Calculus: Analytic Geometry and Calculus, with Vectors

146 Functions, limits, derivatives function, like fl, is continuous except when x = a and x = b. However, f2 has left-hand conti ...

3.4 Continuity^147 The graphs of the signurn and Heaviside functions shown in Figures 3.191 and 3.192 should indicate that these ...

148 Functions, limits, derivatives it follows from the sandwich (or flyswatter) theorem that (3.472) lim q2(x) = 0 = q2(0), x-0 ...

3.4 Continuity 149 is continuous. Prove the statement by filling in the intermediate steps in the formula lim P(x) P(a) a-4a and ...

(^150) Functions, limits, derivatives where x is the distance from the center of the earth to the particle and R is the radius o ...

3.4 Continuity 151 As is easy to guess, the vector function r is said to be continuous at t if (5) and we write (6) if w is a ve ...

152 Functions, limits, derivatives nature of the graph of g. Show that g is discontinuous at each x for which x is rational and ...

3.5 Difference quotients and derivatives 153 subtract to obtain AY = AX + Ax) - AX), and then divide by Ax to obtain (3.52) AY = ...

154 Functions, limits, derivatives the manner in which it is applied can be (or should be) remembered with the aid of the famous ...

3.5 Difference quotients and derivatives 155 as rapidly as we canwrite. Thus scientists differentiate polynomials with gusto. Us ...

156 Functions, limits, derivatives Therefore, an application of a theorem on limits gives TX = C TX +c,TX* As has been remarked, ...

3.5 Difference quotients and derivatives 157 have y = x and must prove that This is true because if y = x, then Ay = Ox, so Ay/A ...

158 Functions, limits, derivatives Very much more about derivatives remains to be learned, and we give a modest but important co ...

3.5 Difference quotients and derivatives 159 Problems 3.59 1 Give the definition of the derivative of f at x. 4nr.: (3.53) or (3 ...

160 Functions, limits, derivatives 6 Calculate dx(x2 + 3)(x2 - 2) by use of the product formula. Then multiply the given factors ...

3.5 Difference quotients and derivatives 161 9 Calculate d 1 - x2 d dx- (^1) + x2' dx(1 + x2) by the quotient formula and by the ...

162 Functions, limits, derivatives 15 A long time ago it was discovered that if P is a polynomial, then between each pair of val ...

3.5 Difference quotients and derivatives 163 and hence (5) f(x + .x) - f(x) = g(x + Ex)h(x + Ax) - g(x)h(x). To make the right s ...

164 Functions, limits, derivatives We can observe that the Newton notation in the formula (2) uses functional notation in a thor ...

3.6 The chain rule and differentiation of elementary functions 165 formulas (3.61) dx xn = xn-1, x sin x =cos x, x cos x = - sin ...