From 0279ee5bf442d4ba13aaf9ef2cf8246c3898605d Mon Sep 17 00:00:00 2001 From: hugh Date: Sat, 28 Mar 2026 10:48:46 -0500 Subject: [PATCH] Add exam_study_guide_01.md --- exam_study_guide_01.md | 76 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 76 insertions(+) create mode 100644 exam_study_guide_01.md diff --git a/exam_study_guide_01.md b/exam_study_guide_01.md new file mode 100644 index 0000000..c762832 --- /dev/null +++ b/exam_study_guide_01.md @@ -0,0 +1,76 @@ +This is exactly the right mindset. To survive a 57-question sprint in 90 minutes, you need to stop "doing math" and start "running protocols." You have about 95 seconds per problem. If you spend 5 minutes on one algebra expansion, you've lost 3 other potential points. + +Here is your **Exam 1 Master Strategy Guide**. It combines everything we've worked through—from the first limit we solved to the last difference quotient. + +*** + +# The 90-Minute Blitz Strategy: The 3-Pass System + +* **Pass 1 (0-30 mins): The Speed Run.** Solve every Linear Rate of Change, Direct Substitution Limit, and Infinity (Tug of War) problem. These should take 20 seconds each. If a problem looks like it requires a lot of writing, skip it for now. +* **Pass 2 (30-70 mins): The Mechanics.** Solve the 0/0 Limits (Factoring), Piecewise Continuity checks, and Difference Quotients. These take 1–2 minutes of algebraic focus. +* **Pass 3 (70-90 mins): The Grind.** Go back to the Square Root (Conjugate) limits and the complex word problems. These are the "time sinks." + +--- + +# Module 1 & 2: Limits and Continuity + +### The Limit "Scan" Checklist +When you see $\lim$, look at the target value immediately. + +1. **If $x \to \infty$ (The Tug of War):** Compare the highest exponents (degrees). + * **Bottom Wins:** (e.g., $x^2 / x^3$) Answer is **0**. + * **Top Wins:** (e.g., $x^4 / x^2$) Answer is **$\infty$** or **$-\infty$**. + * **It's a Tie:** (e.g., $5x^2 / 2x^2$) Answer is the fraction of the front numbers: **5/2**. + + +2. **If $x \to a$ (Direct Substitution):** Plug the number in first. + * **Normal Number:** You're done. + * **Number / 0:** Limit **Does Not Exist (DNE)**. It's a vertical wall. + * **0 / 0:** It's a **Hole**. You must Factor or use a Conjugate to "rescue" it. + +### The Continuity "3-Step Proof" +Use this for the "Is $f(x)$ continuous at $x = c$?" problems. +1. **Step 1:** Does the point exist? (Look for the $\le$ or $\ge$ sign). +2. **Step 2:** Does the limit exist? (Do the left trail and right trail meet at the same altitude?). +3. **Step 3:** Do they match? (Point = Limit). + + +--- + +# Module 3: Average Rates of Change & Difference Quotient + +### The Average Rate of Change (ARC) +* **ARC = Slope.** It is always $\frac{y_2 - y_1}{x_2 - x_1}$. +* **The Linear Cheat Code:** If the function is linear (like $f(x) = 5x + 2$), the ARC is always just the number in front of $x$ (in this case, **5**). Don't calculate it! +* **Graph Problems:** Identify the two years/points on the x-axis, find their heights on the y-axis, and run the slope formula. + +### The Difference Quotient Mastery +Formula: $\frac{f(x+h) - f(x)}{h}$ + +* **The Goal:** Every term in your final answer will likely still have an $x$ or an $h$, but the $h$ on the *bottom* must be cancelled out. +* **The "H-Rule" (Error Detection):** After you subtract the original function, **every remaining term on top must have an $h$**. If you have a term like "+7" with no $h$ attached, you made a FOIL error. Go back and check your expansion. + + +--- + +# Instant Identification Matrix + +| Problem Type | Recognition Key | The Cheat Code / Action | +| :--- | :--- | :--- | +| **Limit at Infinity** | $x \to \infty$ | **Tug of War.** Ignore everything but the highest exponents. | +| **Algebraic Limit** | $x \to$ number | **Plug it in.** If you get 0/0, factor $(x - \text{target})$. | +| **Piecewise Limit** | Function has brackets $\{ $ | **Check both sides.** Plug the x-value into both equations. | +| **Continuity Find** | "Find $k$ to make it continuous" | **The Bridge.** Set the top equation = bottom equation and solve. | +| **Average Rate** | "Over the interval $[a, b]$" | **Slope formula.** Find $f(b)$ and $f(a)$, then $(f(b) - f(a)) / (b - a)$. | +| **Difference Quotient** | Contains $h$ | **Algebraic Cleanout.** Expand, subtract original, divide by $h$. | + +--- + +# Final "Pro-Tips" for the Exam 1 Speed Run + +1. **Watch the Signs:** In the Difference Quotient, when you "Subtract $f(x)$", you must subtract the **whole** thing. This usually means flipping the signs of every term in the original function. +2. **Don't Over-Simplify:** If MyMathLab doesn't explicitly ask for a simplified fraction, sometimes it will accept the unreduced version, saving you 30 seconds of division. +3. **The Square Root Shortcut:** If you have $\lim_{x \to a} \sqrt{f(x)}$, just find the limit of $f(x)$ first, then take the square root of your answer at the very end. +4. **$0$ vs $DNE$:** Remember, $0$ on top is fine (Answer = $0$). $0$ on the bottom is a crash (Answer = $DNE$). + +You have the tools and the templates. Copy this into your master guide, and when you open that exam, spend the first 60 seconds just scrolling through to identify which problems belong to which "Cheat Code." You've got this! \ No newline at end of file