Fundamental Theorem of Calculus

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The first part of the Fundamental Theorem of Calculus and the surprising effect of differentiation on integration.

The Fundamental Theorem of Calculus has far-reaching applications, making sense of reality from physics to finance.

The first part of the Fundamental Theorem is needed throughout the sciences. For example, physicists use it to define and explain the relationship between work and power.

Economists can relate marginal revenue and total revenue using the Fundamental Theorem of Calculus.

Statisticians need the first part of the Fundamental Theorem to explain the relationship between cumulative probability and probability density.

FUNDAMENTAL THEOREM OF CALCULUS

f (x) = 2 {(x - 1.5)}^{3} - 2 x + 7 0 \leq x \leq 3

f (x) = x \sin (x - \frac{π}{3}) + 5 - \frac{3 π}{2} \leq x \leq 2 π

f (x) = \{\begin{matrix} \frac{2}{5} x^{2} - 1 0 \leq x \leq 2.5 \\ \frac{3}{4} x - \frac{3}{8} 2.5 < x \leq 4 \\ 2 \sqrt{x - 3.9} + 3 4 < x \leq 6 \end{matrix}

F (t) = \int_{0}^{t} f (x) d x = \int_{0}^{t} (2 {(x - 1.5)}^{3} - 2 x + 7) d x \approx Approx value

F (t) = \int_{- \frac{3 π}{2}}^{t} f (x) d x = \int_{- \frac{3 π}{2}}^{t} (x \sin (x - \frac{π}{3}) + 5) d x \approx Approx value

F (t) = \int_{0}^{t} f (x) d x = \int_{0}^{t} (\frac{2}{5} x^{2} - 1) d x \approx Approx Value

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Drag the slider to adjust the value of t and answer the following questions.

Drag the slider to adjust the value of t and answer the following questions.

Drag the slider to adjust the value of t and answer the following questions.