The design and development department associated with a

Question

Analytical methods for engineers(calculus) tasks 1,2 and 3

Assignment brief

Scenario

The design and development department associated with a power generation plant has various issues Mich require resolving, you are now asked to provide an analysis of these issues with respect to the questions detailed below:

Task 1:

a. Differentiate the following using product rule

(i) y = xlnx   (ii) y = x³sin 2x

b. Differentiate the following using quotient rule

(i) y = x/1-x   (ii) y = 3ln5t/cos2t

c. Differentiate the following using function of a function rule

(i) y = 2/√(x² - 3x + 5)  (ii)  y = 4 tan³4x

d. Differentiate y = x³ln3x/e^xsin2x - with respect to x

Task 2

a. If y = 3x⁴ + 2x³ - 3x + 2

find

(i) dy/dx (ii) d²y/dx² (iii) d³y/dx³

b. Find dy/dx and d²y/dx² when y = ln(x² -1)

c. Find dy/dx and d²y/dx² when y = sin^-16x

d. Find dy/dx and d²y/dx² when y = tanh^-14x

Task 3

Determine the following integrals

(a) ∫xsinx dx using integration by parts

(b) ∫2et/√(e^t + 4)dt using integration by substitution

(c) ∫4x-3/(x + 1)²dt using integration by partial fractions