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Applications of Derivatives

CSCA Applications of Derivatives study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Applications of Derivatives

1. Core Logic & Visualization

The derivative $f'(x)$ acts as a "barometer" for the function $f(x)$, indicating its trend:

* **Sign of $f'(x)$** $\longleftrightarrow$ **Monotonicity (Increasing/Decreasing)**

* **Zeros of $f'(x)$** $\longleftrightarrow$ **Extrema (Max/Min points)**

| Sign of $f'(x)$ | Behavior of $f(x)$ | Visual |

| :--- | :--- | :--- |

| **Positive (+)** | **Increasing** | 📈 Uphill |

| **Negative (-)** | **Decreasing** | 📉 Downhill |

| **Zero (0)** | **Stationary** | ➖ Flat (Possible Extrema) |

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2. Key Topics & Methods

#### Topic 1: Finding Intervals of Monotonicity

**Steps**:

1. **Domain**: Always consider the domain first.

2. **Differentiate**: Find $f'(x)$.

3. **Solve Inequalities**:

* $f'(x) > 0 \Rightarrow$ **Intervals of Increase**.

* $f'(x) < 0 \Rightarrow$ **Intervals of Decrease**.

#### Topic 2: Finding Local Extrema

**Steps**:

1. Find **Critical Points**: Where $f'(x)=0$ or is undefined.

2. **Sign Chart (First Derivative Test)**: Check the sign change of $f'(x)$ around critical points.

* **+ to -** $\Rightarrow$ **Local Maximum** (Peak)

* **- to +** $\Rightarrow$ **Local Minimum** (Valley)

* No change $\Rightarrow$ No extrema.

#### Topic 3: Absolute Extrema on Closed Interval $[a, b]$

**Steps**:

1. Find all critical points inside $(a, b)$.

2. Evaluate the function at these critical points **AND** at endpoints $a$ and $b$.

3. **Compare**: The highest value is the Absolute Max; the lowest is the Absolute Min.

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3. Examples

**Example 1: Monotonicity & Extrema**

Analyze $f(x) = x^3 - 3x$.

**Solution**:

1. **Derivative**: $f'(x) = 3x^2 - 3 = 3(x+1)(x-1)$.

2. **Critical Points**: $x = \pm 1$.

3. **Sign Chart**:

| $x$ | $(-\infty, -1)$ | $-1$ | $(-1, 1)$ | $1$ | $(1, +\infty)$ |

| :--- | :---: | :---: | :---: | :---: | :---: |

| $f'(x)$ | $+$ | $0$ | $-$ | $0$ | $+$ |

| $f(x)$ | $\nearrow$ | **Max (2)** | $\searrow$ | **Min (-2)** | $\nearrow$ |

* Increasing: $(-\infty, -1]$ and $[1, +\infty)$

* Decreasing: $[-1, 1]$

* Local Max: $f(-1)=2$; Local Min: $f(1)=-2$.

**Example 2: Absolute Extrema**

Find max/min of $f(x) = x^3 - 3x + 1$ on $[-2, 2]$.

**Solution**:

1. Critical points $x=\pm 1$ are inside the interval.

2. Evaluate:

* Endpoints: $f(-2) = -1$, $f(2) = 3$

* Critical Pts: $f(-1) = 3$, $f(1) = -1$

3. Compare: Absolute Max = 3, Absolute Min = -1.

**Example 3: Optimization**

Maximize the area of a rectangle with perimeter 20.

**Solution**:

1. **Model**: Let length be $x$, width is $10-x$. Area $S(x) = 10x - x^2$ ($0<x<10$).

2. **Derivative**: $S'(x) = 10 - 2x$.

3. **Solve**: $S'(x)=0 \Rightarrow x=5$.

4. **Test**: Since graph is a parabola opening down, vertex ($x=5$) is the max.

5. **Result**: Area is max ($25$) when $x=5$ (Square).

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4. Common Pitfalls

* **Notation**: Do not use the union symbol ($\cup$) to combine monotonic intervals. List them separately.

* **Local vs. Absolute**: A local max is a peak; the absolute max is the highest point overall. Always check **endpoints** for closed interval problems.

* **Domain**: In word problems (optimization), ensure your answer lies within the physical constraints (e.g., length must be positive).