Calc 1

1. Functions, Limits, and Continuity

1.1 Functions and Their Representations

• A function $f$ is a rule that assigns to each element $x$ in a set $D$ (domain) exactly one element $f(x)$ in a set $E$ (range).

• Vertical Line Test: A curve in the $xy$-plane is the graph of a function of $x$ if and only if no vertical line intersects the curve more than once.

• Types of Functions:

  • Polynomials: $p(x) = a<em>n x^n + a</em>{n-1} x^{n-1} + \dots + a<em>1 x + a</em>0$

  • Rational functions: $r(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials.

  • Trigonometric functions: $\sin(x)$, $\cos(x)$, $\tan(x)$.

1.2 The Limit of a Function

• Definition: We write $\lim_{x \to a} f(x) = L$ if we can make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ to be sufficiently close to $a$, but not equal to $a$.

• One-Sided Limits:

  • Left-hand limit: $\lim_{x \to a^-} f(x) = L$

  • Right-hand limit: $\lim_{x \to a^+} f(x) = L$

  • $\lim<em>{x \to a} f(x) = L$ if and only if $\lim</em>{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = L$.

• Limit Laws: If $\lim<em>{x \to a} f(x) = L$ and $\lim</em>{x \to a} g(x) = M$ exist:

  • Sum Rule: $\lim_{x \to a} [f(x) + g(x)] = L + M$

  • Product Rule: $\lim_{x \to a} [f(x) g(x)] = L \times M$

  • Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$ provided $M \neq 0$

• Squeeze Theorem: If $f(x) \le g(x) \le h(x)$ when $x$ is near $a$, and $\lim<em>{x \to a} f(x) = \lim</em>{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$.

1.3 Continuity

• A function $f$ is continuous at a number $a$ if:

  1. $f(a)$ is defined (that is, $a$ is in the domain of $f$).

  2. $\lim_{x \to a} f(x)$ exists.

  3. $\lim_{x \to a} f(x) = f(a)$.

• Intermediate Value Theorem (IVT): Suppose that $f$ is continuous on the closed interval $[a, b]$ and let $N$ be any number between $f(a)$ and $f(b)$, where $f(a) \neq f(b)$. Then there exists a number $c$ in $(a, b)$ such that $f(c) = N$.

2. Derivatives

2.1 The Derivative as a Function

• Definition: The derivative of a function $f(x)$ at $x = a$ is given by:
  $\lim<em>{h \to 0} \frac{f(a+h) - f(a)}{h}$ or equivalently, $\lim</em>{x \to a} \frac{f(x) - f(a)}{x - a}$

• Interpretation: Represents the instantaneous rate of change of $y = f(x)$ with respect to $x$, or the slope of the tangent line to the curve at $(a, f(a))$.

2.2 Differentiation Rules

• Power Rule: $\frac{d}{dx}(x^n) = n x^{n-1}$

• Constant Multiple Rule: $\frac{d}{dx}[c f(x)] = c f'(x)$

• Sum Rule: I $\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

• Product Rule: $\frac{d}{dx}[f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$

• Quotient Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x) g(x) - f(x) g'(x)}{(g(x))^2}$

• Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \times g'(x)$

2.3 Derivatives of Special Functions

• Trigonometric Functions:

  • $\frac{d}{dx}(\sin(x)) = \cos(x)$

  • $\frac{d}{dx}(\cos(x)) = -\sin(x)$

  • $\frac{d}{dx}(\tan(x)) = \sec^2(x)$

• Exponential and Logarithmic Functions:

  • $\frac{d}{dx}(e^x) = e^x$

  • $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$

2.4 Implicit Differentiation & Related Rates

• Implicit Differentiation: Used when an equation defines $y$ implicitly as a function of $x$. Differentiate both sides of the equation with respect to $x$, applying the Chain Rule to terms involving $y$, and solve for $\frac{dy}{dx}$.

• Related Rates: Procedure for solving rate-of-change problems:

  1. Identify given quantities and rates of change.

  2. Assign symbols and sketch a diagram if applicable.

  3. Formulate an equation relating the variables.

  4. Differentiate with respect to time $t$ using the Chain Rule.

  5. Substitute known values and solve for the unknown rate.