Chap5_Bisection - Chapter 5 Roots In this chapter were interested in looking for solutions of equations that look like f(x = 0 where x is real The

Chap5_Bisection - Chapter 5 Roots In this chapter were...

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Chapter 5: RootsIn this chapter we’re interested in looking for solutions of equations that looklikef(x) = 0wherexis real.The solutions to such equations are calledrootsoff(x) = 0. They are alsocalled zeros off(x).What does it mean ifx*is a root off(x) = 0? Examples: Find a root of (a) f ( x ) = 2 x - 4, (b) f ( x ) = cos( x ), (c) f ( x ) = x 2 + x - 2, (d) f ( x ) = x 2 + 3 x - 5, (e) f ( x ) = x 4 - 3 x 3 + 7 x 2 - 6 x - 2
Applications Zero-finding Iterative methods Using fzero Example: Range of a cannonball To what elevation should the cannon be raised to hit the target? R θ * V 0 y x Parameters: g = acceleration of gravity ( ms 2 ) : known V 0 = initial speed ( ms 1 ) : known R = distance to target ( m ) : known θ = required elevation (radians): unknown Determine elevation θ needed to hit target using known values of parameters V 0 , R , and g adapted from notes by D. Aruliah Nonlinear Equations MATH 2070U 5 / 37 Applications Zero-finding Iterative methods Using fzero Example: Range of a cannonball Coordinates of cannonball at time t are ( x ( t ) , y ( t )) Motion of cannonball determined by Newton’s 2nd law braceleftBigg x ′′ ( t )= 0, x ( 0 )= 0, x ( 0 )= V 0 cos θ y ′′ ( t )= g , y ( 0 )= 0, y ( 0 )= V 0 sin θ ODE system integrates directly to yield x ( t )=( V 0 cos θ ) t y ( t )=( V 0 sin θ ) t 1 2 g t 2 adapted from notes by D. Aruliah Nonlinear Equations MATH 2070U 6 / 37
Applications Zero-finding Iterative methods Using fzero Example: Range of a cannonball Want to find time T such that x ( T )= R and y ( T )= 0 If y ( T )= 0, then T = 0 or T = 2 V 0 sin θ g Reject T = 0, so we must have x ( T )=( V 0 cos θ ) parenleftbigg 2 V 0 sin θ g parenrightbigg = R Zero-finding problem: find elevation θ such that f ( θ )= 0, where f ( θ ) : = 2 sin θ cos θ Rg V 0 2 adapted from notes by D. Aruliah Nonlinear Equations MATH 2070U 7 / 37 Applications Zero-finding Iterative methods Using fzero Remarks: Range of a cannonball R θ * V 0 y x f ( θ )= 2 sin θ cos θ Rg V 0 2 = 0 Idealisation: no air resistance No solution if Rg V 0 2 > 1 Solution nonunique (sin & cos periodic) Meaningful θ satisfies 0 < θ < π

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