Vi 111 derivatives and integrals 241 of many

Info icon This preview shows pages 256–259. Sign up to view the full content.

View Full Document Right Arrow Icon
possibility the two forms of the question stated above are equivalent.
Image of page 256

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
[VI : 111] DERIVATIVES AND INTEGRALS 241 of many departments of mathematics in which the idea of a derivative finds an application. Another important application is in dynamics. Suppose that a particle is moving in a straight line in such a way that at time t its distance from a fixed point on the line is s = φ ( t ). Then the ‘velocity of the particle at time t ’ is by definition the limit of φ ( t + h ) - φ ( t ) h as h 0. The notion of ‘velocity’ is in fact merely a special case of that of the derivative of a function. Examples XXXIX. 1. If φ ( x ) is a constant then φ 0 ( x ) = 0. Interpret this result geometrically. 2. If φ ( x ) = ax + b then φ 0 ( x ) = a . Prove this (i) from the formal definition and (ii) by geometrical considerations. 3. If φ ( x ) = x m , where m is a positive integer, then φ 0 ( x ) = mx m - 1 . [For φ 0 ( x ) = lim ( x + h ) m - x m h = lim mx m - 1 + m ( m - 1) 1 · 2 x m - 2 h + · · · + h m - 1 . The reader should observe that this method cannot be applied to x p/q , where p/q is a rational fraction, as we have no means of expressing ( x + h ) p/q as a finite series of powers of h . We shall show later on ( § 118 ) that the result of this example holds for all rational values of m . Meanwhile the reader will find it instructive to determine φ 0 ( x ) when m has some special fractional value ( e.g. 1 2 ), by means of some special device.] 4. If φ ( x ) = sin x , then φ 0 ( x ) = cos x ; and if φ ( x ) = cos x , then φ 0 ( x ) = - sin x . [For example, if φ ( x ) = sin x , we have { φ ( x + h ) - φ ( x ) } /h = { 2 sin 1 2 h cos( x + 1 2 h ) } /h, the limit of which, when h 0, is cos x , since lim cos( x + 1 2 h ) = cos x (the cosine being a continuous function) and lim { (sin 1 2 h ) / 1 2 h } = 1 ( Ex. xxxvi . 13).]
Image of page 257
[VI : 112] DERIVATIVES AND INTEGRALS 242 5. Equations of the tangent and normal to a curve y = φ ( x ) . The tangent to the curve at the point ( x 0 , y 0 ) is the line through ( x 0 , y 0 ) which makes with OX an angle ψ , where tan ψ = φ 0 ( x 0 ). Its equation is therefore y - y 0 = ( x - x 0 ) φ 0 ( x 0 ); and the equation of the normal (the perpendicular to the tangent at the point of contact) is ( y - y 0 ) φ 0 ( x 0 ) + x - x 0 = 0 . We have assumed that the tangent is not parallel to the axis of y . In this special case it is obvious that the tangent and normal are x = x 0 and y = y 0 respectively. 6. Write down the equations of the tangent and normal at any point of the parabola x 2 = 4 ay . Show that if x 0 = 2 a/m , y 0 = a/m 2 , then the tangent at ( x 0 , y 0 ) is x = my + ( a/m ). 112. We have seen that if φ ( x ) is not continuous for a value of x then it cannot possibly have a derivative for that value of x . Thus such functions as 1 /x or sin(1 /x ), which are not defined for x = 0, and so necessarily discontinuous for x = 0, cannot have derivatives for x = 0. Or again the function [ x ], which is discontinuous for every integral value of x , has no derivative for any such value of x . Example. Since [ x ] is constant between every two integral values of x , its derivative, whenever it exists, has the value zero. Thus the derivative of [ x ], which we may represent by [ x ] 0 , is a function equal to zero for all values of x save integral values and undefined for integral values. It is interesting to note that the function 1 - sin πx sin πx has exactly the same properties.
Image of page 258

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 259
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern