Algebra and analysis review
Answers are for parents and teachers.
1
Solve log₃(2x − 1) = 2.
2x − 1 = 9, x = 5; 2 · 5 − 1 = 9 > 0, so the root is in the domain.
2
Solve x² − 4x − 5 > 0.
The roots are −1 and 5 and the parabola opens upward: (−∞, −1) ∪ (5, ∞).
3
Find the greatest and least values of f(x) = x³ − 12x on [−3, 3].
f′ = 3x² − 12 = 0, so x = ±2. f(−2) = 16, f(2) = −16, f(−3) = 9, f(3) = −9. The greatest is 16 and the least is −16.
4
Find the area bounded by the parabola y = 4 − x² and the Ox axis.
∫(4 − x²)dx from −2 to 2 = 2 · (8 − 8/3) = 32/3.
5
Why does the derivative of (2x + 1)⁵ contain the extra factor 2?
This is a composite function: the outer function is u⁵ and the inner one is u = 2x + 1. By the chain rule we multiply 5u⁴ by the inner derivative u′ = 2: 10(2x + 1)⁴.