PureMath.pdf

# The tangent of the angle which the tangent at p makes

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the tangent of the angle which the tangent at P makes with OX , and { φ ( b ) - φ ( a ) } / ( b - a ) the tangent of the angle which AB makes with OX . It is easy to give a strict analytical proof. Consider the function φ ( b ) - φ ( x ) - b - x b - a { φ ( b ) - φ ( a ) } ,

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[VI : 125] DERIVATIVES AND INTEGRALS 275 O X Y a b A B φ ( a ) φ ( b ) P Fig. 43. which vanishes when x = a and x = b . It follows from Theorem B of § 121 that there is a value ξ for which its derivative vanishes. But this derivative is φ ( b ) - φ ( a ) b - a - φ 0 ( x ); which proves the theorem. It should be observed that it has not been assumed in this proof that φ 0 ( x ) is continuous. It is often convenient to express the Mean Value Theorem in the form φ ( b ) = φ ( a ) + ( b - a ) φ 0 { a + θ ( b - a ) } , where θ is a number lying between 0 and 1. Of course a + θ ( b - a ) is merely another way of writing ‘some number ξ between a and b ’. If we put b = a + h we obtain φ ( a + h ) = φ ( a ) + 0 ( a + θh ) , which is the form in which the theorem is most often quoted. Examples XLVII. 1. Show that φ ( b ) - φ ( x ) - b - x b - a { φ ( b ) - φ ( a ) } is the difference between the ordinates of a point on the curve and the corre- sponding point on the chord.
[VI : 127] DERIVATIVES AND INTEGRALS 276 2. Verify the theorem when φ ( x ) = x 2 and when φ ( x ) = x 3 . [In the latter case we have to prove that ( b 3 - a 3 ) / ( b - a ) = 3 ξ 2 , where a < ξ < b ; i.e. that if 1 3 ( b 2 + ab + a 2 ) = ξ 2 then ξ lies between a and b .] 3. Establish the theorem stated at the end of § 124 by means of the Mean Value Theorem. [Since φ 0 (0) = c , we can find a small positive value of x such that { φ ( x ) - φ (0) } /x is nearly equal to c ; and therefore, by the theorem, a small positive value of ξ such that φ 0 ( ξ ) is nearly equal to c , which is inconsistent with lim x +0 φ 0 ( x ) = a , unless a = c . Similarly b = c .] 4. Use the Mean Value Theorem to prove Theorem (6) of § 113 , assuming that the derivatives which occur are continuous. [The derivative of F { f ( x ) } is by definition lim F { f ( x + h ) } - F { f ( x ) } h . But, by the Mean Value Theorem, f ( x + h ) = f ( x )+ hf 0 ( ξ ), where ξ is a number lying between x and x + h . And F { f ( x ) + hf 0 ( ξ ) } = F { f ( x ) } + hf 0 ( ξ ) F 0 ( ξ 1 ) , where ξ 1 is a number lying between f ( x ) and f ( x )+ hf 0 ( ξ ). Hence the derivative of F { f ( x ) } is lim f 0 ( ξ ) F 0 ( ξ 1 ) = f 0 ( x ) F 0 { f ( x ) } , since ξ x and ξ 1 f ( x ) as h 0.] 126. The Mean Value Theorem furnishes us with a proof of a result which is of great importance in what follows: if φ 0 ( x ) = 0 , throughout a certain interval of values of x , then φ ( x ) is constant throughout that interval. For, if a and b are any two values of x in the interval, then φ ( b ) - φ ( a ) = ( b - a ) φ 0 { a + θ ( b - a ) } = 0 . An immediate corollary is that if φ 0 ( x ) = ψ 0 ( x ), throughout a certain interval, then the functions φ ( x ) and ψ ( x ) differ throughout that interval by a constant.

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[VI : 127] DERIVATIVES AND INTEGRALS 277 127. Integration. We have in this chapter seen how we can find the derivative of a given function φ ( x ) in a variety of cases, including all those of the commonest occurrence. It is natural to consider the converse question, that of determining a function whose derivative is a given function .
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