Understanding Engineering Mathematics

Motivation

You may need the material of this chapter for:

• finding and interpreting rates of change such as velocity, acceleration, etc.


• interpreting solutions of differential equations


• finding optimal values of various engineering variables


• sketching the graph of an engineering function


• finding areas, centroids, etc.


• evaluating volumes, moments of inertia, etc.


• evaluating mean and rms values of alternating currents


• evaluating probability, the mean and standard deviation in statistics

10.1 Review

10.1.1 The derivative as a gradient and rate of change ➤292 309➤➤

For each of the following functions find:

(a) The gradient or slope of the graph of the function at the point specified

(b) The rate of change of the function at the point specified

(i) y=x^2 , x= 1 (ii)y=cosx, x=π

10.1.2 Tangent and normal to a curve ➤293 310➤➤

Find the equations of the (i) tangent and (ii) the normal to the curvey=x^2 −x−1at

the point (2, 1).

10.1.3 Stationary points and points of inflection ➤294 310➤➤

Locate and classify any stationary points and points of inflection of the following functions

(i) 16x− 3 x^3 (ii) x+

1

x

(iii) xex

(iv) sinxcosx (v) x^4

10.1.4 Curve sketching in Cartesian coordinates ➤299 310➤➤

Sketch the functions given in Q10.1.3.

10.1.5 Applications of integration – area under a curve ➤304 310➤➤

A.Find the area enclosed between the curve, thex-axis and the limits stated for each of
the following cases:

(i) y= 4 x^2 + 1 x=0, 2 (ii) y=xex x=0, 1

B.Find the area enclosed between the curvesy=x^2 −xandy= 2 x−x^2.