The First Fundamental Theorem of Calculus connects two big ideas: accumulation and rate of change. It says that when a function F(x) is built by adding up signed area under another function f(t), the derivative of F gives back the original function f. This matters because it explains why derivatives and integrals are inverse processes in a precise way.
It also gives a powerful method for understanding changing quantities from their accumulated totals.
Key Facts
- If F(x) = ∫_a^x f(t) dt, then F'(x) = f(x).
- The variable t is a dummy variable of integration, while x controls the moving upper limit.
- F(x) measures signed area from t = a to t = x under the curve y = f(t).
- If f(x) > 0, then F is increasing at x because F'(x) > 0.
- If f(x) < 0, then F is decreasing at x because F'(x) < 0.
- For G(x) = ∫_a^{g(x)} f(t) dt, the chain rule gives G'(x) = f(g(x))g'(x).
Vocabulary
- Accumulation function
- A function defined by an integral with a variable limit that measures the net amount accumulated from a starting point.
- Signed area
- Area counted as positive when the graph is above the axis and negative when the graph is below the axis.
- Dummy variable
- A placeholder variable inside an integral that can be renamed without changing the value of the integral.
- Upper limit of integration
- The endpoint of an integral that can move and determine how much area is included.
- First Fundamental Theorem of Calculus
- The theorem stating that the derivative of an accumulation function F(x) = ∫_a^x f(t) dt is the integrand evaluated at x.
Common Mistakes to Avoid
- Forgetting that ∫_a^x f(t) dt is a function of x. The upper limit moves, so the accumulated area changes as x changes.
- Treating t and x as the same variable. The variable t is only used inside the integral, while x sets the moving endpoint.
- Writing F'(x) = ∫_a^x f'(t) dt. The theorem says differentiating the accumulation function gives f(x), not the integral of the derivative.
- Ignoring negative area below the axis. Area below the t-axis decreases the accumulation function because the integral uses signed area.
Practice Questions
- 1 Let F(x) = ∫_2^x (3t^2 - 4t) dt. Find F'(x) and then evaluate F'(3).
- 2 Let G(x) = ∫_1^{x^2} sqrt(1 + t^3) dt. Use the First Fundamental Theorem and the chain rule to find G'(x).
- 3 Suppose F(x) = ∫_0^x f(t) dt and the graph of f is below the t-axis on the interval 1 < x < 4. Explain what happens to F on that interval and why.