MicroSims Gallery

Interactive educational simulations for exploring calculus concepts. Each MicroSim focuses on a single learning objective and provides immediate visual feedback.

• Accumulation Function Explorer

  

  Visualize how the accumulation function F(x) = integral from a to x of f(t) dt grows as x moves across the interval.

• Additivity Property Visualization

  

  Demonstrates how the area under a curve can be split into sub-regions that sum to the total, illustrating the additivity property of definite integrals.

• Family of Antiderivatives

  

  Interactive visualization showing how different values of the constant C produce a family of parallel antiderivative curves. All antiderivatives of a function differ only by a constant.

• Asymptote Analyzer

  

  Explore vertical and horizontal asymptotes of rational functions.

• Asymptote Behavior Visualization

  

  Interactive visualization showing how rational functions behave near vertical asymptotes and as x approaches infinity, featuring Delta the robot traveling along curves.

• All Three Asymptote Types

  

  Interactive comparison of vertical, horizontal, and oblique asymptotes showing their defining characteristics and behavior.

• Average Rate of Change Explorer

  

  An interactive MicroSim that allows students to explore average rate of change by dragging two points along a curve and observing the secant line and slope calculation update in real-time.

• Average Value of a Function

  

  Interactive visualization showing the average value of a function as the height of a rectangle with equal area to the region under the curve.

• Balloon Inflation Simulator

  

  An interactive MicroSim that helps students compare how the rate of radius change varies with different radii and volume flow rates, discovering the inverse relationship between dr/dt and r squared.

• Box Volume Optimizer

  

  Interactive MicroSim demonstrating how cutting squares from cardboard corners affects the volume of the resulting box. Students examine the optimization problem V(x) = x(L-2x)(W-2x).

• Can Optimizer

  

  An interactive MicroSim demonstrating optimization by exploring how cylinder radius affects surface area for a fixed volume, helping students calculate optimal can dimensions.

• Candidates Test Calculator

  

  An interactive step-by-step guide through the closed interval method for finding global extrema on closed intervals.

• Chain Rule Steps

  

  Interactive step-by-step guide to applying the chain rule for differentiating composite functions with color-coded visualization of inside and outside functions.

• Changing Bounds in u-Substitution

  

  Interactive visualization showing how integration bounds transform from x-domain to u-domain during u-substitution, with side-by-side graphs confirming area equality.

• Closed Interval Method

  

  Interactive MicroSim demonstrating the Closed Interval Method for finding absolute extrema on closed intervals through a step-by-step guided process.

• Closest Point Finder

  

  An interactive MicroSim that finds the minimum distance from a draggable target point to various curves, demonstrating that at the closest point, the connecting line is perpendicular to the tangent.

• Complete Curve Sketch

  

  Interactive MicroSim demonstrating the complete curve sketching process, building graphs step-by-step with intercepts, asymptotes, extrema, and concavity.

• Composite Function Identifier

  

  An interactive MicroSim for training students to recognize composite functions and decompose them into inside (g) and outside (f) functions, preparing them for the chain rule.

• Concavity Explorer

  

  Interactive MicroSim exploring concave up and concave down regions by dragging a point along a curve and observing how the tangent line slope changes in relation to the second derivative.

• Conical Tank Draining

  

  Interactive visualization of related rates for water draining from a conical tank, showing similar triangles relationship between radius and height.

• Three Conditions for Continuity

  

  An interactive MicroSim that helps students explain how each of the three continuity conditions corresponds to visual features of a function graph.

• Coordinate System Explorer

  

  Interactive exploration of the Cartesian coordinate plane with point placement and quadrant identification.

• Critical Point Finder

  

  Interactive MicroSim to help students identify critical points by finding where f'(x) = 0 or f'(x) does not exist, with step-by-step solutions.

• Curve Analysis Dashboard

  

  Complete interactive visualization showing how to fully analyze a function using derivatives, including critical points, inflection points, concavity, and increasing/decreasing intervals.

• Derivative from Graph

  

  Practice estimating derivatives by drawing tangent lines on curves and comparing your estimated slope to the actual derivative value.

