Exam Cram Sheet: Functions, Graphs, and Mathematical Foundations

This guide covers the essential foundations for Calculus I, focusing on the mechanics of functions, limits, and the transition into derivatives.

## Topic Weighting

## Key Concepts Summary

• The Function Rule: Every input $x$ in the Domain maps to exactly one output $y$ in the Range.
• Transcendental vs. Algebraic: Algebraic functions involve poly/roots; Transcendental functions (Trig, Logs, Exponents) "transcend" algebra.
• The Limit Existence: $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$.
• The Derivative: Represents the instantaneous rate of change or the slope of the tangent line at a point.

[!IMPORTANT] Intermediate Value Theorem (IVT): If $f$ is continuous on $[a, b]$, it must take every value between $f(a)$ and $f(b)$ at least once.

## Common Pitfalls

• Inverse Trig Ranges: Forgetting that $\arcsin(x)$ is restricted to $[-\pi/2, \pi/2]$ and $\arccos(x)$ to $[0, \pi]$.
• Log Rules: Confusing $\ln(x+y)$ with $\ln(x) + \ln(y)$. (Note: $\ln(xy) = \ln x + \ln y$ is correct).
• Chain Rule Neglect: Forgetting to multiply by the "inside" derivative, e.g., $\frac{d}{dx}(\sin(x^2)) = 2x\cos(x^2)$, not just $\cos(x^2)$.
• Vertical Asymptotes: Assuming every zero of the denominator is an asymptote (check if it's a hole/removable discontinuity first).

## Mnemonics / Memory Triggers

• SOH CAH TOA: $\sin = \frac{O}{H}, \cos = \frac{A}{H}, \tan = \frac{O}{A}$.
• The CAST Rule: Quadrants where trig functions are positive: Cos (IV), All (I), Sin (II), Tan (III).
• Quotient Rule Song: "Low d-High minus High d-Low, over the square of what's below."
  • $\frac{d}{dx}[\frac{u}{v}] = \frac{v du - u dv}{v^2}$
• Log Power Rule: "The Log brings the power down to the front." $\ln(x^n) = n\ln(x)$.

## Formula / Equation Sheet

Trigonometric Identities

Exponential & Logarithmic Rules

• Change of Base: $\log_b(x) = \frac{\ln x}{\ln b}$
• Natural Base: $e = \lim_{n \to \infty} (1 + \frac{1}{n})^n \approx 2.718$
• Inverses: $e^{\ln x} = x$ and $\ln(e^x) = x$

Core Derivatives

1. Power Rule: $\frac{d}{dx} x^n = nx^{n-1}$
2. Exponentials: $\frac{d}{dx} e^{g(x)} = e^{g(x)} \cdot g'(x)$
3. Logarithms: $\frac{d}{dx} \ln(x) = \frac{1}{x}$
4. Trig: $\frac{d}{dx} \sin x = \cos x$; $\frac{d}{dx} \cos x = -\sin x$

## Practice Set

1. Inverse Property: Given $f(x) = \frac{3x - 1}{x + 2}$, find the domain of $f^{-1}(x)$.
   • Hint: The domain of the inverse is the range of the original function. Check the horizontal asymptote.
2. Limit Calculation: Evaluate $\lim_{x \to 0} \frac{e^{2x} - 1}{\sin(x)}$.
   • Check for indeterminate form 0/0 and apply L'Hôpital's Rule.
3. Factor Identification: A polynomial $f(x)$ passes through $(-5, 0)$, $(1, 0)$, and $(4, 0)$. Which must be a factor of $f(x)$?
   • A) $(x+1)$ | B) $(x-1)$ | C) $(x-5)$ | D) $(x+4)$
4. Chain Rule: Differentiate $y = \ln(\cos(x^3))$.
   • Apply layers: Natural log $\rightarrow$ Cosine $\rightarrow$ Cubic.
5. Logarithmic Equation: Solve for $x$ in $2\ln(x) - \ln(x-1) = \ln(4)$.

▶Click to view Practice Set Answers

1. Range of f: $y \neq 3$ (H.A. is $3/1). Therefore, Domain of f^{-1}$ is $x \neq 3$.
2. $\frac{d}{dx}$ top: $2e^{2x}$; bottom: $\cos x$. At $x=0$, $\frac{2(1)}{1} = \mathbf{2}$.
3. Since $f(1) = 0$, $(x-1)$ must be a factor.
4. $y'$= \frac{1}{\cos(x^3)} \cdot (-\sin(x^3)) \cdot 3x^2 = \mathbf{-3x^2\tan(x^3)}$$.
5. $\ln(\frac{x^2}{x-1}) = \ln(4) \Rightarrow x^2 = 4x - 4 \Rightarrow x^2 - 4x + 4 = 0 \Rightarrow (x-2)^2 = 0 \Rightarrow \mathbf{x=2}$.