Interactive Calculus Visualizer

Derivatives & Differentiation

Select Function

Enter Custom Function (use x as variable)

Supported: +, -, *, /, ^, sin, cos, tan, exp, log, sqrt, abs

X-Axis Range: [-5, 5]

Y-Axis Range: [-10, 10]

Point of Interest (x-value)

x = 1.0

Limit h-value (for difference quotient)

h = 0.50

Tangent Point (x-value)

x = 1.0

Show Secant Line Approximation

Tangent Line Equation

The equation of the tangent line at point (x₀, f(x₀)) is:

y - f(x₀) = f'(x₀)(x - x₀)

Where f'(x₀) is the derivative at x₀.

Understanding Derivatives

The derivative f'(x) represents the instantaneous rate of change of f(x) at point x. Geometrically, it's the slope of the tangent line to the curve at that point.

\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]

Integrals & Area Under Curves

Function to Integrate

Lower Bound (a)

a = -2.0

Upper Bound (b)

b = 2.0

Number of Rectangles (n)

Riemann Sum Type

Riemann Sum

A Riemann sum approximates the definite integral by dividing the area into rectangles:

\[\int_a^b f(x)dx \approx \sum_{i=1}^n f(x_i^*) \Delta x\]

where \(\Delta x = \frac{b-a}{n}\) and \(x_i^*\) is a sample point in the i-th subinterval.

The Fundamental Theorem of Calculus connects differentiation and integration:

\[\int_a^b f(x)dx = F(b) - F(a)\]

where F is an antiderivative of f (i.e., F' = f).

Variable Upper Limit

F(x) = ∫₀ˣ f(t) dt, where x = 0.0

FTC Part 1

If \(F(x) = \int_a^x f(t)dt\), then \(F'(x) = f(x)\).

FTC Part 2

If F is an antiderivative of f, then \(\int_a^b f(x)dx = F(b) - F(a)\).

Understanding Integrals

The definite integral \(\int_a^b f(x)dx\) represents the signed area between the curve f(x) and the x-axis from x=a to x=b.

\[\int_{-2}^{2} x^2 dx = \frac{16}{3} \approx 5.333\]

Calculus Concepts Explained

Calculus is the mathematical study of continuous change, much like geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

Differential Calculus

Concerned with the concept of a derivative, which represents the rate of change of a function. The derivative describes how a function changes as its input changes.

Slope of a curve at a point
Instantaneous rate of change
Velocity as derivative of position
Optimization problems

Integral Calculus

Concerned with the concept of an integral, which represents the accumulation of quantities. The integral gives the area under a curve or the total accumulated value.

Area under a curve
Accumulation of quantities
Distance as integral of velocity
Volumes of solids of revolution

The two branches are related by the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations.

\[\frac{d}{dx} \int_a^x f(t)dt = f(x) \quad \text{and} \quad \int_a^b F'(x)dx = F(b) - F(a)\]

Use the interactive graphs above to explore these concepts. Try changing functions, adjusting parameters, and visualizing how derivatives relate to slopes and integrals relate to areas.