• Derivative Information Flow

  

  Interactive diagram showing how information about function behavior flows from the second derivative to the first derivative to the original function.

• Derivative Interpretation Selector

  

  Interactive MicroSim showing the derivative as both slope (graphical) and rate of change (contextual).

• Derivative Test Comparison

  

  An interactive comparison tool showing the First and Second Derivative Tests side-by-side for classifying critical points.

• First vs Second Derivative Test

  

  An interactive MicroSim comparing the First Derivative Test and Second Derivative Test methods for classifying critical points of functions.

• Differentiability vs Continuity

  

  Interactive Venn diagram showing the set relationship between differentiable and continuous functions.

• Differentiability Checker

  

  Step-through tool for checking differentiability at a point.

• Direct Substitution Decision Tree

  

  Interactive flowchart guiding students through the decision process for evaluating limits.

• Direction of Motion Analyzer

  

  Interactive MicroSim analyzing when an object moves in positive vs negative direction based on the sign of velocity.

• Discontinuity Classification

  

  An interactive MicroSim for classifying discontinuities by analyzing limit behavior at specific points.

• Domain and Range Visualizer

  

  An interactive MicroSim that helps students interpret domain and range graphically by showing function graphs with highlighted number lines.

• End Behavior Explorer

  

  Interactive MicroSim for examining how leading term characteristics determine the end behavior of polynomial and rational functions.

• Even and Odd Function Integral Symmetry

  

  Interactive visualization showing how symmetry properties of even and odd functions affect definite integrals over symmetric intervals.

• EVT Conditions Explorer

  

  Interactive visualization showing why the Extreme Value Theorem requires both continuity and a closed interval, with counterexamples.

• Exponential and Logarithm Relationship

  

  Compare exponential and logarithmic functions as inverses reflected across y = x.

• e^x is Its Own Derivative

  

  Interactive MicroSim demonstrating that the exponential function e^x is unique because its derivative equals the function itself.

• Exponential Integrals

  

  Interactive visualization demonstrating how the integral of a^x depends on the base a and why dividing by ln(a) is necessary.

• Global vs Local Extrema Visualizer

  

  Interactive MicroSim helping students distinguish between global and local maxima/minima on function graphs.

• Factoring Technique for Limits

  

  Visualize how factoring and canceling common factors reveals limit values.

• Fencing Problem Optimizer

  

  An interactive MicroSim for visualizing and solving the classic fencing optimization problem.

• FTC Connection Visualization

  

  Visualize how FTC Part 1 and Part 2 are two sides of the same relationship between differentiation and integration.

• FTC Part 1 Calculator

  

  Practice applying the Fundamental Theorem of Calculus Part 1 to find derivatives of accumulation functions.

• FTC Part 2 Step-by-Step Evaluator

  

  Walk through the FTC Part 2 evaluation process step by step to evaluate definite integrals.

• Function Composition Visualizer

  

  Interactive visualization showing how composite functions work by tracing values through two function machines in sequence.

• Function and Derivative Comparison

  

  Interactive side-by-side visualization showing a function and its derivative with synchronized point and tangent line displays.

• Function Machine

  

  An interactive MicroSim demonstrating how functions process inputs to produce exactly one output.

• Learning Graph Viewer

  

  Interactive visualization of the course concept dependency learning graph.

• Graphical Inverse Derivatives

  

  Interactive visualization showing the relationship between a function and its inverse, with tangent lines demonstrating the reciprocal slope property.

• Growth Rate Comparison

  

  Compare growth rates of logarithmic, polynomial, and exponential functions to understand which dominates as x approaches infinity.

• Implicit Curves Explorer

  

  Interactive visualization of implicit curves with tangent lines.

• Implicit Differentiation Steps

  

  Interactive step-by-step visualization of the implicit differentiation process with color-coded terms.

• Implicit Tangent Line Explorer

  

  Interactive exploration of tangent lines to curves defined by implicit equations, including circles, ellipses, hyperbolas, and the Folium of Descartes.

• Integration Strategy Decision Flowchart

  

  An interactive decision tree that guides students through choosing the right integration technique.

