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Differential Calculus Simulations

A collection of interactive 3D visualizations and simulations to help you master concepts in differential calculus.

Functions, Limits, and Continuity - Theory & Concepts - Limit Explorer

Master the foundational concepts of calculus: basic function properties, domain and range, limits, limit laws, the Squeeze Theorem, and continuity.

Limit Explorer

-13
x0.5000
f(x)1.5000
Target (L)2

Notice how f(x) gets closer to L as x approaches c, even if f(c) is undefined.

Functions, Limits, and Continuity - Theory & Concepts - Epsilon Delta

Master the foundational concepts of calculus: basic function properties, domain and range, limits, limit laws, the Squeeze Theorem, and continuity.

Epsilon-Delta Visualization

Adjust the output tolerance (ϵ\epsilon) to see the required input window (δ\delta). The function is f(x)=2x+1f(x) = 2x + 1 with a limit of L=5L = 5 at c=2c = 2. Notice that δ=ϵ/2\delta = \epsilon / 2.

Values:
Limit L=5L = 5
L+ϵL + \epsilon = 6.00
L−ϵL - \epsilon = 4.00
Input c=2c = 2
Required δ\delta = 0.50
c+δc + \delta = 2.50
c−δc - \delta = 1.50
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Functions, Limits, and Continuity - Theory & Concepts - Continuity Explorer

Master the foundational concepts of calculus: basic function properties, domain and range, limits, limit laws, the Squeeze Theorem, and continuity.

Continuity & Discontinuity Explorer

Analysis at x = 2

The function is continuous at x = 2. The limit as x approaches 2 exists and is equal to the function value f(2) = 4.

Formula
f(x)=x2f(x) = x^2
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The Derivative - Theory & Concepts - Secant Tangent

Master the core of differential calculus: the historical context, one-sided derivatives, differentiability, differentiation rules, and implicit differentiation.

Secant Line to Tangent Line

Adjust the distance h between the two points. As h → 0, the secant line (connecting two points) converges into the tangent line (touching one point), demonstrating the fundamental definition of the derivative. The curve is f(x) = x²/4 + 1.

Calculations:
Point 1: (x, f(x)) = (1, 1.25)
Point 2: (x+h, f(x+h)) = (3.00, 3.25)
Secant Slope: [f(x+h) - f(x)] / h = 1.000
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Derivatives of Transcendental Functions - Theory & Concepts - Transcendental Explorer

Expand your differentiation skills to trigonometric, exponential, logarithmic, and hyperbolic functions, including inverse hyperbolic derivatives.

Trigonometric Derivative

Observe the derivative graph and tangent slope changes dynamically.

ddx[sin⁡x]=cos⁡x\frac{d}{dx}[\sin x] = \cos x
-3.16.3
f(x) = sin(x)0.8415
f'(x) = cos(x)0.5403
Tangent Slope0.5403
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Derivatives of Transcendental Functions - Theory & Concepts - Exponential Growth

Expand your differentiation skills to trigonometric, exponential, logarithmic, and hyperbolic functions, including inverse hyperbolic derivatives.

Exponential Growth: P(t)=P0ertP(t) = P_0 e^{rt}

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Observation: The solid blue line represents the population P(t)P(t), and the dashed red line is its derivative P′(t)P'(t). Notice how P′(t)P'(t) is always a constant multiple of P(t)P(t).

Theorems of Calculus - Theory & Concepts

Understand the foundational theorems that bridge the gap between continuous functions and derivatives: the Extreme Value Theorem, Rolle's Theorem, the Mean Value Theorem, and Cauchy's MVT.

Mean Value Theorem

There exists at least one point cc in (a,b)(a, b) where the tangent line is parallel to the secant line.

f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}
03
Secant Slope2.0000
Tangent Slope-1.0000
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Theorems of Calculus - Theory & Concepts - L Hopital Rule

Understand the foundational theorems that bridge the gap between continuous functions and derivatives: the Extreme Value Theorem, Rolle's Theorem, the Mean Value Theorem, and Cauchy's MVT.

