Advanced Algebra, Functions, and Calculus Study Notes

Logarithmic to Linear Conversion

• To convert a non-linear function to a linear equation of the form $y = mx + c$, apply logarithms to both sides.

• Power Law: $y = Bx^n \rightarrow \log(y) = \log(B) + n\log(x)$. Here, mapping $Y = \log(y)$ against $X = \log(x)$ yields gradient $m = n$ and intercept $c = \log(B)$.

• Exponential Law: $y = Ab^{-x} \rightarrow \log(y) = \log(A) - x\log(b)$. Mapping $\log(y)$ against $x$ yields gradient $m = -\log(b)$ and intercept $c = \log(A)$.

Rational Expressions and Algebra

• To simplify rational fractions: Factor denominators to find the Least Common Denominator (LCD), create equivalent fractions, and add/subtract numerators.

• Example: Dr. Ahmed Sengab demonstrates $\frac{1}{x^2 - 4} + \frac{1}{x - 2} = \frac{1 + (x + 2)}{(x + 2)(x - 2)} = \frac{x + 3}{(x + 2)(x - 2)}$.

Functions: Definitions and Types

• Function: A relation where every input has exactly one output.

• One-to-One: Each $x$ produces a unique $y$.

• Many-to-One: Multiple $x$ values produce the same $y$ value (e.g., $y = x^2$).

• Inverse Function ($f^{-1}(x)$): Exists only for one-to-one functions. To find it, replace $f(x)$ with $y$, swap $x$ and $y$, and solve for $y$.

• Composite Functions: $fg(x)$ means substituting the function $g(x)$ into $f(x)$. If $fg(x) = x$, then $g(x) = f^{-1}(x)$.

Geometric Transformations

• X-axis: $f(x \pm a)$ (Translation), $f(ax)$ (Stretching), $f(-x)$ (Reflection on y-axis).

• Y-axis: $f(x) \pm a$ (Translation), $af(x)$ (Stretching), $-f(x)$ (Reflection on x-axis).

• Modulus:   - $y = |f(x)|$: Reflect the negative y-part above the x-axis.   - $y = f(|x|)$: Erase the negative x-part and reflect the positive x-part across the y-axis.

Exponential and Natural Logarithm (Ln) Laws

• Exponential ($e^x$): Growth function. $e^{-\infty} = 0$, $e^0 = 1$, and any number divided by $\infty$ is zero.

• Natural Log (Ln): Base-e logarithm. $\ln(a) = b \iff e^b = a$.

• Rules:   - $\ln(a) + \ln(b) = \ln(ab)$   - $\ln(a) - \ln(b) = \ln(\frac{a}{b})$   - $\ln(a^x) = x\ln(a)$   - $\ln(e^x) = x$

Trigonometric Identities and the R-Form

• Reciprocals: $\sec(x) = \frac{1}{\cos(x)}$, $\csc(x) = \frac{1}{\sin(x)}$, $\cot(x) = \frac{\cos(x)}{\sin(x)} = \frac{1}{\tan(x)}$.

• Pythagorean: $\sin^2(x) + \cos^2(x) = 1$, $1 + \cot^2(x) = \csc^2(x)$, $\tan^2(x) + 1 = \sec^2(x)$.

• Double Angles: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$; $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$.

• R-Form: Used to combine sum/difference of sine and cosine into a single term:   - a\sin(x) \pm b\cos(x) = R\sin(x \pm ̑)   - a\cos(x) ∓ b\sin(x) = R\cos(x \pm ̑)   - Where $R = \sqrt{a^2 + b^2}$ and ̑ = \tan^{-1}(\frac{b}{a}).

Differentiation Rules

• Chain Rule: $\frac{d}{dx}[(g(x))^n] = n(g(x))^{n-1} \times g'(x)$.

• Exponential/Log: $\frac{d}{dx}[e^{g(x)}] = g'(x)e^{g(x)}$ and $\frac{d}{dx}[\ln(g(x))] = \frac{g'(x)}{g(x)}$.

• Product Rule: $(uv)' = u'v + uv'$.

• Quotient Rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$.

• Trigonometric: $\sin(x) \rightarrow \cos(x)$; $\cos(x) \rightarrow -\sin(x)$; $\tan(x) \rightarrow \sec^2(x)$.

Integration Methods

• Power Rule: $\int (ax + b)^n dx = \frac{(ax + b)^{n+1}}{a(n+1)} + c$.

• Exponential/Log: $\int e^{ax+b} dx = \frac{e^{ax+b}}{a} + c$ and $\int \frac{1}{ax+b} dx = \frac{\ln|ax+b|}{a} + c$.

• Trigonometric:   - $\int \sin(x) dx = -\cos(x) + c$   - $\int \cos(x) dx = \sin(x) + c$   - $\int \sec^2(x) dx = \tan(x) + c$

• Special Case for Squares: Use identities to integrate squared functions, e.g., $\sin^2(x) = \frac{1}{2} - \frac{\cos(2x)}{2}$.