• Integration Verifier

  

  Interactive tool to verify integration results by differentiating the proposed antiderivative and comparing to the original integrand.

• Derivative of Inverse Functions

  

  Interactive visualization showing geometrically why the derivative of an inverse function equals the reciprocal of the original derivative.

• Inverse Function Reflector

  

  Compare functions and their inverses graphically, seeing the reflection relationship across y = x.

• Inverse Trig Derivatives Quiz

  

  Interactive flashcard-style quiz to help students memorize and apply the six inverse trigonometric derivative formulas.

• Inverse Trig Domains

  

  Visualize why domain restrictions are necessary for inverse trigonometric functions using the horizontal line test.

• Inverse Trig Integral Pattern Matcher

  

  Match integrands to their corresponding inverse trigonometric antiderivative types through interactive pattern recognition.

• Intermediate Value Theorem Visualization

  

  Interactive visualization demonstrating the IVT with Delta robot traveling along continuous functions.

• Ladder Problem Explorer

  

  Interactive visualization of the classic related rates ladder problem.

• Left vs Right Riemann Sums

  

  Interactive comparison of left and right Riemann sum approximations showing when each method overestimates or underestimates.

• L'Hospital's Rule Visualizer

  

  Interactive visualization showing how L'Hospital's Rule transforms indeterminate limits.

• Limit Laws Practice

  

  Interactive practice problems for applying limit laws to evaluate limits algebraically.

• Limit Visualization with Hole

  

  Interactive visualization showing how function values approach a limit even when the function has a hole.

• Limits from Graphs Practice

  

  Practice reading left-hand, right-hand, and two-sided limits from graphical representations.

• Linear Approximation Error

  

  Assess the accuracy of linear approximations by comparing the approximation to the actual function value.

• ln(x) as Area Under 1/t

  

  Interactive visualization showing the natural logarithm as the accumulated area under the curve y = 1/t.

• Local Linearity

  

  Interactive MicroSim demonstrating how curves appear linear when zoomed in sufficiently.

• Logarithmic Differentiation

  

  Step-by-step guide through the logarithmic differentiation process with color-coded transformations.

• Marginal Analysis Calculator

  

  Interactive visualization for calculating and interpreting marginal cost, revenue, and profit.

• Marginal Cost Analyzer

  

  Interactive MicroSim that calculates and interprets marginal cost as the derivative of the cost function.

• Motion Analysis

  

  Interactive visualization showing the relationship between position, velocity, and acceleration.

• Multi-Rule Derivative Builder

  

  An interactive step-by-step tool showing how multiple derivative rules combine to differentiate complex functions.

• Mean Value Theorem Explorer

  

  Interactive visualization of the Mean Value Theorem showing secant lines, tangent lines, and the values of c.

• Nested Chain Rule Unwrap

  

  An interactive visualization of nested function differentiation using the chain rule.

• Net Signed Area Visualizer

  

  Demonstrates that definite integrals compute net signed area, with regions below the x-axis contributing negatively.

• Non-Differentiable Points Gallery

  

  Interactive gallery showing the three types of non-differentiable points with animated secant lines.

• One-Sided Derivatives

  

  Interactive visualization showing how left and right secant lines approach a point.

• One-Sided Limits Comparison

  

  Compare left-hand and right-hand limits to determine whether a two-sided limit exists.

• Optimization Problem Setup Flowchart

  

  An interactive flowchart showing the step-by-step process for setting up optimization problems.

• Piecewise Function Builder

  

  Construct piecewise functions by defining different rules for different domains.

• Piecewise Continuity Explorer

  

  Interactive MicroSim for determining whether piecewise functions are continuous at boundary points.

• Polynomial Explorer

  

  Examine how polynomial degree and leading coefficient affect graph shape and end behavior.

• Polynomial Integration

  

  Interactive step-by-step guide to integrating polynomial functions using the power rule.

• Power Rule Explorer

  

  Interactive visualization of the power rule showing f(x) = x^n and its derivative f'(x) = nx^(n-1).

• Power Rule Integration

  

  Interactive visualization showing the relationship between derivative and integral power rules.

• Product Rule Visualization

  

  Interactive geometric visualization of the product rule showing why d(fg) = fdg + gdf.

