Calculus I: Derivatives, Rates of Change, and Applications – Study Guide

Exam Structure and Academic Policies

Exam Logistics and Academic Honesty

This section outlines the expectations and rules for the upcoming Calculus exam, including accommodations, permitted tools, and academic integrity requirements.

• Accommodations: Students with CLASS office arrangements will have adjusted exam times as needed.

• Permitted Tools: Desmos Test Mode is required; students must use their own device and submit with the timer visible.

• Handwritten Solutions: All solutions must be written by hand, with full justification for each step.

• Academic Honesty: Strict adherence to the Academic Honesty Policy is required. Sharing exam content is prohibited.

Core Calculus Topics

1. Average Rate of Change

The average rate of change of a function over an interval measures how much the function's output changes per unit change in input.

• Definition: For a function f(x) over the interval [a, b], the average rate of change is:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

• Secant Line: The average rate of change corresponds to the slope of the secant line connecting (a, f(a)) and (b, f(b)) on the graph.

• Interpretation: Useful for estimating how a quantity changes over a specific interval.

• Example: If f(x) represents position at time x, the average rate of change is the average velocity over [a, b].

2. Definition of the Derivative

The derivative of a function at a point measures the instantaneous rate of change, or the slope of the tangent line at that point.

• Limit Definition:

$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Geometric Interpretation: The derivative at a point is the slope of the tangent line to the graph at that point.

• Equation of Tangent Line: At x = a, the tangent line is:

$y = f(a) + f'(a)(x - a)$

• Units: The units of the derivative are the units of f(x) per unit of x.

• Significance: A positive derivative indicates increasing function, negative indicates decreasing, and zero indicates a horizontal tangent (possible local max/min).

• Non-differentiability: Functions may not be differentiable at points with sharp corners, cusps, vertical tangents, or discontinuities.

3. Working with Derivatives

Derivatives can be estimated, computed, and interpreted in various ways, both graphically and numerically.

• Estimating from Graphs: The slope of the tangent line at a point on the graph gives the derivative.

• Numerical Estimation: Using a table of values, approximate the derivative by calculating average rates of change over small intervals.

• Graphing Derivatives: Sketch the derivative function based on the original function's graph (e.g., where the original is increasing, the derivative is positive).

• Shortcuts: Use differentiation rules (power, product, quotient, chain) for polynomials and other functions.

• Interpreting Signs: The sign of the derivative indicates the function's behavior (increasing/decreasing).

• Linear Approximation: Use the tangent line to estimate function values near a point.

• Applications: Derivatives are used to determine rates of change in context, such as velocity, growth rates, or marginal cost.

4. Applications: Supply and Demand, Exponential and Logarithmic Functions

Derivatives are applied in various real-world contexts, including economics and modeling with exponential/logarithmic functions.

• Supply and Demand: Systems of linear equations can model market equilibrium; derivatives can analyze marginal changes.

• Exponential/Logarithmic Functions: Used to model growth and decay; derivatives describe rates of change in these contexts.

5. Optimization (Closed Interval Method)

Optimization involves finding the maximum or minimum values of a function, often within a specified interval.

• Closed Interval Method: To find absolute extrema on [a, b]:

  1. Find critical points inside (a, b) where $f'(x) = 0$ or $f'(x)$ is undefined.

  2. Evaluate $f(x)$ at critical points and endpoints a, b.

  3. Compare values to determine the maximum and minimum.

• Applications: Used in economics, engineering, and science to optimize quantities such as cost, area, or profit.

Soft Skills for Calculus Exams

• Graphing: Use Desmos or hand sketching to plot functions, label axes, and highlight key points.

• Time Management: Practice pacing to ensure completion of all exam sections.

Summary Table: Key Concepts and Skills

Concept	Definition/Skill	Example/Application
Average Rate of Change	Change in function value per unit input over an interval	Average velocity over a time interval
Derivative (Limit Definition)	Instantaneous rate of change; slope of tangent line	Velocity at a specific instant
Graphical Interpretation	Estimating derivative from graph or table	Sketching tangent lines, identifying increasing/decreasing intervals
Optimization	Finding max/min values on a closed interval	Maximizing area, minimizing cost

Additional info:

• Some content was inferred and expanded for clarity and completeness, such as the explicit steps for the closed interval method and the summary table.

• References to "Desmos Test Mode" and "Supply and Demand" labs suggest integration of technology and real-world applications in the course.