Case 1: sin(x) / x

Explore how both curves become linear under high zoom, showing that their ratio approaches the ratio of their tangent lines.

lim⁡x→0sin⁡xx[00]\lim_{x \to 0} \frac{\sin x}{x} \quad \left[\frac{0}{0}\right]
-1.5Target c = 01.5
1x (Wide View)50x (Linear Zoom)
Limit Approximations
f(x) = \sin x0.479426
g(x) = x0.500000
Ratio f(x)/g(x)0.958851
Slopes f'(x)/g'(x)0.877583
Blue: f(x)Orange: g(x)Vertical line: Current x position
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As you zoom to 50x, observe how both curves look perfectly straight (tangent line approximation) and intersect exactly at y=0y = 0 at x=0x = 0.

Theorems of Calculus - Theory & Concepts - Newtons Method

Understand the foundational theorems that bridge the gap between continuous functions and derivatives: the Extreme Value Theorem, Rolle's Theorem, the Mean Value Theorem, and Cauchy's MVT.

Newton's Method Interactive Visualization

Finding the positive root of f(x) = x² - 4 (The root is at x=2).

Iteration Progress

    3.0000
    Current Estimate

    Applications of the Derivative - Theory & Concepts - Ladder

    Apply differentiation to solve problems in motion, optimization, geometric analysis, and marginal cost.

    Related Rates: Sliding Ladder

    Length (L)10 m
    x6.00 m
    y8.00 m

    Assume dx/dt = 2 m/s

    dy/dt-1.50 m/s

    Notice how dy/dt (speed of top of ladder) increases dramatically as x approaches L (bottom pulls away).

    FloorWallL = 10xy

    Applications of the Derivative - Theory & Concepts - Cone Related Rates

    Apply differentiation to solve problems in motion, optimization, geometric analysis, and marginal cost.

    Hydraulic Related Rates: Conical Reservoir

    A standard civil engineering application: analyze how depth hh changes over time as water flows into a conical tank at a constant rate dV/dtdV/dt.

    0.1m10m (Full)
    Governing Equation
    dhdt=dV/dtπr2\frac{dh}{dt} = \frac{dV/dt}{\pi r^2}
    Current Radius $r$:
    2.00 m
    Water Volume $V$:
    16.8 m³
    Rate dh/dt:0.1194 m/min
    Reservoir Geometry Cross-Section
    R = 5mH = 10mr = 2.00mh = 4.0mQin = 1.5 m³/min
    Depth vs. Depth Rate (dh/dt)
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    Key Insight: Observe how the green curve decreases exponentially. As the depth hh increases, the cross-sectional area expands, meaning the rate of vertical rise dh/dtdh/dt slows down drastically for the same constant inflow!

    Applications of the Derivative - Theory & Concepts - Canal Optimization

    Apply differentiation to solve problems in motion, optimization, geometric analysis, and marginal cost.

    Canal Optimization

    Adjust the depth yy of the trapezoidal canal. The total area is held constant at 50 m². Notice how the wetted perimeter PP changes, reaching a minimum when the cross-section approaches a half-hexagon.

    Cross-Section Properties:
    Area A: 50.0 m² (Constant)
    Depth y: 5.37 m
    Bottom Width b: 6.21 m
    Wetted Perimeter P: 18.61 m
    Optimal Design (Minimum P)
    b = 6.2my = 5.4m
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    Differentials and Approximations - Theory & Concepts - Linear Approximation

    Learn how to use differentials to approximate function values and calculate errors, and discover Newton's Method and Taylor/Maclaurin Polynomials.

    Linear Approximation: f(x)=xf(x) = \sqrt{x}

    Target xx26.0
    Actual f(x)f(x)5.0990
    Approx L(x)L(x)5.1000
    Error0.0010
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    Observation: The green dot is the base point aa. The closer Δx\Delta x is to 00, the closer the linear approximation L(x)L(x) (red line) is to the actual function f(x)f(x) (blue line).

    Partial Differentiation - Theory & Concepts - Partial Deriv

    Expand calculus to multivariable functions, partial derivatives, the gradient vector, the second partials test, and Lagrange Multipliers.

    Partial Derivatives

    Visualizing Surface: z=0.5(x2−y2)z = 0.5(x^2 - y^2)
    Red Arrow: Slope along x-axis (∂z/∂x\partial z / \partial x)
    Green Arrow: Slope along y-axis (∂z/∂y\partial z / \partial y)

    0.50
    0.50
    z value:0.00
    Slope X (∂z/∂x\partial z/\partial x):0.50
    Slope Y (∂z/∂y\partial z/\partial y):-0.50
    Drag to Rotate | Scroll to Zoom

    Partial Differentiation - Theory & Concepts - Gradient Vector

    Expand calculus to multivariable functions, partial derivatives, the gradient vector, the second partials test, and Lagrange Multipliers.

