Fundamental Theorem of Calculus, Part 1
\( \displaystyle\frac{d}{dx}\int_a^x f(t)\,dt = f(x) \)
\( x \)
1.50
\( h \)
1.00
▶ h → 0
Reset
\( f(x) = \sin(x) + 1.5 \)
\( \int_x^{x+h} f(t)\,dt \)
Rectangle \( h \cdot f(x+h) \)
Comparison 1: Rectangle vs Integral (Area)
Quantity
Value
\( h \cdot f(x+h) \)
—
\( \displaystyle\int_x^{x+h} f(t)\,dt \)
—
Difference:
—
Comparison 2: Derivative Convergence
Quantity
Value
\( \displaystyle\frac{1}{h}\int_x^{x+h} f(t)\,dt \)
—
\( f(x+h) \)
—
Difference
—
↓ h → 0
\( f(x) \)
—
Logic Flow
\[ \frac{d}{dx}\int_a^x f(t)\,dt = \lim_{h \to 0} \frac{1}{h}\int_x^{x+h} f(t)\,dt = \lim_{h \to 0} \frac{h \cdot f(x+h)}{h} = \lim_{h \to 0} f(x+h) = f(x) \]