Mathematics Functions and Calculus Flashcards

What is the maximum value of f(x) = −2x² + 8x − 3?

Answer: 5

The vertex is at x = −b/(2a) = 2; f(2) = −8 + 16 − 3 = 5.

For f(x) = x³ − 3x, the x-values of the turning points are:

Answer: x = ±1

Setting f′(x) = 3x² − 3 = 0 gives x² = 1, so x = ±1.

Which of the following equals ∫4x³ dx?

Answer: x⁴ + C

∫4x³ dx = 4 · x⁴/4 + C = x⁴ + C by the power rule for integration.

The asymptotes of f(x) = 2/(x − 3) + 1 are:

Answer: x = 3 and y = 1

The vertical asymptote occurs where the denominator is zero (x = 3) and the horizontal asymptote is the constant being added (y = 1).

The average rate of change of f(x) = x² from x = 1 to x = 3 is:

Answer: 4

[f(3) − f(1)] / (3 − 1) = (9 − 1) / 2 = 8/2 = 4.

If f(x) = log₂(x), then f(8) equals:

Answer: 3

log₂(8) = 3 because 2³ = 8.

The transformation that maps f(x) to f(x + 2) − 3 is:

Answer: Shift left 2 and down 3

Replacing x with x+2 shifts the graph 2 units left; subtracting 3 outside shifts it 3 units down.