    Gradient Vector Visualization

    Surface: f(x, y) = x² + y²

    Current State

    Position (x, y)(0.00, 0.00)
    ∇f = <2x, 2y><0.00, 0.00>
    Magnitude |∇f|0.00

    Move your mouse over the grid. Notice how the red gradient vector always points directly outward from the origin, perpendicular to the circular level curves. This shows the direction of steepest ascent on the paraboloid surface.

    Radius of Curvature - Theory & Concepts - Superelevation Curvature

    Understand the curvature of a function, the radius of curvature, parametric/polar forms, and the osculating circle, essential concepts for highway engineering and structural analysis.

    Superelevation Design: Curvature Applications

    In civil engineering highway alignment, the radius of curvature RR directly governs the road banking angle ee (superelevation) to ensure vehicles navigate safely at speed vv.

    50m (Sharp)500m (Gentle)
    30 km/h120 km/h
    0% (Flat)10% (Banked)
    Design Governing Equation
    e+f=v2gR=v2κge + f = \frac{v^2}{g R} = \frac{v^2 \kappa}{g}
    Curvature $\kappa$:0.00667 m⁻¹
    Centrifugal Force:1.85 m/s²
    Required Pavement Friction $f$:0.129
    ✅ SAFE: Curve coordinates satisfy all AASHTO design guidelines!
    Rear-View Pavement Force Diagram
    NFfFg = mgFc = mv²/RRoad Angle θ = 3.4° (e = 0.06)
    Curvature (κ × 10³) vs. Comfortable Superelevation (e %)
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    Calculus Application: Curvature κ=lim⁡Δs→0ΔθΔs\kappa = \lim_{\Delta s \to 0} \frac{\Delta \theta}{\Delta s} changes dynamically along highway easement spiral curves. Superelevation is gradually increased in direct linear proportion to curvature to maintain a safe, slip-free ride!

    Radius of Curvature - Theory & Concepts

    Understand the curvature of a function, the radius of curvature, parametric/polar forms, and the osculating circle, essential concepts for highway engineering and structural analysis.

    Osculating Circle Simulation

    Move the slider to see how the radius of curvature and the osculating circle change along the parabola y=x2y = x^2.

    Point (x, y):(0.50, 0.25)
    First Derivative (y'):1.00
    Radius of Curvature (R):1.41

    Derivatives of Parametric and Polar Curves - Theory & Concepts - Calculus Projectile Parametric

    Learn how to find derivatives, slopes, and tangency angles for curves defined parametrically and in polar coordinates.

    Parametric Trajectory: Projectile Motion

    Explore how parametric derivatives govern vertical and horizontal velocity rates, constructing the instantaneous tangent slope vector dy/dxdy/dx.

    0.00sPeak: 2.16sImpact: 4.32s
    Parametric Derivatives
    x(t) position:45.82 m
    y(t) position:22.94 m
    dx/dt (Horizontal rate):21.21 m/s
    dy/dt (Vertical rate):0.02 m/s
    Tangent Slope dy/dx:0.0011
    Concavity d²y/dx²:-0.02180 m⁻¹
    Dynamic Motion Path & Tangent Vector
    Peak h=22.9mRange=91.7mvxvyVTangent dy/dx = 0.001
    Observation: Horizontal velocity vx=dx/dtv_x = dx/dt is constant throughout flight (ignoring air drag). Vertical velocity vy=dy/dtv_y = dy/dt decreases linearly from positive to negative due to gravity. The combined velocity vector is always exactly tangent to the curve, represented by the parametric derivative dy/dxdy/dx!

    Derivatives of Parametric and Polar Curves - Theory & Concepts - Parametric Polar

    Learn how to find derivatives, slopes, and tangency angles for curves defined parametrically and in polar coordinates.

    Parametric & Polar Tangent Explorer

    The tangent line slope is given by:

    dydx=dy/dtdx/dt\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

    For polar, substitute x=r cos(θ) and y=r sin(θ) first.