AP Precalculus - Advanced Concepts & Hard Scenarios

Front

Justify the existence of a local maximum in a polynomial function without calculus.

Back

A polynomial is continuous on a closed interval, so by the Extreme Value Theorem it must attain an absolute maximum there. If that maximum occurs at an interior point of the interval, then it is a local maximum.

Front

Analyze the concavity change at a point of inflection for a cubic function.

Back

For a cubic function f(x), the rate of change changes from increasing to decreasing (or vice versa) at the point of inflection. Graphically, this is where the curve transitions from concave up to concave down. The sign of the second derivative (or the behavior of the slope values) shifts.

Front

Determine the function type given constant second differences of output values.

Back

If the second differences of output values over equal-length intervals are constant (and non-zero), the function is quadratic. This is because the discrete analogue of the second derivative of a quadratic function (ax^2+bx+c) is constant (2a). If second differences are zero, the function is linear.

Front

Compare Average Rate of Change (AROC) for f(x)=sin(x) on [0, π/2] vs [π/2, π].

Back

AROC on [0, π/2] is (sin(π/2)-sin(0))/(π/2-0) = 2/π ≈ 0.637. AROC on [π/2, π] is (sin(π)-sin(π/2))/(π-π/2) = -2/π ≈ -0.637. The magnitudes are equal because the sine function is symmetric about the line x=π/2; the rise on the way up equals the fall on the way down.

Front

Identify the domain of f(x) = log_2(x+3) + log_2(x-1).

Back

To recognize or establish the identity of someone or something. From Latin idem meaning “the same,” via Late Latin identificare (“to make identical”), related to identitas (“identity”).