Mathematics Functions and Calculus Flashcards

What is the derivative of f(x) = sin(x)?

Answer: cos(x)

By the standard trigonometric derivative rule, d/dx[sin(x)] = cos(x).

The area under f(x) = 2x from x = 0 to x = 3 is:

Answer: 9

∫₀³ 2x dx = [x²]₀³ = 9 − 0 = 9.

If f(x) = eˣ, then f′(x) equals:

Answer: eˣ

The exponential function eˣ is its own derivative: d/dx[eˣ] = eˣ.

A function f(x) is concave down on an interval when:

Answer: f″(x) < 0

A negative second derivative means the rate of change is decreasing, indicating the graph curves downward (concave down).

The y-intercept of f(x) = 3x² − 5x + 2 is:

Answer: 2

The y-intercept is found by setting x = 0: f(0) = 0 − 0 + 2 = 2.

The period of f(x) = sin(2x) is:

Answer: π

The period of sin(bx) is 2π/b; with b = 2 the period is 2π/2 = π.

If f′(c) = 0 and f″(c) > 0, the point x = c is:

Answer: A local minimum

By the second derivative test, f′(c) = 0 with f″(c) > 0 indicates the curve is concave up at c, confirming a local minimum.