AP Precalculus Unit 1 & 2 Review

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What is the average rate of change of a function on an interval?

Back

The average rate of change of a function f(x) on the interval [a, b] is given by the formula: \( \frac{f(b) - f(a)}{b - a} \). It represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

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What does it mean for a function to be increasing on an interval?

Back

A function f(x) is said to be increasing on an interval (a, b) if for any two points x1 and x2 in (a, b), where x1 < x2, it holds that f(x1) < f(x2).

Front

What is the definition of a positive function?

Back

A function f(x) is positive on an interval (a, b) if for all x in (a, b), f(x) > 0.

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How do you determine the intervals where a function is increasing or decreasing?

Back

To determine the intervals where a function is increasing or decreasing, find the derivative f'(x). The function is increasing where f'(x) > 0 and decreasing where f'(x) < 0.

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What is a root of a polynomial function?

Back

A root of a polynomial function is a value of x for which the polynomial evaluates to zero, i.e., f(x) = 0.

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What is the factored form of a polynomial?

Back

The factored form of a polynomial expresses it as a product of its linear and/or irreducible quadratic factors, based on its roots.

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What is an extremum in calculus?

Back

An extremum is a point in the domain of a function where the function takes on a maximum or minimum value.

Front

How do you find the number of extrema of a function?

Back

To find the number of extrema, analyze the critical points by setting the derivative f'(x) to zero and determining the sign changes.

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What is the significance of complex roots in polynomial functions?

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Complex roots occur in conjugate pairs and indicate that the polynomial does not cross the x-axis at those points, affecting the graph's behavior.

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What is the relationship between the degree of a polynomial and the number of roots?

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A polynomial of degree n has exactly n roots, counting multiplicities, which may be real or complex.