Math Problem Statement

Une primitive pour calculer une intégrale. On considère la fonction racine carrée définie sur [1;4], noteé f(x) = √x, et on veut démontrer une propriété liée à la primitive de cette fonction.

Solution

The image contains a mathematical problem related to integral calculus, specifically focusing on using a primitive function to compute an integral. The problem involves the square root function and visualizations using rectangles for approximations.

Would you like a step-by-step explanation of the solution? Here are five related questions to deepen your understanding:

  1. How do we define a primitive function for a given function?
  2. Why is the integral of a function related to the area under its curve?
  3. How can we approximate an integral using rectangles?
  4. What is the significance of the Fundamental Theorem of Calculus in this problem?
  5. How can we justify that F(x)=23x3/2F(x) = \frac{2}{3} x^{3/2} is a primitive of f(x)=xf(x) = \sqrt{x}?

Tip: A primitive function F(x)F(x) of f(x)f(x) satisfies F′(x)=f(x)F'(x) = f(x), meaning its derivative gives back the original function.

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Primitive Functions
Square Root Function
Area under a Curve

Formulas

F(x) = ∫ f(x) dx
F'(x) = f(x)
F(x) = (2/3) x^(3/2)

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12

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