TMUA Calculus 4 — Questions and Answers
Question 1: The tangent to y = x² − 4x + 3 at x = 1 has equation:
- y = −2x + 2 (Correct answer)
- y = −2x
- y = 2x − 2
- y = x
Correct answer: y = −2x + 2
y(1) = 0, y′(1) = 2(1) − 4 = −2; tangent: y = −2(x − 1) = −2x + 2.
Question 2: What is ∫₁ᵉ (1/x) dx?
- 0
- 1 (Correct answer)
- e
- e − 1
Correct answer: 1
[ln x]₁ᵉ = ln e − ln 1 = 1 − 0 = 1.
Question 3: For which x does f(x) = x³ − 6x² + 9x have a point of inflection?
- x = 1
- x = 2 (Correct answer)
- x = 3
- x = 0
Correct answer: x = 2
f″(x) = 6x − 12 = 0 gives x = 2, and the sign of f″ changes there, confirming an inflection point.
Question 4: Differentiate y = tan(x²) with respect to x.
- sec²(x²)
- 2x·sec²(x²) (Correct answer)
- 2 sec²(x)·tan(x)
- sec²(2x)
Correct answer: 2x·sec²(x²)
By the chain rule, dy/dx = sec²(x²) · 2x.
Question 5: Which statement about f(x) = |x| at x = 0 is correct?
- f is differentiable but not continuous
- f is not continuous at x = 0
- f is continuous but not differentiable at x = 0 (Correct answer)
- f has a local maximum at x = 0
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.
Question 6: If f(x) = cos²(x) + sin²(x), then f′(x) = ?
- 2 cos(x)sin(x)
- 0 (Correct answer)
- −2 sin(x)cos(x)
- 1
Correct answer: 0
cos²(x) + sin²(x) = 1 (the Pythagorean identity), so f is constant and f′(x) = 0.
Question 7: Find the x-coordinate of the stationary point of f(x) = x² − 6x + 8.
- x = 2
- x = 3 (Correct answer)
- x = 4
- x = 6
Correct answer: x = 3
f′(x) = 2x − 6 = 0 gives x = 3; f″(3) = 2 > 0 so this is a minimum.
TMUA Calculus 4
The tangent to y = x² − 4x + 3 at x = 1 has equation: