Behaviour of Accumulation Functions, Area, Graphical, Numerical, Analytical - Calculus

Accumulation functions represent the running total, or the cumulative area, under the graph of another function, often over a fixed interval starting at a constant point 'a' and extending to a variable point 'x'. For a given function f(t), the accumulation function F(x) = ∫[from a to x] f(t) dt measures this total accumulated quantity, such as distance traveled or total rainfall. The First Fundamental Theorem of Calculus shows that the accumulation function F(x) is an antiderivative of f(x), meaning its rate of change, F'(x), is equal to the original function f(x).

💡Key Characteristics
• Definition: An accumulation function F(x) is defined as a definite integral where the upper limit is a variable, such as F(x) = ∫[from a to x] f(t) dt.
• Meaning: It provides a cumulative measure of the quantity represented by f(t) over the interval from 'a' to 'x'.
• Relationship to the Original Function: According to the First Fundamental Theorem of Calculus, the derivative of an accumulation function F(x) is the original function f(x).
• Behavior:
◦ F(x) increases when f(x) is positive.
◦ F(x) decreases when f(x) is negative.
◦ F(x) has a maximum or minimum value when f(x) = 0, representing points where the area accumulation changes direction.
• Applications: Accumulation functions are used to model quantities that change over time or intervals, such as:
◦ Total distance traveled by a car.
◦ Total amount of water in a tank after a certain time.
◦ Total rainfall over a given period.

💡Example
Consider the function f(t) = 2t representing the velocity of an object at time t. The accumulation function for distance, D(x), would be:
D(x) = ∫[from 0 to x] 2t dt This function D(x) gives the total distance the object has traveled from time 0 to time x, and D'(x) = 2x, which is the original velocity function.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link
• Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link

💡Chapters:
00:00 Fundamental theorem of calculus, part 1, visual proof

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