Lecture note 1 Numerical Analysis Computer cannot operate on functions directly. Example:What is the integral ofedt. 。Many more.… 1.2 Calculus Review In the following,we assume f is a function defined on a set X of real numbers. 1.2.1 Some definitions Definition 1 (Limit)f has the limit L at ro,written lim f(x)=L I-T0 if,given any real number e>0,there erists a real number o>0 such that lf(x)-L<e,whenever x∈Xand‖x-xol<6. Definition 2(Continuous)f is continuous at ro if lim f(x)=f(xo). Remarks: We say f is continuous on the set X if it is continuous at each number in X. The limit of a sequence is defined in a similar manner. f is continuous at to If {In}is any sequence in X converging to ro,then limn→of(xn)=f(xo. Definition 3(Derivative)If f is defined in an open interval containing ro.f is differentiable at ro if f'(xo）=lim f()-f(xo) x→x0工-T0 erists.The number f'(ro)is called the derivative of f at ro. Remarks: f is differentiable on X if f is differentiable at each number in X. Geometrically,the derivative of f at zo is the slope of the tangent line to the graph at f at (ro,f(ro)). If the function f is differentiable at zo,then f is continuous at xo.Lecture note 1 Numerical Analysis • Computer cannot operate on functions directly. Example: What is the integral of R x −∞ e −t 2 dt. • Many more... 1.2 Calculus Review In the following, we assume f is a function defined on a set X of real numbers. 1.2.1 Some definitions Definition 1 (Limit) f has the limit L at x0, written limx→x0 f(x) = L if, given any real number ǫ > 0, there exists a real number δ > 0 such that kf(x) − Lk < ǫ, whenever x ∈ X and kx − x0k < δ. Definition 2 (Continuous) f is continuous at x0 if limx→x0 f(x) = f(x0). Remarks: • We say f is continuous on the set X if it is continuous at each number in X. • The limit of a sequence is defined in a similar manner. • f is continuous at x0 ⇔ If {xn} is any sequence in X converging to x0, then limn→∞ f(xn) = f(x0). Definition 3 (Derivative) If f is defined in an open interval containing x0. f is differentiable at x0 if f ′ (x0) = limx→x0 f(x) − f(x0) x − x0 exists. The number f ′ (x0) is called the derivative of f at x0. Remarks: • f is differentiable on X if f is differentiable at each number in X. • Geometrically, the derivative of f at x0 is the slope of the tangent line to the graph at f at (x0, f(x0)). • If the function f is differentiable at x0, then f is continuous at x0. 3