Calculus Flashcards

The tangent to y = x² − 4x + 3 at x = 1 has equation:

Answer: y = −2x + 2

y(1) = 0, y′(1) = 2(1) − 4 = −2; tangent: y = −2(x − 1) = −2x + 2.

What is ∫₁ᵉ (1/x) dx?

Answer: 1

[ln x]₁ᵉ = ln e − ln 1 = 1 − 0 = 1.

For which x does f(x) = x³ − 6x² + 9x have a point of inflection?

Answer: x = 2

f″(x) = 6x − 12 = 0 gives x = 2, and the sign of f″ changes there, confirming an inflection point.

Differentiate y = tan(x²) with respect to x.

Answer: 2x·sec²(x²)

By the chain rule, dy/dx = sec²(x²) · 2x.

Which statement about f(x) = |x| at x = 0 is correct?

Answer: f is continuous but not differentiable at x = 0

|x| is continuous everywhere, but the left derivative is −1 and right derivative is +1 at x = 0, so it is not differentiable there.

If f(x) = cos²(x) + sin²(x), then f′(x) = ?

Answer: 0

cos²(x) + sin²(x) = 1 (the Pythagorean identity), so f is constant and f′(x) = 0.

Find the x-coordinate of the stationary point of f(x) = x² − 6x + 8.

Answer: x = 3

f′(x) = 2x − 6 = 0 gives x = 3; f″(3) = 2 > 0 so this is a minimum.