Understanding Engineering Mathematics

(iv) ex 2 (cosx+sinx) (v) x^2 sinx+ 2 xcosx−2sinx (vi) x^2 2 ln|x|− x^2 4 (vii) 1 2 (x^2 − 1 )ex 2 9.3.11 Choice of integration ...

10 Applications of Differentiation and Integration In general terms, the usefulness of calculus in elementary engineering maths ...

Motivation You may need the material of this chapter for: finding and interpreting rates of change such as velocity, accelerati ...

10.1.6 Volume of a solid of revolution ➤308 311➤➤ Find the volume of the solid of revolution formed when the positive area enclo ...

10.2.2 Tangent and normal to a curve ➤ 291 310➤ The derivative at a point (a,b)onacurvey=f(x)will give us the slope,m,ofthe tang ...

so the equation of the normal is y− 1 =−^13 (x− 2 ) or x+ 3 y− 5 = 0 10.2.3 Stationary points and points of inflection ➤ 291 310 ...

globalminimum of the function. Similarly for a maximum. Indeed, for a function such asy=x+ 1 x the local minimum actually has a ...

and so in this case the derivative isdecreasingas we increasexthroughx=x 0 and thus amaximum pointx=x 0 is characterised by f′(x ...

Solution to review question 10.1.3 (i) For the functionf(x)= 16 x− 3 x^3 we havef′(x)= 16 − 9 x^2. Solving f′(x)= 16 − 9 x^2 = 0 ...

In this casef′′(x)is never zero for finitex, so we have no points of inflection. Notice that in this case the local minimum valu ...

10.2.4 Curve sketching in Cartesian coordinates ➤ 291 310➤ Curve sketching is exactly that – it does not meanplottingthe curve, ...

half the curve. This may reduce considerably the amount of work we have to do in sketching the curve. G.Look for points where th ...

H.Points of particular importance include maximum and minimum values(humps and hollows)and also points of inflection. While we d ...

No particular specialpointsspring to mind, except thestationary points and points of inflection. We know from Review Question 10 ...

− 10 y y = x + y^ = x (^1) x 1 x Figure 10.8Sketch ofy=x+ 1 x . (iii)y=xex Symmetry:none Gateways:(0, 0) only Restrictions:none ...

0 y x y = sin x cos x = sin 2x p −p 1 2 Figure 10.10Sketch ofy=sinxcosx. (v) y=x^4 We have already discussed this relatively sim ...

an area split into strips a curve split into line segments a solid body split into very small particle like elements The rest ...

the definite integral interpreted as thearea under a curveas illustrated in Figure 10.13. A A = ∫ f(x) dx y = f(x) y (^0) a b ...

or A(x+h)−A(x) h = 1 2 [f(x+h)+f(x)] and this approximation improves ashgets smaller. Indeed, taking the limith→0weget lim h→ 0 ...

0 3 2 , x y = x^2 − x y^ = 2 x −^ x 2 y (1,0) 3 4 Figure 10.15Area betweeny= 2 x−x^2 andy=x^2 −x. However, this will still be ac ...