106, 107] 
DERIVATIVES AND INTEGRALS 
215 
be a point, such as P, where the tangent is parallel to AB. For 
ft (£) is the tangent of the angle which the tangent at P makes 
with OX, and {<^> (&) — </>(«-)}/(& — a.) the tangent of the angle which 
AB makes with OX. 
It is easy to give a strict analytical proof. Consider the 
function 
0(4)-*(«)-^-{*(6)-*(«)), 
which vanishes when x = a and x = b. It follows from Theorem B 
above that there is a value £ for which its derivative vanishes. 
But this derivative is 
which proves the theorem. It should be observed that in this 
proof it has not been assumed that ft (x) is continuous. 
Examples XLIX. 1. Show that the expression 
-</>(«)} 
is the difference between the ordinates of a point on the curve and the 
corresponding point on the chord. 
2. Verify the theorem for (x) = x 2 and for <£ (x) = x 3 . 
[In the latter case we have to prove that (b 3 — a 3 )/{b - a) — 3£ 2 , where. 
a<£<5; i.e. that if J(6 2 + «6 + a 2 ) = £ 2 , then £ lies between a and 6.] 
3. Prove the result of Ex. xlviii. 45 by means of the mean value 
theorem. 
[Since <£'(0) = c we can find a small positive value of x such that 
{cj){x)-(p(0)}lx is nearly equal to c; and therefore, by the theorem, a small 
positive value of £ such that (£) is nearly equal to c, which is inconsistent 
with lim ft{x) — a, unless a — c. Similarly b = c.] 
*-»-+0 
107. Another form of the Mean Value Theorem. It is 
often convenient to express the Mean Value Theorem in the form 
</> (¿>) = c^> (a) + {b — a) ft {a + 6 {b — a)| 
where 6 is a number lying between 0 and 1. Of course 
a + 0{b — a) is merely another way of writing ‘ some number £ 
between a and 6.’ If we put b = a + h we obtain 
<p (cb + K) = (j) (<z) + h<p (ft + Oh) 
which is the form in which the theorem is most often quoted.