Understanding Engineering Mathematics

d((x 1 ,y 1 ), (x 2 ,y 2 ))= √ (x 2 −x 1 )^2 +(y 2 −y 1 )^2 (i) d(( 0 , 0 ), ( 1 , 1 ))= √ ( 1 − 0 )^2 +( 1 − 0 )^2 = √ 2 (ii) d ...

A 0 x 1 x 2 x 2 − x 1 y 2 − y 1 y x M Q P B C (x 2 , y 2 ) (x 1 , y 1 ) Figure 7.6Midpoint of a line segment. Similarly for they ...

Solution to review question 7.1.3 (a) The midpoints of the line segment joining (x 1 ,y 1 )and(x 2 ,y 2 )has coordinates ( x 1 + ...

(iv) For (−1,−1), (−2,−3), the gradient is m= − 3 + 1 − 2 + 1 = 2 7.2.4 Equation of a straight line ➤ 204 221➤ Knowing that theg ...

The gradient ism=AP/BP=tanθ. Two special cases require comment. For a horizontal line (i.e. parallel to thex-axis) the equation ...

B. Rewriting the lines in the form y=mx+c the gradient inm, the intercept on they-axis isc(x= 0 ,y=c)and the intercept on thex-a ...

y 0 AD x B C ab l 2 l 1 Figure 7.9Gradients of perpendicular lines multiply to−1. On the other hand the gradient ofl 2 is − BD C ...

so c=^32 and the equation isy=^12 ( 3 −x)orx+ 2 y− 3 =0. 7.2.6 Intersecting lines ➤ 205 222➤ If two lines intersect, then their ...

The second equation is actually the same as the first – just cancel 2 throughout the equation. So these equations actually repre ...

Often, it is the latter form of the equation of a circle that is given, and you are asked to find the centre and radius. To do t ...

7.2.8 Parametric representation of curves ➤ 205 222➤ Sometimes, instead of writing an equation in the formy= f(x),i.e.intermsofx ...

represent a circle with centre (a,b) and radiusr, since: (x−a)^2 +(y−b)^2 =r^2 The parameterθ can be regarded as the angle made ...

B.Plot the points with polar coordinates (i) ( 3 2 , 3 π 4 ) (ii) (1,π/3) (iii) (1, 120°)(iv)(2,− 60 °) (v) (2,π/2) (vi) (3,π) ( ...

7.3.5 Parallel and perpendicular lines ➤➤ 205 214 ➤ For the lines in RE7.3.4B determine: (a) lines parallel to each of them thro ...

7.4 Applications 1.In simplelinear programmingwe consider a set of linear relations that constrain two variablesx,yin the form a ...

Answers to reinforcement exercises 7.3.1 Coordinate systems in a plane A. y − 4 − 3 − 2 − 10 1 2 3 4 x 3 4 (−2,3) (−1,−1) (0,−2) ...

7.3.2 Distance between two points (i) √ 10 (ii) 2 (iii) √ 2 (iv) 2 (v) 1 (vi) √ 2 7.3.3 Midpoint and gradient of a line A. (i) ( ...

B.(i) ( 1 , 1 2 ) , √ 21 2 (ii) ( − 3 2 , 1 ) , √ 41 2 (iii) ( 0 , − 1 2 ) , √ 13 2 . C.x+y− 2 − 2 √ 2 =0, x−y+ 2 + 2 √ 2 = 0 Th ...

8 Techniques of Differentiation Differentiation is a relatively straightforward side of calculus. We only have to remember a doz ...

parametric differentiation higher order derivatives Motivation You may need the material of this chapter for: modelling rates ...