TMUA Calculus 3

Question 1

For f(x) = e^(x²), what is f′(x)?

Accepted Answer

2x·e^(x²)

Answer

By the chain rule, d/dx[e^(x²)] = e^(x²) · 2x.

Question 2

The area enclosed between y = x² and y = x over [0, 1] is:

Accepted Answer

1/6

Answer

∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.

Question 3

If h(x) = sin(3x), what is h″(x)?

Accepted Answer

−9 sin(3x)

Answer

h′(x) = 3 cos(3x) and h″(x) = −9 sin(3x).

Question 4

Which theorem guarantees that a continuous function on a closed interval [a, b] attains a maximum and a minimum?

Accepted Answer

Extreme Value Theorem

Answer

The Extreme Value Theorem states that a continuous function on a closed, bounded interval attains its maximum and minimum.

Question 5

Evaluate lim_{x→∞} (3x² + 1)/(x² − 5).

Accepted Answer

3

Answer

Dividing numerator and denominator by x² gives (3 + 1/x²)/(1 − 5/x²) → 3 as x → ∞.

(TMUA) Test of Mathematics for University Admission Practice Test