx} &= f'(x) \\
|
||
df &= f'(x)dx \\
|
||
&= \frac{1}{2\sqrt{81}}(-3) = -\frac{1}{6} \\
|
||
f(78) &= \sqrt{81}-\frac{1}{6} \\
|
||
&= \frac{53}{54}
|
||
\end{align*}
|
||
$$
|
||
|
||
### Curve sketching
|
||
|
||
Please see [SL Math - Analysis and Approaches 1#5.2 - Increasing and decreasing functions](/g11/mhf4u7/#52-increasing-and-decreasing-functions) for more information.
|
||
|
||
## Integrals
|
||
|
||
Please see [SL Math - Analysis and Approaches 2#Integration](/g11/mhf4u7/#52-increasing-and-decreasing-functions) for more information.
|
||
|
||
### More integration rules
|
||
|
||
- $\int a^xdx = \frac{a^x}{\ln a} + C$
|
||
- $\int\sec^2xdx=\tan x+C$
|
||
- $\int\text{cosh } xdx = \text{sinh } x + C$
|
||
- $\int\text{sinh } xdx = \text{cosh } x + C$
|
||
- $\int\frac{1}{\sqrt{1-x^2}}dx = \sin^{-1}x+C$
|
||
- $\int\csc^2xdx = -\cot x+C$
|
||
- $\int\sec x\tan x dx = \sec x + C$
|
||
- $\int\csc x\cot xdx = -\csc x + C$
|
||
- $\int\frac{1}{1+x^2}dx=\tan^{-1}x+C$
|
||
- $\int\sec xdx = \ln|\sec x + \tan x| + C$
|
||
- $\int\csc x dx = -\ln|\csc x + \cot x| + C$
|
||
|
||
### Integration by parts
|
||
|
||
IBP lets you replace an integration problem with a different, potentially easier one.
|
||
|
||
$$
|
||
\int u\ dv = uv-\int v\ du
|
||
$$
|
||
|
||
or, in function notation:
|
||
|
||
$$
|
||
\int u(x)v'(x)dx = u(x)v(x)-\int v(x)u'(x)dx
|
||
$$
|
||
|
||
Effectively, a product of two factors should be made simpler such that one is differentiable and the other is integratable. While there are integrals on both sides, the constant $C$ can be cancelled out for simplicity.
|
||
|
||
Heuristics to be used:
|
||
|
||
- $dv$ must be differentiable
|
||
- $u$ should be simpler when differentiated
|
||
- IBP might need to be used repeatedly
|
||
- IBP and u-substitution might be needed together
|
||
|
||
!!! example
|
||
To solve $\int xe^xdx$:
|
||
|
||
Let $u=x$, $dv=e^xdx$:
|
||
|
||
$\therefore du=dx, v=e^x + C$
|
||
|
||
via IBP:
|
||
|
||
$$
|
||
\begin{align*}
|
||
\int udv &= xe^x - \int e^xdx \\
|
||
&= xe^x-e^x + K
|
||
\end{align*}
|
||
$$
|
||
|
||
Please see [SL Math - Analysis and Approaches 2#Area between two curves](/g11/mcv4u7/#area-between-two-curves) for more information.
|
||
|
||
- A **Type 1** region is bounded by functions of $x$ — it's open-ended in the x-axis.
|
||
- A **Type 2** region is bounded by functions of $y$, which can be solved by integrating $y$.
|
||
- A **Type 3** region can be viewed as either Type 1 or 2.
|
||
|
||
Substituting $u=\cos\theta$, $du=-\sin\theta d\theta$ is common.
|
||
|
||
### Mean values
|
||
|
||
|
||
The **mean value** of a continuous function $f(x)$ in $[a, b]$ is equal to:
|
||
|
||
$$\text{m.v.} (f) = \frac{1}{b-a}\int_a^b f(x)dx$$
|
||
|
||
The **root mean square** is equal to the square root of the mean value for each point:
|
||
|
||
$$\text{r.m.s.} (f) = \sqrt{\frac{1}{b-a}\int_a^b f(x)^2dx}$$
|
||
|
||
### Trigonometric substitution
|
||
|
||
If $a\in\mathbb R$, functions of the form $\sqrt{x^2\pm a^2}$ or $\sqrt{a^2-x^2}$ can be rearranged in the form of a trig function.
|
||
|
||
- In $\sqrt{x^2 + a^2} \rightarrow x=a\tan\theta$
|
||
- In $\sqrt{x^2-a^2} \rightarrow x=a\sec\theta$
|
||
- In $\sqrt{a^2-x^2} \rightarrow x=a\sin\theta$
|
||
|
||
…which can be used to derive other trig identities to be integrated.
|
||
|
||
### Rational integrals
|
||
|
||
All integrals of rational functions are expressible as more rational functions, ln, and arctan.
|
||
|
||
Partial fraction decomposition is useful here.
