TMUA Calculus 5

Question 1

The volume of the solid formed by rotating y = √x about the x-axis from x = 0 to x = 4 is:

Accepted Answer

8π

Answer

V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = 8π.

Question 2

Which of the following is an antiderivative of 1/(1 + x²)?

Accepted Answer

arctan(x)

Answer

d/dx[arctan(x)] = 1/(1 + x²), so arctan(x) + C is the antiderivative.

Question 3

Using implicit differentiation, find dy/dx if x² + y² = 25.

Accepted Answer

−x/y

Answer

Differentiating: 2x + 2y·(dy/dx) = 0, so dy/dx = −x/y.

Question 4

lim_{h→0} [f(x+h) − f(x)]/h defines:

Accepted Answer

The derivative f′(x)

Answer

This is the limit definition of the derivative f′(x) at a point x.

Question 5

f(x) = x⁴ − 8x² + 3 has local minima at:

Accepted Answer

x = 2 and x = −2

Answer

f′(x) = 4x(x² − 4) = 0 at x = 0, ±2; f″(±2) = 32 > 0 confirms minima at x = ±2.

(TMUA) Test of Mathematics for University Admission Practice Test