Business Calculus Exam 1 Study Guide: Derivatives, Rates of Change, and Curve Analysis

Slope and Linear Functions

Slope of a Straight Line

The slope of a straight line measures its steepness and direction. It is a fundamental concept in calculus and business applications, such as cost and revenue analysis.

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Parallel Lines: Two lines are parallel if their slopes are equal ($m_2 = m_1$).

• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is $-1$ ($m_2 = -\frac{1}{m_1}$).

• Equation of a Line: $y = mx + b$ or $y - y_1 = m(x - x_1)$

Example: Find the equation of the line parallel to $3x - y = -2$ with an x-intercept of 5.

• Rewrite: $y = 3x + 2$

• Parallel slope: $m = 3$

• Point: $(5, 0)$

• Equation: $y - 0 = 3(x - 5) \Rightarrow y = 3x - 15$

Example (Business Application): Estimating sales using a linear function $G(x)$ for gallons sold at price $x$.

• Points: $(3.00, 20,000)$ and $(6.00, 17,000)$

• Slope: $m = \frac{17,000 - 20,000}{6.00 - 3.00} = -1000$

• Equation: $G(x) = -1000x + 23,000$

• Estimate: $G(3.50) = 19,500$ gallons

Derivatives and Tangent Lines

Derivative Notation and Basic Rules

The derivative measures the rate of change of a function. It is denoted by $f'(x)$, $y'$, $\frac{dy}{dx}$, or $\frac{d}{dx}[f(x)]$.

• Constant Function: $f(x) = b \Rightarrow f'(x) = 0$

• Linear Function: $f(x) = mx + b \Rightarrow f'(x) = m$

• Power Rule: $f(x) = x^r \Rightarrow f'(x) = r x^{r-1}$

Equation of Tangent Line: $y - f(a) = f'(a)(x - a)$

Example: Find the derivative of $y = \frac{1}{\sqrt{x^{15}}}$.

• Rewrite: $y = x^{-\frac{15}{2}}$

• Derivative: $y' = -\frac{15}{2} x^{-\frac{17}{2}} = -\frac{15}{2\sqrt{x^{17}}}$

Example: $f(x) = \frac{1}{x^4}$, $f(2) = \frac{1}{16}$, $f'(x) = -4x^{-5} = -\frac{4}{x^5}$, $f'(2) = -\frac{1}{8}$

Example: Tangent to $y = x^5$ at $x = 2$:

• $f'(x) = 5x^4$, $f'(2) = 80$

• Point: $(2, 32)$

• Equation: $y - 32 = 80(x - 2)$

Rules for Differentiation

Constant Multiple, Sum, and General Power Rules

These rules simplify the process of finding derivatives for more complex functions.

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

Example: $\frac{d}{dx}[90x^{-6}] = -540x^{-7} = -\frac{540}{x^7}$

Example: $y = 16\sqrt[4]{4x^4 + 4}$, $y' = 64x^3(4x^4 + 4)^{-\frac{3}{4}}$

Example: $\frac{d}{dx}[3x + (5x + 7)^4] = 3 + 20(5x + 7)^3$

Second Derivatives and Marginal Analysis

Second Derivatives, Rates of Change, Marginal Costs/Revenues/Profits

The second derivative measures the rate of change of the rate of change (concavity). Marginal analysis uses derivatives to estimate changes in cost, revenue, or profit for small changes in production.

• Second Derivative Notation: $f''(x)$, $y''$, $\frac{d^2y}{dx^2}$

• Estimating Change: $f(a + 1) \approx f(a) + f'(a)$, $f(a + h) \approx f(a) + f'(a) \cdot h$

• Marginal Cost: $C'(x)$, Marginal Revenue: $R'(x)$, Marginal Profit: $P'(x)$

Example: If $C(60) = 3000$ and $C'(60) = 25$, then $C(61) \approx 3025$

Example: $R(x) = 5x - 0.02x^2$, $R'(x) = 5 - 0.04x$, $R'(20) = 4.20$ per unit

Derivatives as Rates of Change

Velocity and Acceleration

In business and science, derivatives are used to model rates of change such as velocity and acceleration.

• Velocity: $v(t) = s'(t)$

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

Example: $s(t) = 176t - 16t^2$

• Initial velocity: $v(0) = 176$ feet/sec

• Acceleration: $a(10) = -32$ feet/sec$^2$

• Rocket hits ground at $t = 11$ sec

• Velocity at impact: $v(11) = -176$ feet/sec

Additional Example: Estimating drug concentration and sales using derivatives.

• $f(2.25) \approx 2.825$ mg

• $f(14.25) \approx 59,000$ computers

Curve Analysis: Increasing, Decreasing, Maximums, Minimums, Concavity

Analyzing Functions Using Derivatives

Derivatives help determine where functions increase, decrease, reach maximum or minimum values, and change concavity.

• Increasing: Where $f'(x) > 0$

• Decreasing: Where $f'(x) < 0$

• Relative Maximum: Where $f'(x)$ changes from positive to negative

• Relative Minimum: Where $f'(x)$ changes from negative to positive

• Concave Up: Where $f''(x) > 0$

• Concave Down: Where $f''(x) < 0$

• Inflection Point: Where $f''(x)$ changes sign

Example: For the function shown below:

• Increasing: $x < 1$, $x > 4$

• Decreasing: $1 < x < 4$

• Relative Maximum: $x = 1$

• Relative Minimum: $x = 4$

• Concave Up: $x > 2.5$

• Concave Down: $x < 2.5$

• Inflection Point: $x = 2.5$

First and Second Derivative Tests, Curve Sketching

Tests for Local Extreme Points and Inflection Points

The first derivative test and second derivative test are used to classify critical points and inflection points.

• First Derivative Test: If $f'(a) = 0$ and $f'$ changes sign at $a$, $f$ has a local max or min.

• Second Derivative Test: If $f'(a) = 0$ and $f''(a) < 0$, local max; if $f''(a) > 0$, local min.

• Inflection Point: If $f''(a) = 0$ and $f''$ changes sign at $a$.

5-Step Method for Curve Analysis:

1. Compute $f'(x)$ and $f''(x)$

2. Find critical points ($f'(x) = 0$)

3. Classify points using derivative tests

4. Find inflection points ($f''(x) = 0$)

5. Determine intercepts, asymptotes, and end behavior

Example: $f(x) = x^3 + 12x^2 + 36x + 1$

• $f'(x) = 3x^2 + 24x + 36 = 3(x + 6)(x + 2)$

• Critical Points: $x = -6$, $x = -2$

• $f''(x) = 6x + 24$

• $f''(-6) = -12$ (Local Max), $f''(-2) = 12$ (Local Min)

• Inflection Point: $x = -4$

• Values: $f(-6) = 1$, $f(-2) = -31$, $f(-4) = -15$

• Local Max: $(-6, 1)$, Local Min: $(-2, -31)$, Inflection Point: $(-4, -15)$