Fundamental Theorem of Calculus
Connect signed accumulated area A(x)=integral from a to x of (t^2-c) dt with A'(x)=x^2-c. Use the exact cubic antiderivative.
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The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration. It shows that accumulating area under a curve f(t) creates a new function A(x), whose derivative is the original function: A'(x) = f(x).
Worked Example
Let the integrand be f(t) = t2 - c. The exact antiderivative is F(t) = t3/3 - ct. We define the accumulated area from a to x as:
Suppose a = 0, x = 1, and c = 0. We find:
- F(0) = 0
- F(1) = 1/3 - 0 = 1/3
- The area A(1) = 1/3 - 0 = 1/3.
The exact derivative of the accumulation function at x is simply the value of the integrand evaluated at x:
A numerical approximation of the derivative can be obtained using the central secant formula:
[A(x+h) - A(x-h)] / (2h).
For a quadratic integrand, this expands exactly to x2 - c + h2/3. With h = 0.5, the secant is:
Interactive Experiment
Use the Fundamental Theorem of Calculus module to visually confirm this relationship:
- Set Lower bound a to 0 to fix the start of our accumulation.
- Set Offset c to 0 so the integrand is f(t) = t2.
- Set Difference step h to 0.5 to see the secant approximation.
- Set Upper position x to 1. Observe the shaded area representing 1/3.
- Notice that the Exact Derivative is exactly 1, while the Central Secant gives 13/12 (≈1.083333).
- Change Offset c to 1 and Lower bound a to -1. If you set Difference step h to 0.25, the exact area becomes -4/3 and the exact derivative becomes 0, with the central secant reflecting a small error of 1/48.
- Set Lower bound a and Upper position x both to 1 with Offset c = 0. Signed accumulation is 0 but Exact derivative remains 1.
- Set Lower bound a = 0, Upper position x = 1 and Offset c = 2. Read signed accumulation −5/3 and derivative −1; the graphs retain the signs.
- Clear Upper position x to see the error replace the dependent results. Enter 1 again to recover the calculation. Drag a plot and choose Reset view; all inputs stay unchanged.
Limitations
The visualization uses a simple quadratic polynomial f(t) = t2 - c to clearly demonstrate the relationship without overwhelming analytical complexity. While the polynomial is defined for all real numbers (including out-of-bounds evaluated sampled points at x ± h), the module rejects text inputs outside the interval [-2, 2] to keep the graphical plots bounded and readable on screen. The finite secant approximation exhibits a non-zero error proportional to h2/3 because the function is not linear; this explicitly demonstrates that the secant is an approximation, unlike the exact analytical derivative.
References
OpenStax Calculus Volume 1, The Fundamental Theorem of Calculus explains the connection between definite integrals and antiderivatives, underpinning the calculations shown in this module.
Experience it interactively
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