• Profit Optimizer

  

  Interactive MicroSim showing the relationship between revenue, cost, and profit functions.

• Rate Interpretation Dashboard

  

  Interactive dashboard for interpreting derivative values in context across multiple real-world scenarios.

• Four Riemann Sum Methods Comparison

  

  Compare left, right, midpoint, and trapezoidal Riemann sum methods.

• Rolle's Theorem Visualizer

  

  Interactive visualization of Rolle's Theorem showing guaranteed existence of horizontal tangent lines.

• Differentiation Rule Selector

  

  An interactive decision tree that helps students choose the appropriate differentiation rule.

• Secant Lines Approaching Tangent Line

  

  Interactive visualization showing how secant lines approach the tangent line as h approaches 0.

• Second Derivative Explorer

  

  Interactive visualization showing f(x), f'(x), and f''(x) in three synchronized panels.

• Sigma Notation Visualizer

  

  Interactive MicroSim that helps students understand sigma notation by expanding sums and calculating totals step by step.

• Sign Chart Builder

  

  Construct sign charts for derivatives to determine intervals of increase and decrease.

• Sin(x)/x Visualization

  

  Understand the fundamental trigonometric limit using unit circle and graph visualizations.

• Sketch from Derivative Info

  

  Students construct function graphs given only derivative information.

• Speed Change Analyzer

  

  Interactive MicroSim showing when an object speeds up vs slows down based on velocity and acceleration signs.

• Squeeze Theorem Visualization

  

  Interactive demonstration of how the Squeeze Theorem pins down limit values.

• Tangent Line Calculator

  

  Given a function and a point, shows step-by-step calculation of the tangent line equation.

• Term by Term Differentiation

  

  An interactive MicroSim showing how polynomial differentiation works term by term.

• Three Connected Graphs Explorer

  

  Interactive visualization showing f(x), f'(x), and f''(x) simultaneously with synchronized cursor.

• Timeline of Calculus History

  

  An interactive timeline showing 2300 years of calculus development.

• Transcendental Integral Practice

  

  Interactive practice with transcendental integrals including trig, exponential, logarithmic, and inverse trig functions.

• Transformation Playground

  

  Manipulate a, b, h, k parameters to transform parent functions and see the effects in real-time.

• Trig Derivative Cycle

  

  Interactive visualization showing the cyclic pattern of trigonometric derivatives.

• Trig Derivatives Flashcards

  

  Interactive flashcard-style reference for memorizing derivatives of the six trigonometric functions.

• Trig Graph Transformer

  

  Manipulate amplitude, period, phase shift, and vertical shift for trigonometric functions.

• Trig Identity Visualizer

  

  See the Pythagorean identity sin^2 + cos^2 = 1 geometrically using the unit circle.

• Trig Integral Reference

  

  Interactive reference showing all six basic trig integrals with visual connections to their derivatives.

• Two Moving Objects Distance

  

  Interactive visualization examining how the rate of distance change between two moving objects depends on their positions and velocities.

• u-Substitution Steps

  

  Interactive step-by-step guide to applying u-substitution for evaluating integrals.

• Interactive Unit Circle

  

  Explore the relationship between angles and sine/cosine values with connected graph visualizations.

• Verification Methods Comparison

  

  An interactive MicroSim that helps students judge which verification method is most appropriate for different optimization scenarios.

• Vertical Asymptote Explorer

  

  Interactive visualization showing how one-sided limits determine function behavior near vertical asymptotes.

• Vertical Line Test

  

  Interactive tool for testing whether graphs represent functions using the vertical line test.

• Zoom to Linear

  

  An interactive MicroSim demonstrating that sufficiently zoomed-in views of differentiable functions appear linear.

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About MicroSims

MicroSims are lightweight, interactive educational simulations designed for browser-based learning. Each MicroSim:

• Focuses on a single learning objective
• Provides immediate visual feedback
• Works on any device with a modern browser
• Can be embedded in any webpage via iframe

All MicroSims in this course use open-source JavaScript libraries (primarily p5.js) so students can view the source code and learn from the implementations.

Total MicroSims: 123