Test 2 Key Concepts - Test 2 Key Concepts Newtons Method 4.8 If given a function f(x use x(n 1)=xn(f(xn f(xn Eulers Method 9.2 If given the initial

Test 2 Key Concepts - Test 2 Key Concepts Newtons Method...

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Unformatted text preview: Test 2 Key Concepts: Newton’s Method 4.8 If given a function, f(x) use: x(n+1)=xn-­‐ (f(xn) / f’(xn)) Euler’s Method 9.2 If given the initial value : y’=F(x,y), y(x0)=y0, step-­‐size, h, use: Yn=yn-­‐1+hF(xn-­‐1,yn-­‐1) Xn=xn-­‐1+h *For n=1,2,3… Separable Differential Equations 9.3 If given dy/dx= g(x)f(y) : Separate equation into: dy/f(y)=g(x)/dx Then integrate to get equation into correct form Monotonic Sequence Theorem 11.1 • Sequence must be increasing/decreasing: o Start with a base case to determine if increasing or decreasing o Prove using induction ! want an+1> an or an+1<an • Sequence must converge to a value: o Set the limit as an approaches infinity = L o Therefore the limit as an+1 approaches infinity = L ! Solve for L Alternations Series Estimation 11.5 • Check that a function passes the alternating series test • Steps To Find Error o Determine an o |error|=an ! Solve for n Tricks/Tips: • If a series contains (-­‐1)^n , check for convergence using alternating series test • If a series contains both x and n, check for convergence using ratio tests • If a series contains a function to the nth power, check for convergence using root tests • A harmonic series is always divergent ...
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