|
||
|
||
$$\int \frac{1}{x^2+a^2}dx=\frac{1}{a}tan^{-1}\left(\frac{x}{a}\right)+C$$
|
||
|
||
## Summary of all integration rules
|
||
|
||
- $\int x^n\ dx = \frac{1}{n+1}x^{n+1} + C,n\neq -1$
|
||
- $\int \frac{1}{x}dx = \ln|x| + C$
|
||
- $\int e^x\ dx = e^x + C$
|
||
- $\int a^x\ dx = \frac{1}{\ln a} a^x + C$
|
||
- $\int\cos x\ dx = \sin x + C$
|
||
- $\int\sin x\ dx = -\cos x + C$
|
||
- $\int\sec^2 x\ dx = \tan x + C$
|
||
- $\int\csc^2 x\ dx = -\cot x + C$
|
||
- $\int\sec x\tan x\ dx = \sec x + C$
|
||
- $\int\csc x\cot x\ dx = -\csc x + C$
|
||
- $\int\text{cosh}\ x\ dx = \text{sinh}\ x + C$
|
||
- $\int\text{sinh}\ x\ dx = \text{cosh}\ x + C$
|
||
- $\int\text{sech}^2\ x\ dx = \text{tanh}\ x + C$
|
||
- $\int\text{sech}\ x\text{tanh}\ x\ dx = \text{sech}\ x + C$
|
||
- $\int\frac{1}{1+x^2}dx=\tan^{-1}x+C$
|
||
- $\int\frac{1}{a^2+x^2}dx=\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C$
|
||
- $\int\frac{1}{\sqrt{1-x^2}}dx=\sin^{-1}x+C$
|
||
- $\int\frac{1}{x\sqrt{x^2-1}}dx=\sec^{-1}x+C$
|
||
- $\int\sec x\ dx = \ln|\sec x+\tan x|+C$
|
||
- $\int\csc x\ dx = -\ln|\csc x + \cot x|+C$
|
||
|
||
## Applications of integration
|
||
|
||
The length of a curve over a given interval is equal to:
|
||
|
||
$$L=\int^b_a\sqrt{1+\left(\frac{dy}{dx}\right)^2\ dx}$$
|
||
|
||
For curves bounded by functions of $y$:
|
||
|
||
$$L(y)=\int^b_a\sqrt{1+\left(\frac{dx}{dy}\right)^2\ dy}$$
|
||
|
||
### Solids of revolution
|
||
|
||
Please see [SL Math - Analysis and Approaches 2#Volumes of solids of revolution](/g11/mcv4u7/#volumes-of-solids-of-revolution) for more information.
|
||
|
||
The **parallel axis theorem can be used** to shift the axis of the solid to $y=k$:
|
||
|
||
$$V=\pi\int^b_a [f(x)^2 + 2kf(x)]\ dx$$
|
||
|
||
Around the vertical axis about the origin with a function that is bounded by $y$:
|
||
|
||
$$V=\int^b_a2\pixf(x)\ dx$$
|
||
|
||
Around the vertical axis about the origin with functions bounded by $x$:
|
||
|
||
$$V=\int^b_a2\pi(x-k)[f(x)-g(x)]\ dx$$
|
||
|
||
The **frustrum** is the sesction bounded by two parallel plates.
|
||
|
||
The surface area of the solids are as follows:
|
||
|
||
$$SA=\int^b_a2\pi f(x)\sqrt{1+f'(x)^2}\ dx$$
|
||
|
||
Around the vertical axis about the origin:
|
||
|
||
$$SA=\int^b_a2\pi x\sqrt{1+f'(x)^2}\ dx$$
|
||
|
||
### Improper integrals
|
||
|
||
An improper integral is a definite integral where only one bound is defined:
|
||
|
||
!!! example
|
||
$\int_2^\infty$ or $\int_a^b$, where only $a$ is defined.
|
||
|
||
These can be expanded into limits:
|
||
|
||
$$\int_a^\infty f(x)\ dx = \lim_{t\to\infty}\int_a^t f(x)\ dx$$
|
||
|
||
The integral converges to a value if the limit exists.
|
||
|
||
$$\int_{-\infty}^a f(x)\ dx = \lim_{t\to-\infty}\int^a_tf(x)\ dx$$
|
||
|
||
Discontinuities can be simply dodged. If there is a discontinuity:
|
||
|
||
- at $b$: $\int_a^{b^-}f(x)\ dx$
|
||
- at $a$: $\int_{a^+}^b f(x)\ dx$
|
||
- at $a<c<b$: $\int_a^cf(x)\ dx + \int_c^bf(x)\ dx$
|
||
|
||
Limits to both infinities must be broken up because they may not approach them at the same rate.
|
||
|
||
$$\int^\infty_{-\infty}x\ dx = \int^0_{-\infty} x\ dx + \int^\infty_0 x\ dx$$
|
||
|
||
## Polar form
|
||
|
||
Please see [MATH 115: Linear Algebra#Polar form](/ce1/math115/#polar-form) for more information.
|
||
|
||
Instead of $r$ and $\theta$, engineers use $\rho$ and $\phi$.
|
||
|
||
For $\rho \geq 0$, these basic conversions go between the two forms:
|
||
|
||
- $x=\rho\cos\phi$
|
||
- $y=\rho\sin\phi$
|
||
- $\phi=\sqrt{x^2+y^2}$
|
||
- $\phi=\tan^{-1}\left(\frac{y}{x}\right) + 2k\pi,k\in\mathbb Z$
|
||
|
||
Polar form allows for simpler representations such as $x^2+y^2=4 \iff \rho=2$
|
||
|
||
Functions are described in the form $\rho=f(\phi)$, such as $\rho=\sin\phi+2$.
|
||
|
||
### Area under curves
|
||
|
||
From the axis to the curve:
|
||
|
||
$$A=\int^\beta_\alpha\frac{1}{2}[f(\phi)]^2\ d\phi$$
|
||
|
||
Between two curves:
|
||
|
||
$$A=\int^\beta_\alpha\frac{1}{2}[f(\phi)^2-g(\phi)^2]\ d\phi$$
|
||
|
||
Arc length:
|
||
|
||
$$L=\int^\beta_\alpha\sqrt{f'(\phi)^2 + f(\phi)^2}\ d\phi = \int^\beta_\alpha\sqrt{\left(\frac{d\rho}{d\phi}\right)^2+\rho^2}\ d\phi$$
|
||
|
||
## Complex numbers
|
||
|
||
Please see [MATH 115: Linear Algebra#Complex Numbers](/ce1/math115/#complex-numbers) for more information.
|