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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

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+ϵ=(L+epsilon).toFixed(2)L + \epsilon = {(L + epsilon).toFixed(2)}
L−ϵ=(L−epsilon).toFixed(2)L - \epsilon = {(L - epsilon).toFixed(2)}
Input c=2c = 2
Required δ=delta.toFixed(2)\delta = {delta.toFixed(2)}
c+δ=(c+delta).toFixed(2)c + \delta = {(c + delta).toFixed(2)}
c−δ=(c−delta).toFixed(2)c - \delta = {(c - delta).toFixed(2)}
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The Derivative - Theory & Concepts

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

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 Visualization

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c = 0.0c = 3.0
Secant Line (Average Rate)
a=0,b=3a = 0, b = 3f(a)=2.0,f(b)=8.0f(a) = 2.0, f(b) = 8.0msec=f(b)−f(a)b−a=2.00m_{sec} = \frac{f(b)-f(a)}{b-a} = 2.00
Tangent Line (Inst. Rate)
c=1.00c = 1.00f(c)=2.00f(c) = 2.00f′(c)=−1.00f'(c) = -1.00

Applications of the Derivative - Theory & Concepts

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

Differentials and Approximations - Theory & Concepts

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

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

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

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.