57 lines
3.4 KiB
Markdown
57 lines
3.4 KiB
Markdown
To help you hit that 95-second-per-problem target, I have designed this "Drill Test." Instead of 57 problems, we are going to do a **12-problem representative sample**.
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To simulate the real exam pressure, set a timer for **19 minutes** (this keeps the ~95 second average). Your goal isn't just to get them right—it's to identify the "Cheat Code" from your guide within the first 5 seconds.
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---
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### Phase 1: The Speed Run (Target: 30 Seconds Each)
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*Identify the pattern and execute immediately.*
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**1.** Evaluate: $\lim_{x \to \infty} \frac{8x^{2} - 10x}{2x^{2} + 5}$
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**2.** Find the Average Rate of Change (ARC) for $f(x) = 15x - 7$ on the interval $[4, 10]$.
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**3.** Evaluate: $\lim_{x \to 4} (x^{2} - 3x + 1)$
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**4.** Evaluate: $\lim_{x \to \infty} \frac{5x + 100}{x^{3}}$
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### Phase 2: The Mechanics (Target: 90 Seconds Each)
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*Requires a bit of "scratch paper" work.*
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**5.** Evaluate: $\lim_{x \to 6} \frac{x^{2} - 36}{x - 6}$
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**6.** Find the value of $k$ that makes $f(x)$ continuous at $x = 3$:
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$f(x) = \begin{cases} 4x + k & x \le 3 \\ 15 & x > 3 \end{cases}$
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**7.** Find the ARC for $f(x) = x^{2} + 1$ from $x = 1$ to $x = 4$.
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**8.** Simplify the Difference Quotient for $f(x) = 5x^{2}$.
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### Phase 3: The Grind (Target: 2-3 Minutes Each)
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*These are your "Pass 3" problems. Don't let them eat your time early.*
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**9.** Evaluate: $\lim_{x \to 0} \frac{\sqrt{x + 9} - 3}{x}$
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**10.** Simplify the Difference Quotient for $f(x) = \frac{2}{x}$.
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**11.** Is $f(x)$ continuous at $x = 2$? Prove why/why not:
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$f(x) = \begin{cases} x^{2} & x < 2 \\ 5 & x = 2 \\ 4 & x > 2 \end{cases}$
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**12.** Find the ARC for $f(x) = \sqrt{x}$ on the interval $[16, 25]$.
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---
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### The Strategy Key (Read this AFTER you try the problems)
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| Problem | The Cheat Code Used | Why it's a "Speed" or "Grind" problem |
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| :--- | :--- | :--- |
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| **1** | **Tug of War (Tie)** | Degrees match ($x^2$). Just take $8/2$. Answer: **4**. |
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| **2** | **Linear Cheat Code** | $f(x)$ is linear. ARC is always the slope. Answer: **15**. |
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| **3** | **Direct Substitution** | No denominator. Just plug in 4. Answer: **5**. |
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| **4** | **Tug of War (Bottom Wins)** | $x^3$ on bottom is bigger than $x$ on top. Answer: **0**. |
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| **5** | **0/0 Rescue (Factor)** | Plugging 6 gives 0/0. Factor top to $(x-6)(x+6)$. Answer: **12**. |
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| **6** | **The Bridge** | Set $4(3) + k = 15$. $12 + k = 15$. Answer: **k = 3**. |
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| **7** | **Standard ARC** | Find $f(4)=17$ and $f(1)=2$. $(17-2)/(4-1) = 15/3$. Answer: **5**. |
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| **8** | **"H" Cleanout** | $5(x+h)^2 - 5x^2 = 10xh + 5h^2$. Divide by $h$. Answer: **10x + 5h**. |
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| **9** | **Conjugate Rescue** | Multiply top/bottom by $(\sqrt{x+9} + 3)$. Answer: **1/6**. |
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| **10** | **LCD Method** | Requires common denominators. Answer: **-2 / (x(x+h))**. |
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| **11** | **3-Step Proof** | Point = 5. Limit = 4. They don't match! Answer: **No (Step 3 fail)**. |
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| **12** | **Root ARC** | $(\sqrt{25} - \sqrt{16}) / (25 - 16) = (5-4)/9$. Answer: **1/9**. |
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### How to use this for the 90-minute Exam:
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If you encounter a problem like **#9 or #10** on the exam, and you've already spent 60 seconds and are feeling stuck—**SKIP IT.** Mark it and move on. In the time it takes to struggle through one rational difference quotient, you could have finished four "Tug of War" or "Linear ARC" problems. You want to secure all the 30-second points first to build a "time bank" for the harder ones at the end.
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How many of these were you able to identify correctly before looking at the key? |