Foundations of Calculus: Algebra, Functions, and Trigonometry

Review of Quadratic Equations

Factoring and Roots

Quadratic equations are polynomial equations of degree two, commonly written as $ax^2 + bx + c = 0$. The solutions (roots) can be found by factoring or using the quadratic formula.

• Standard Form: $ax^2 + bx + c$

• Factored Form: $(a_1x + \beta_1)(a_2x + \beta_2)$, where $a_1a_2 = a$, $a_1\beta_2 + a_2\beta_1 = b$, $\beta_1\beta_2 = c$

• Quadratic Formula: $x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example

• Solve $x^2 + bx + c = 0$ using the quadratic formula.

Logarithmic Expressions

Definition and Properties

A logarithm is the inverse operation to exponentiation. The logarithm $\log_a b = x$ means $a^x = b$.

• Definition: $\log_a b = x \iff a^x = b$

• Properties:

  1. $\log_a 1 = 0$

  2. $\log_a a = 1$

  3. $\log_a (bc) = \log_a b + \log_a c$

  4. $\log_a \left(\frac{b}{c}\right) = \log_a b - \log_a c$

  5. $\log_a (b^k) = k \log_a b$

  6. $\log_a b = \frac{\log_c b}{\log_c a}$

Examples

• $\log(2x) = \log 2 + \log x$

• $\log x^2 = 2 \log x$

• $\log_{25} 5 = \frac{1}{2}$

Additional info: Logarithms with base $e$ are called natural logarithms, denoted $\ln x$.

Exponential Expressions

Properties of Exponents

Exponential expressions involve repeated multiplication of a base. The following properties are fundamental:

• $a^x \cdot a^y = a^{x+y}$

• $(a^x)^y = a^{xy}$

• $a^x \cdot b^x = (ab)^x$

• $a^0 = 1$

• Inverse Relationship: $\log_a(a^x) = x$

Examples

• Solve $2^x = 16 \implies x = 4$

• Solve $3^x = 20 \implies x = \frac{\log 20}{\log 3}$

Functions

Definition and Notation

A function $f$ from a set $A$ to $\mathbb{R}$ assigns to each element $x$ in $A$ a unique real number $f(x)$.

• Domain: The set of all input values for which the function is defined.

• Range: The set of all possible output values of the function.

Example

• For $f(x) = x^2$, domain is $\mathbb{R}$, range is $[0, \infty)$.

Substituting Values and Expressions

To evaluate a function, substitute the given value or expression for $x$.

• Given $f(x) = x^2$, $f(1) = 1$, $f(2) = 4$, $f(3) = 9$

• For $f(x+h) = (x+h)^2 = x^2 + 2xh + h^2$

Difference Quotient

The difference quotient is used to compute the average rate of change of a function:

• $\frac{f(x+h) - f(x)}{h}$

• For $f(x) = x^2$, $\frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Linear Functions

Definition and Graph

A linear function has the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Its graph is a straight line.

• Slope ($m$): Measures the steepness of the line.

• Y-intercept ($b$): The point where the line crosses the y-axis.

Equation of a Line

• General Form: $ax + by = d$

• Slope-Intercept Form: $y = mx + b$

• Vertical Line: $x = p$

• Horizontal Line: $y = k$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

Example

• Find the equation of the line passing through $(2,3)$ and $(4,9)$: Slope: $m = \frac{9-3}{4-2} = 3$ Equation: $y = 3x + b$ Substitute $(2,3)$: $3 = 3 \cdot 2 + b \implies b = -3$ Final equation: $y = 3x - 3$ Or, using point-slope form: $y - 3 = 3(x - 2)$

Trigonometric Functions

Degrees and Radians Conversion

Angles can be measured in degrees or radians. To convert between them:

• Degrees to radians: multiply by $\frac{\pi}{180}$

• Radians to degrees: multiply by $\frac{180}{\pi}$

Unit Circle and Trigonometric Ratios

The unit circle helps define the sine, cosine, and tangent functions.

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• $\sec \theta = \frac{1}{\cos \theta}$

• $\csc \theta = \frac{1}{\sin \theta}$

• $\cot \theta = \frac{1}{\tan \theta}$

Graphs of Trigonometric Functions

• Sine Function: $y = \sin x$, domain $(-\infty, \infty)$, range $[-1, 1]$

• Cosine Function: $y = \cos x$, domain $(-\infty, \infty)$, range $[-1, 1]$

• Tangent Function: $y = \tan x$, domain $x \neq \frac{\pi}{2} + k\pi$, $k \in \mathbb{Z}$, range $(-\infty, \infty)$

Special Values

• $\sin 0 = 0$, $\sin \frac{\pi}{2} = 1$, $\sin \pi = 0$

• $\cos 0 = 1$, $\cos \frac{\pi}{2} = 0$, $\cos \pi = -1$

• $\tan 0 = 0$, $\tan \frac{\pi}{4} = 1$, $\tan \frac{\pi}{2}$ does not exist (DNE)

Additional info: These foundational topics are essential for understanding calculus concepts such as limits, derivatives, and integrals.