Calculus 1 Lecture 1.5 | Slope of a Curve, Velocity, and Rates of Change

The video “Calculus 1 Lecture 1.5: Slope of a Curve, Velocity, and Rates of Change” by Professor Leonard dives into the fundamental calculus concept of finding the slope of a curve at a point using limits, connecting it to velocity and rates of change.

Key Concepts Explained:

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The Goal: Slope of a Curve at a Point 📈

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• The core idea is to find the slope of the tangent line to a curve at a specific point.
• Unlike linear functions, curves change slope continuously, so the slope varies at each point.
• To describe the slope at one point, we start by considering the secant line between two nearby points and then make the points infinitely close.

Introducing Points and Notation:

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• Fix point $P$ at $x = x_0$, and let point $Q$ be at $x_0 + h$, where $h$ is a small increment.
• The function values at these points are $f(x_0)$ and $f(x_0 + h)$, respectively.

Secant Line and Slope:

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• The slope of the secant line between $P$ and $Q$ is:

$$ \frac{f(x_0 + h) - f(x_0)}{h} $$

• This is called the difference quotient.

Using Limits to Find the Tangent Line Slope:

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• To find the tangent slope, let $h$ approach 0 (meaning $Q$ gets arbitrarily close to $P$).
• The slope of the tangent is the limit of the difference quotient as $h \to 0$:

$$ \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} $$

• This limit, if it exists, is the instantaneous rate of change at $x_0$.

Example Calculation (Function: $y = x^2$ at $x=1$):

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1. Compute $f(1 + h) = (1 + h)^2 = 1 + 2h + h^2$.
2. Substitute into the difference quotient:

$$ \frac{(1 + 2h + h^2) - 1}{h} = \frac{2h + h^2}{h} = 2 + h $$

3. Take the limit as $h \to 0$:

$$ \lim_{h \to 0} (2 + h) = 2 $$

4. The tangent slope at $x=1$ is 2.

Equation of the Tangent Line:

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• Using point-slope form:

$$ y - y_0 = m(x - x_0) $$

• With $x_0=1$, $y_0=1$, and slope $m=2$:

$$ y - 1 = 2(x - 1) \rightarrow y = 2x - 1 $$

Instantaneous Velocity and Rates of Change:

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• The same concept applies to velocity, where the position function is $s(t)$, and the instantaneous velocity is the derivative (limit of average velocity as time interval shrinks to zero).
• Rates of change in general can be seen as slopes of functions, connecting algebraic computations to real-world physical meanings.

Summary:

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• This lecture demonstrates how limits formalize the intuitive idea of a slope at a point.
• Introduces the difference quotient as a stepping stone towards the derivative.
• Shows practical computation of tangent slopes using limits.
• Highlights the foundational importance for understanding rates of change and motion in calculus.

This video marks a pivotal moment in transitioning from algebraic approximations to precise calculus definitions, essential for all further studies in the subject. ¹