alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

0 continuous 44 meter s sec 3 example 3 vt is

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Unformatted text preview: ial of the for m a 0 + a 1 t + a 2 t 2 -2 1 with coefficients a 0 = 0 , a 1 = u . s i n θ a nd a2 = -- -- g . So the height 2 function y(t) is single valued and continuous . single continuous It is important to know that polynomials are to functions what rational numbers are to real numbers. We can approximate any real number to any required degree of accuracy by a suitable rational number. Likewise we can approximate almost any real value function by a suitable polynomial. For this we have Lagrange’s method of interpolation. 42 Example 1: is f(x) continuous at x = a ? 1 continuous f(x) a f (x ) = { x for 0 ≤ x ≤ a for 2a -- x for a ≤ x ≤ 2 a a (0,0 0,0) (0,0 ) Approaching a from the left : f(x) = x left Near a and to the left of a : left 2a x x = a − δx L imit f(x) = Limit { a − δx} = a δx → 0 x → a− Approaching a from the right : f(x) = 2a − x right Near a and to the right of a : x = a + δx right L imit f(x) = Limit { 2a −(a +δx)} = Limit { 2a −a −δx } δx → 0 δx → 0 x → a+ Limit = δx → 0 { a − δx } = a Hence, L imita− f(x) = a = L imita+ f(x) x→ x→ Value f(x) = f(a) = a x=a Since, L imita− f(x) = L imita+ f(x) = Value f(x) = a x→ x→ x=a we say : f(x) is continuous at x = a. continuous 43 Example 2 : is f(x) = |x| continuous at x = 0 ? continuous f(x) f(x) = |x| (0 , 0 ) Approaching 0 from the left : f(x) = |x| = −x left Near 0 and to the left of 0 : left L imit x → 0− x x = 0 − δx Limit Limit f(x) = δx → 0 { − (0 − δx)} = δx → 0 { +δx} = 0 Approaching 0 from the right : f(x) = x right Near 0 and to the right of 0 : x = 0 + δx right L imit x → 0+ Limit Limit f(x) = δx → 0 { (0 +δx) = δx → 0 { + δx } = 0 Hence, L imit − f(x) = 0 = L imit + f(x) x→0 x→0 Value x = 0 f(x) = f(0) = 0 Value Since, L imit0− f(x) = L imit + f(x) = x = 0 f(x) = 0 x→ x→0 we say : f(x) is continuous at x = 0. continuous 44 [meter s / sec] 3: Example 3: v(t) is the speed of a car weighing one ton that star ts fro...
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