Understanding Engineering Mathematics

8.3.6 Parametric differentiation

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A.Ifx=e^2 t,y=et+1, evaluate

dy

dx

and

d^2 y

dx^2

as functions oftby two different methods

and compare your results.

B.Obtain

dy

dx

and

d^2 y

dx^2

for each of the following parametric forms:

(i) x=3cost,y=3sint (ii) x=t^2 +3,y= 2 t+1 (iii) x=etsint,y=et

(iv) x=2cosht,y=2sinht

8.3.7 Higher order derivatives

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A.Evaluate the second derivatives of each of the following functions:

(i) x^2 + 2 x+ 1 (ii) ex

2

(iii) exsinx

(iv)

x− 1

(x+ 1 )(x+ 2 )

B.Evaluate the 20th derivative of each of the following functions:

(i) x^17 + 3 x^15 + 2 x^5 + 3 x^2 −x+ 1 (ii) ex−^1

(iii) e^3 x (iv)

1

x− 1

(v)

x

(x− 1 )(x+ 2 )

8.4 Applications

1.Calculus was invented (by Newton and Liebnitz) partly to handle the mechanics of

moving particles, culminating in Newton’s laws of motion, which to this day forms one

of the major foundations of engineering science. This question is an open invitation to

link the topics of this chapter with what you know about dynamics. Things you might

think about are:

(i) the definitions oftime, position, displacement, speedandvelocity, acceleration

and their mathematical expressions in terms of derivatives of each other, with

respect to each other

(ii) Newton’s second law isF=mawhere F is force,mmass and a accelera-

tion. SupposeF is given as a function of position and velocity. Then it would

be convenient to have a derivative expression for acceleration that involves just

these two variables. Use the function of a function rule to show that, with usual

notation:

d^2 s

dt^2

=v

dv

ds