Maths of M2 & Function (20)

1. Let y = cos 3x, use the definition of differentiation (the first principle) to show that y′ = −3 sin 3x.

2.Determine the open intervals where the function is increasing or decreasing y = (1/2)(x^2) − ln x.

3. Verify that the formula ∫1 / [(x^2)(1+x^2)^(1/2)] dx = − (1+x^2) / x + C
is correct by using the definition of integration.

4. Evaluate the areas of the region R bounded by the x-axis and the curve
y = 1 + sin x between x = 0 and x = 3π/2

5. Use definite integrals to compute the area of the shaded region. Here
f(x) = 4 – x^2, g(x) = 2 − x, and −2 ≤ x ≤ 3.

Thx