Exam Cram: Application of Derivatives

This guide focuses on the practical application of the derivative to analyze function behavior, solve real-world optimization problems, and relate rates of change across multiple variables.

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Topic Weighting

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Key Concepts Summary

• Critical Points: Occur where $f'(c) = 0$ or $f'(c)$ is undefined. These are the only candidates for local extrema.
• The Extreme Value Theorem (EVT): If $f is continuous on a closed interval [a, b]$, it must have an absolute maximum and minimum. Check critical points AND endpoints.
• First Derivative Test:
  • $f'(x)$ changes from $+$ to $-$ $\rightarrow$ Local Max.
  • $f'(x)$ changes from $-$ to $+$ $\rightarrow$ Local Min.
• Concavity & Points of Inflection:
  • $f''(x) > 0 \rightarrow$ Concave Up (CCU).
  • $f''(x) < 0 \rightarrow$ Concave Down (CCD).
  • Inflection Point: Where $f''(x)$ changes sign.
• Related Rates: Differentiating equations with respect to time ($t) using the Chain Rule (e.g., \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$).
• L'Hôpital's Rule: Used for indeterminate limits $\frac{0}{0}$ or $\frac{\infty}{\infty}$. $\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)}$.

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Common Pitfalls

[!WARNING] Don't Forget the Endpoints! When finding absolute extrema on $[a, b], students often find critical points but forget to test f(a)$ and $f(b)$.

• Confusion between $f'(x)$ and $f''(x)$: $f'(x) = 0 does NOT guarantee a max/min (it could be a terrace point like y=x^3$ at $x=0$). Always check for a sign change.
• Implicit Differentiation Errors: In Related Rates, forgetting to multiply by the "inner" derivative (e.g., writing $2r$ instead of $2r \frac{dr}{dt}$).
• L'Hôpital Abuse: Do not apply L'Hôpital's rule if the limit is not indeterminate. For example, $\lim_{x\to 0} \frac{x+1}{x}$ is not $\frac{0}{0}$, so L'Hôpital does not apply.
• Average vs. Instantaneous: Average rate is $\frac{f(b)-f(a)}{b-a}$. Instantaneous rate is $f'(c)$.

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Mnemonics / Memory Triggers

• The Smiley Face Rule (Concavity):
  • $f''(x) > 0$ (Positive) $\rightarrow$ Smile (Concave Up $\cup$)
  • $f''(x) < 0$ (Negative) $\rightarrow$ Frown (Concave Down $\cap$)
• R.O.C.S. (Related Rates Strategy):
  1. Read the problem.
  2. Outline (Draw a diagram/variables).
  3. Construct the equation.
  4. Solve (Differentiate with respect to $t$).

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Formula / Equation Sheet

Visualizing the Mean Value Theorem: The tangent at some point $c$ is parallel to the secant line.

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Practice Set

1. Optimization: A rectangular garden is to be fenced against a wall. If you have 100m of fencing, find the maximum area. (Hint: $A = xy$, $2x + y = 100$).
2. Related Rates: A 10ft ladder leans against a wall. The bottom slides away at 2ft/s. How fast is the top sliding down when the base is 6ft from the wall?
3. L'Hôpital's Rule: Evaluate $\lim_{x\to 0} \frac{\sin(x) - x}{x^3}$.
4. Mean Value Theorem: Given $f(x) = x^2$ on $[0, 2], find the value of c$ that satisfies the MVT.
5. Function Analysis: If $f'(x) = (x-1)(x-3)$, identify the intervals of increase/decrease and locate local extrema.

▶Click for Answers

1. $x=25, y=50$, Area = $1250 m^2$.
2. $-1.5 ft/s$ (using $x^2 + y^2 = 10^2$).
3. $-1/6$ (requires L'Hôpital three times).
4. $c = 1$.
5. Inc: (-\infty$, 1) $\cup (3, \infty); Dec: $(1, 3)$. Local Max at $x=1$, Local Min at $x=3$.