AP Calculus AB 2.1 Instantaneous Rate of Change Using Limits (Example 1 with Quadratics)

### AP Calculus AB 2.1: Instantaneous Rate of Change Using Limits

**Overview:**
In AP Calculus AB, Section 2.1 introduces the concept of the **instantaneous rate of change**, which is the derivative at a specific point. This is calculated using **limits**, a fundamental concept in calculus. The instantaneous rate of change measures how a function is changing at a single point, as opposed to the average rate of change over an interval.

### Key Concept: Instantaneous Rate of Change
The **instantaneous rate of change** of a function \( f(x) \) at a specific point \( x = c \) is defined as the limit of the average rate of change as the interval around \( c \) becomes infinitesimally small. Mathematically, this is expressed as:
\[
\text{Instantaneous Rate of Change} = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}
\]
This is the **definition of the derivative** at \( x = c \).

### Example 1: Instantaneous Rate of Change of a Quadratic Function

Let’s use the quadratic function \( f(x) = x^2 + 2x \) to calculate the instantaneous rate of change at \( x = 3 \).

#### Step 1: Set Up the Limit Definition of the Derivative
Using the limit definition of the derivative, we want to find:
\[
\lim_{h \to 0} \frac{f(3 + h) - f(3)}{h}
\]
This expression will give us the instantaneous rate of change at \( x = 3 \).

#### Step 2: Evaluate the Function at \( x = 3 + h \) and \( x = 3 \)
1. **Find \( f(3 + h) \):**
\[
f(3 + h) = (3 + h)^2 + 2(3 + h) = (9 + 6h + h^2) + (6 + 2h) = h^2 + 8h + 15
\]

2. **Find \( f(3) \):**
\[
f(3) = 3^2 + 2(3) = 9 + 6 = 15
\]

#### Step 3: Substitute into the Limit Expression
Now, substitute \( f(3 + h) \) and \( f(3) \) into the limit formula:
\[
\lim_{h \to 0} \frac{(h^2 + 8h + 15) - 15}{h} = \lim_{h \to 0} \frac{h^2 + 8h}{h}
\]

#### Step 4: Simplify the Expression
Factor out \( h \) in the numerator:
\[
\lim_{h \to 0} \frac{h(h + 8)}{h} = \lim_{h \to 0} (h + 8)
\]
As \( h \to 0 \), the expression simplifies to:
\[
8
\]

#### Step 5: Interpret the Result
The instantaneous rate of change of \( f(x) = x^2 + 2x \) at \( x = 3 \) is **8**. This means that at the point \( x = 3 \), the function is increasing at a rate of 8 units per unit of \( x \).

### Conclusion
In AP Calculus AB, the **instantaneous rate of change** is found by applying the **limit definition of the derivative**. For quadratic functions like \( f(x) = x^2 + 2x \), this involves evaluating the function at \( x + h \), subtracting the value of the function at \( x \), and simplifying the resulting expression before taking the limit as \( h \to 0 \). The result gives the slope of the tangent line to the curve at that specific point, representing how the function is changing at that moment.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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