ANALYTICAL METHODS FOR ENGINEERS ASSIGNMENT - CALCULUS

This assignment assesses Outcome - Analyse and model engineering situations and solve problems using calculus.

Questions -

Q1. Differentiate the following functions with respect to x.

(a) Use the quotient rule to differentiate

y = (2x⁴ - 3x)/(4x - 1)

(b) Use the Chain rule to differentiate

y = 6cos(x³ + 3)

(c) Select an appropriate rule to differentiate

y = (4x² - e^2x)sin3x

Q2. The angular displacement, θ radians, of the spoke of a wheel is given by the expression θ = 0.5t⁴ - t³, where t is time in seconds. Find:

(a) The angular velocity after 2 seconds.

(b) The angular acceleration after 3 seconds.

(c) The time when the angular acceleration is zero.

Q3. From a rectangular sheet of metal measuring 120 mm by 75mm, equal squares of side x are cut from each of the four corners. The remaining flaps are then folded upwards to form an open box.

(a) Draw a neat and simple diagram of the rectangular sheet of metal (not drawn to scale) and show the dimensions given including the squares of side x.

(b) Show that the volume of the box is given by: V = 9000x - 390x² + 4x³

(c) Find the value of x such that the volume is a maximum.

Q4. Find the following integrals.

(a) ∫(5x² + √x - 4/x²)dx

(b) ∫[cos(x/2) - sin(3x/2)]dx

(c) ₁∫⁵ s/√(s²+4) ds

Q5. (a) Find the area bounded by the curve y = 1/x between x = 2 and x = 6. (Give your answer to 3 decimal places.)

(b) Sketch the curve for y = sin(x) between x = 0 and x = 2π, then find the total area enclosed by the curve y = sin x and the X axis between x = 0 and x = 1.7π.

Q6. If the instantaneous rate of change of a population is 50t² - 100t^3/2 (measured in individuals per year) and the initial population is 25000 then

(a) What is the population after t years?

(b) What is the population after 25 years?

Q7. Use integration by parts to solve the following integral

∫5xcos(4x)dx