By Harley Flanders; Justin J Price
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Let us try an example: y = x 2 at P (1, 1). See Fig. 2. nearby point to (1, 1) is Q,, = (1 + h, (1 + h)2� where h is a small number, positive or negative. By the slope formula, A = (1 h)2 - 1 ( l + h)2 l - + sIope of PQ . ( l + h) I h So far so good; now we must test some small values of h and see if a message comes through. l. • The word 1ecut simply means a line through two or more distinct points of a curve. 65 2. DE RIVATIVE S 66 ,. > ' = \"" (I Fis. 1 slope • (I Fig. 2 As Q. P. the slope of PQ.
Domain Is y defined for all real x or is there some restriction on x? For example, y = Jl=XI is defined only for Ix I :5 1 and y = 1/(x - 1 )(x - 4) is not defined for x = 1 or x = 4. Range Is there some limitation on y? For example, if y = 1/(1 + x2� then by inspection, 0 < y s 1 . The graph does not extend above the level y = 1 or below the level y = 0. Sign ofy Can you tell where y > 0 or y < O? For example y = 1/x is positive for x > 0 and negative for x < 0. Also y is never 0. Increasing or decreasing?
The answer makes good sense geometrically. The slope Ja2 is always positive, except that it is zero at a = As I a I increases, Ja2 increases rapidly, so the curve Ii slope of PQ,, difference quotient P P, Q11 0). It. P y 8), y - --- - - -- ---- - ,1 = ( - 8. _ _ " - Ii- ·- . - 8) " 0, y Answer 0. becomes very steep (Fig. ). f(x), a. f (a)) and Q,. f(a + h)). Compute the slope of the secant through P and Q,. 01 I lr = O. r = f(x) at (-2, 12). ) Now compute the slope of = f (x) at (0, 0). x ) = 2x2 + l'< 9 ,, = -0.
Calculus with analytic geometry by Harley Flanders; Justin J Price