Task -1:

1. Find the differential coefficient for three different functions to demonstrate the use of function of a function and the product and quotient rules

2. use integral calculus to solve two simple engineering problems involving the definite and indefinite integral

3. use differential calculus to find the maximum/minimum for an engineering problem

4. use numerical integration and integral calculus to analyse the results of a complex engineering problem

Task 2:

Find the differential coefficient for three different functions to demonstrate the use of function of a function and the product and quotient rules

Use differential calculus to find the maximum/minimum for an engineering problem

1. Differentiate algebraic and trigonometric functions using the product, quotient and function of function rules

2. Determine higher order derivatives for algebraic, logarithmic, inverse trigonometric and inverse hyperbolic functions

3. Analyse engineering situations and solve engineering problems using calculus

Q 1: Determine the second order derivatives for the following equations.

a. y= ln?(2x² - 5) wrt x

b. y= (x^3-3x+1)/(x^3+3x+1) wrt x

c. y=2 cosh^(-1) t/3 wrt t

Q. 2: The vertical displacement s cm of a mass attached to a stiff spring at time t seconds is given by:

s=ae^-ktsin2πft

Determine the velocity and acceleration of the end of the mass after 2 seconds if a=3.2, k=0.85 and f=25

Q. 3: A cone of height (h) and radius (r) has a slant length (l) of 0.4 cm. These are related by the equation l² = r² + h²

The volume of the cone (V) is given by:

V= 1/3 π r² h

Substitute for r and use differentiation to determine the maximum and minimum volumes.

Task 3:

Use integral calculus to solve two simple engineering problems involving the definite and indefinite integral

Use numerical integration and integral calculus to analyse the results of a complex engineering problem

Integrate functions using the rules, by parts, by substitution and partial fractions

Analyse engineering situations and solve engineering problems using calculus

Q 1. Integrate the following functions using the appropriate method chosen from the list below:

by parts
by substitution
by using partial fractions

a. (x^2 ln?x )dx

b. 1/ (x² + 6x + 34)dx

c. x²/(x-1)(x+2)²dx

Q. 2: State the equation (and show working) that will determine the area under the curve of y = x² + 6x + 10

Q. 3: Find the volume of the shape formed when the function y = 1/2 x² + 1 between x = 1 and x = 4 is rotated about the x-axis.

Q. 4: The discharge current in an RC circuit is given by:

i=10.1e^((- t)/(0.88m ))mA

Plot the discharge curve and use the trapezium rule to determine the total charge, q, leaving the capacitor between t = 0 and t = 2 secs if:

q=∫idt