Spiral Review (March 10 - 14)

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What is a recursive sequence?

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A recursive sequence is a sequence of numbers where each term is defined based on the previous term(s). For example, in the sequence defined by a₁ = -2 and a_n+1 = a_n + 5, each term is generated by adding 5 to the previous term.

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What is an explicit formula for a sequence?

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An explicit formula for a sequence allows you to calculate the nth term directly without needing to know the previous terms. For example, the explicit formula for the sequence defined recursively by a₁ = -2 and a_n+1 = a_n + 5 is a_n = 5n - 7.

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How do you determine if a function is increasing or decreasing on an interval?

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A function is increasing on an interval if its derivative is positive throughout that interval. Conversely, it is decreasing if its derivative is negative. Analyzing the graph can also help determine the behavior of the function.

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What is the revenue function in terms of price changes?

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The revenue function can be expressed as Revenue = (Number of items sold) × (Cost of each item). For Desi's business, if he decreases the price by $0.50 for each x, the revenue function is R(x) = (500 + 40x)(5 - 0.50x).

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What does it mean for a polynomial to have a factor?

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A polynomial has a factor (x + c) if substituting -c into the polynomial yields zero. This is known as the Factor Theorem.

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What is the significance of the vertex form of a quadratic function?

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The vertex form of a quadratic function, y = a(x - h)² + k, allows you to easily identify the vertex (h, k) of the parabola, which is the highest or lowest point depending on the value of a.

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What is the difference between increasing and decreasing intervals on a graph?

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Increasing intervals are sections of the graph where the y-values rise as x-values increase, while decreasing intervals are sections where the y-values fall as x-values increase.

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How do you find the explicit formula for a recursively defined sequence?

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To find the explicit formula, identify the pattern in the recursive definition, then express the nth term in terms of n without referencing previous terms. For example, a_n = 5n - 7 is derived from a₁ = -2 and a_n+1 = a_n + 5.

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What is the role of the derivative in determining function behavior?

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The derivative of a function provides information about the slope of the function at any point. A positive derivative indicates the function is increasing, while a negative derivative indicates it is decreasing.

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What does it mean for a function to be a polynomial?

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A polynomial is a mathematical expression consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Examples include x² + 3x + 2 and 4x³ - x + 1.