Te energy e stored in a spring is given by the formula e

Question

Question 1 -

(a) lim_x→4 (x²-3x-4)/(x²+x-20)

(b) lim_x→∞(x²-17)/(2x²+101)

Question 2 -

(a) d/dx e^{sin(x^2)}

(b) d/dx ln(sin^-1(x³))

Question 3 -

The energy E stored in a spring is given by the formula E = 100x² where x is the compression. The rate of change of the energy with respect to the compression is called the thrust T of the spring.

(a) Express the thrust T of the spring in terms of x;

(b) Find the energy E at x = 1.5;

(c) Find the thrust T at x = 1.5;

(d) Find the compression x when the energy E = 500;

(e) Find the compression x when the thrust T = 100.

Question 4 -

Find dy/dx at the point (0, 8) given that y = 8 + xe^y.

Question 5 -

(a) Find the indefinite integral of f(x)= 6x - 2x³ - x√x + 3e^x.

(b) Hence calculate ₀∫¹6x - 2x³ - x√x + 3e^x dx.

Question 6 -

Evaluate the following integrals using integration by parts:

(a) ∫(ln x)/x⁴ dx

(b) ∫x cos 16x dx