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Final Review 2: Applications • Approximations: √ – 4.04: f (x + ∆x) ≈ f (x) + f 0 (x) · ∆x where ∆x is very small compared to x. – Newton’s Method: To solve the equation f (x) = 0, pick x1 that may be close to the solution, f (xn ) then use the formula xn+1 = xn − 0 repeatedly. f (xn ) – Numerical Integration: left/right/midpoint Riemann Sum. • Sketching the graph: – Increasing(decreasing): f 0 (x) > 0(< 0). – Concave up(down): f 00 (x) > 0(< 0). – (Global) Maximum and Minimum: ∗ Only works for continuous functions on CLOSED interval. ∗ Candidates are critical points: stationary points, endpoints and singular points. ∗ Evaluate the function at all critical points. The largest value(s) is the maximum and the smallest value(s) is the minimum. – Local Maximum and Minimum: ∗ Works for all continuous functions. ∗ A local extreme doesn’t have to be a global extreme. ∗ Candidates are critical points. Usually on open interval, hence no endpoint. ∗ First derivative test: If c is a critical point, test the sign change of f 0 (x) around c. ∗ Second derivative test: If c is a critical point, test the sign of f 00 (c). – To determine the sign of f 0 (x), notice that the sign can only change at the critical points. In the other word, between two adjacent critical points, the signs are all + or all −. Pick any number a in each interval and test the sign of f 0 (a) will determine every sign in that interval. • Differential Equations (Separating the Variable): dy into f 0 (x) and y into f (x) if necessary. dx 2. Put f (x) and f 0 (x) terms on one side and x terms on the other side. 1. Rewrite 3. Take integral on both sides. Don’t forget the constants. 4. Simplify to the form f (x) =(a function in x) if possible. • Length, Area and Volume: Figure out the setting of each formula. Z b – Area = (f (x) − g(x)) dx a Z b π(f 2 (x) − g 2 (x)) dx – Volume = a Z b x · (f (x) − g(x)) dx – Volume = 2π a Z b – Length = p (f 0 (t))2 + (g 0 (t))2 dt a Z – Area = 2π a b p g(t) (f 0 (t))2 + (g 0 (t))2 dt – To find the implicit a and b, solve the equation f (x) = g(x). – Sometimes you have to switch x and y. • Practical Problems: Related rates, problems on extremes, work and center of mass.