Solutions of Equations in One Variable PDF

Chapter two: Solutions of Equations in One Variable (f(x)=0 find x) Definition: An equation that includes linear terms is called a linear equation of order n. That is 𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + ⋯ + 𝑎𝑛 𝑥 𝑛 = 𝑏. Definition: An equation that includes nonlinear terms is call...

Chapter two: Solutions of Equations in One Variable (f(x)=0 find x) Definition: An equation that includes linear terms is called a linear equation of order n. That is 𝑎1 𝑥 + 𝑎2 𝑥 2 + 𝑎3 𝑥 3 + ⋯ + 𝑎𝑛 𝑥 𝑛 = 𝑏. Definition: An equation that includes nonlinear terms is called a nonlinear equation. (Nonlinear terms such as 𝑒 𝑥 , ln(𝑥 ) , log(𝑥 ) , cos(𝑥 ) , tan(𝑥 ) , 𝑥 𝑘 , 𝑎 𝑥 , …) Example: 1. The equation 5𝑥 4 + 2𝑥 + 3 = 0 is linear of order 4. 2. The equation 2 sin(𝑥 ) − 3𝑥 −4 = 0 is nonlinear equation (because of sin(𝑥 ) and 𝑥 −4 ). 3. The equation−2𝑒 𝑥 + 2 ln(𝑥 ) − 1 = 0 is nonlinear equation (because of 𝑒 𝑥 and ln(𝑥 )). Method 1. Braketing metod: Definition: Let 𝑓(𝑥) be continuous function and 𝑓 (𝑐) = 0 for any number 𝑐 ∈ 𝐷𝑜𝑚(𝑓), then 𝑐 is called a root of equation 𝑓(𝑥 ) = 0. Remark: A root of the equation 𝑓(𝑥 ) = 0 is also called a zero of the function 𝑓(𝑥) and a solution of the equation 𝑓(𝑥 ) = 0. Recall that 1. The quadratic equation: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏, 𝑐 are real numbers and 𝑎 ≠ 0. −𝒃±√𝒃𝟐 −𝟒𝒂𝒄 2. The quadratic formula: 𝒙=. 𝟐𝒂  5 Example: Show that the equation 𝑥 2 − 𝑥 + 1 = 0 has two roots using 2 Braketing metod? Sol. Let 𝑎 = 1, 𝑏 = 5\2, 𝑐 = 1. So 𝟐 − 𝟓 ± √(𝟓) − 𝟒 𝟐 𝟐 𝒙= 𝟐 1 Thus 𝑥 = 2 and 𝑥 =. 2 5 1 So 𝑥 2 − 𝑥 + 1 = 0 ⟶ (𝑥 − 2) (𝑥 − ) = 0. 2 2 Remarks: If f is continuous and k=0 is between 𝑓(𝑎) and 𝑓 (𝑏), then f has at least one root. Example: Use the intermediate value theorem to show that there is a 𝜋 solution of the equation 2 tan(𝑥 ) = 1 + 𝑥 in the interval [0, ]. 4 Solution: 1 1 Example: Let 𝑓(𝑥 ) = − + 1. Show that there is a number 𝑐 ∈ 𝑥−1 𝑥−4 (1,4) such that 𝑓 (𝑐) = 1. Solution: 1 1 Since 𝑓 (𝑥 ) = 𝑥−1 − 𝑥−4 + 1 is continuous on [2,3]; 3 1 𝑓(2) = > 1, 𝑓(3) =