Business Calculus Study Notes: The Derivative

Chapter 1: The Derivative

1.1 The Slope of a Straight Line

The concept of slope is foundational in calculus, representing the steepness of a function. Understanding slope in straight lines helps us later interpret slope in curves.

• Slope-Intercept Form: The equation of a nonvertical line is $y = mx + b$, where m is the slope and b is the y-intercept.

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ (rise over run).

• Economic Interpretation: In cost functions like $y = 2x + 1000$, the slope (2) is the marginal cost (cost per additional unit), and the y-intercept (1000) is the fixed cost.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, useful for writing equations given a point and slope.

• Parallel Lines: Same slope, different y-intercepts; never intersect.

• Perpendicular Lines: Slopes are opposite reciprocals; $m_1 \cdot m_2 = -1$.

Example: The equation $y = 30,000 - 400t$ (oil tank) has a y-intercept of 30,000 gallons (initial amount) and a slope of -400 (oil used per day).

1.2 The Slope of a Curve at a Point

To find the slope of a curve at a specific point, we use the concept of a tangent line, which touches the curve at one point and approximates its slope there. The slope of the tangent line is the slope of the curve at that point.

• Secant Line: Connects two points on a curve; its slope approximates the tangent as the points get closer.

• Instantaneous Rate of Change: As the secant line approaches the tangent, we find the instantaneous slope.

Example: For $f(x) = x^2$ at $P = (1,1)$, the slope of the tangent line is 2.

Rate of Change: The slope at a point on a graph represents the rate of change at that instant.

1.3 The Derivative and Limits

The derivative of a function gives the slope of the curve at any point. For $y = f(x)$, the derivative is $f'(x)$, also written as $\frac{dy}{dx}$.

• Linear Functions: The derivative is the slope $m$; for $y = mx + b$, $f'(x) = m$.

• Power Rule: If $f(x) = x^r$, then $f'(x) = r x^{r-1}$.

Example: For $f(x) = x^3$, $f'(x) = 3x^2$; for $f(x) = \sqrt{x}$, $f'(x) = \frac{1}{2\sqrt{x}}$.

Geometric Meaning: $f'(a)$ is the slope of the tangent line to $f(x)$ at $(a, f(a))$.

Point-Slope Equation for Tangent Line: $y - f(a) = f'(a)(x - a)$

• Difference Quotient: $\frac{f(x+h) - f(x)}{h}$ as $h \to 0$ gives the derivative.

1.4 Limits and the Derivative

The concept of a limit is central to calculus. The limit describes the value a function approaches as the input approaches a certain point.

• Definition: $\lim_{x \to a} g(x) = L$ if $g(x)$ gets arbitrarily close to $L$ as $x$ approaches $a$.

• Basic Limits: $\lim_{x \to a} C = C$ (constant), $\lim_{x \to a} x = a$.

• Limit Theorems: Sum, difference, product, quotient, and power rules for limits.

• Difference Quotient: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Limits at Infinity: Analyze behavior as $x \to \infty$ or $x \to -\infty$.

Example: $\lim_{x \to \infty} \frac{1}{x^2 + 1} = 0$

1.5 Differentiability and Continuity

A function is differentiable at a point if its derivative exists there. Common reasons for non-differentiability include discontinuities, abrupt changes, or vertical tangents.

• Continuity: A function is continuous at $x = a$ if $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$.

• Relationship: Differentiability implies continuity, but not vice versa.

Example: Piecewise cost functions may be continuous but not differentiable at certain points.

1.6 Rules of Differentiation

Several rules simplify the process of finding derivatives:

• Constant Multiple Rule: $\frac{d}{dx}[k \cdot f(x)] = k \cdot \frac{d}{dx}f(x)$

• Sum Rule: $\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$

• General Power Rule: $\frac{d}{dx}[g(x)^r] = r \cdot g(x)^{r-1} \cdot \frac{d}{dx}g(x)$

Example: $\frac{d}{dx}(5x^3 + 4x) = 15x^2 + 4$

1.7 More about Derivatives

Derivatives can be taken with respect to any variable, and higher-order derivatives (second, third, etc.) are possible.

• Second Derivative: $f''(x) = \frac{d^2}{dx^2}f(x)$, measures the rate of change of the rate of change.

• Estimation: $f(a+1) \approx f(a) + f'(a)$ for small changes.

Example: Marginal cost and marginal revenue use derivatives to estimate changes in economics.

1.8 The Derivative as a Rate of Change

The derivative represents the instantaneous rate of change, while the secant line gives the average rate of change over an interval.

• Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$

• Instantaneous Rate of Change: $f'(a)$

• Velocity: $v(t) = s'(t)$, where $s(t)$ is position.

• Acceleration: $a(t) = v'(t) = s''(t)$

Example: For $s(t) = -16t^2 + 128t + 5$, velocity and acceleration can be found using derivatives.

Application: Slope as Rate of Change

In practical scenarios, the slope of a graph at a point represents the rate of change of a quantity, such as oil consumption over time.

Example: The slope at point P on the oil level graph is -62.5, indicating a decrease of 62.5 gallons per hour.

Summary Table: Basic Derivative Rules

Rule	Formula	Example
Power Rule	$\frac{d}{dx}x^n = n x^{n-1}$	$\frac{d}{dx}x^3 = 3x^2$
Constant Rule	$\frac{d}{dx}C = 0$	$\frac{d}{dx}5 = 0$
Sum Rule	$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$	$\frac{d}{dx}(x^2 + x) = 2x + 1$
Constant Multiple	$\frac{d}{dx}[k f(x)] = k f'(x)$	$\frac{d}{dx}(3x^2) = 6x$

Additional info: These notes expand on the original material with academic context, definitions, and examples for clarity and completeness.