TMUA Calculus 4

Question 1

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

Accepted Answer

y = −2x + 2

Answer

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

Question 2

What is ∫₁ᵉ (1/x) dx?

Accepted Answer

1

Answer

[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?

Accepted Answer

x = 2

Answer

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.

Accepted Answer

2x·sec²(x²)

Answer

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

Question 5

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

Accepted Answer

f is continuous but not differentiable at x = 0

Answer

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

(TMUA) Test of Mathematics for University Admission Practice Test