1) You are using the Bisection Method to approximate a root of f(x) = x3 -72x - 220 within the interval [9,10]. What is the new interval after one iteration?  a) [8,9] b) [9,9.5] c) [9.5,10] d) [9.25,9.75] 2) Given the data points: (1,1), (-2,1), (3,2)  What is the slope m at the least-square regression line y= mx+b? a) 23/38 b) 15/38 c) 19/23 d) 38/23 3) What is the derivative used in the Newton-Raphson method for f(x) = x2 + 8x +15? a) x+8 b) 2x+8 c) x2+15 d) 2x 4) Which of the following best describes the Newton-Raphson method? a) It uses fixed-point iteration b) It simplifies the equation algebraically c) It always gives exact answers d) It approximates roots using a tangent line and iteration 5) In the bisection method, what condition must be satisfied to begin the method? a) f(x1) = f(x2) b) f(x1) ⋅ f(x2)>0 c) f(x1) =0 d) f(x1) ⋅ f(x2)<0 6) Which method uses linear interpolation to estimate the root? a) Newton-Raphson Method b) Bisection Method c) False-Position Method d) Incremental Search 7) In the graphical method, where is the root located? a) Where the graph intersects the y-axis b) Where the function equals zero c) Where the function has maximum slope d) Where the function is undefined 8) What is the purpose of the least-squares regression method? a) To eliminate all errors b) To solve equations graphically c) To fit a line that minimizes the sum of the squared errors d) To find roots of non-linear equations 9) ​Given the least-squares regression equation y = mx+b, what does m represent? a) Y-intercept b) Mean of y c) Error margin d) Slope 10) If y = xn , then the derivative dy/dx is: a) n ⋅ xn+1 b) xn-1 c) n ⋅ xn-1 d) n ⋅ xn 11) In the bisection method, if f(0.5) = 0.107 and f(10 = -0.632, what can we conclude? a) Root lies between 0.5 and 1 b) Root lies before 0 c) No root in the interval d) Root lies exactly at 0.5 12) ​How does the False-Position method calculate the next approximation? a) Midpoint between left and right b) Newton-Raphson formula c) Linear interpolation between xL and xR d) Graphical estimation 13) GIVEN: (-1,0), (0,2), (1,4), (k,5) with m= 1.7, b= 1.9 What is the value of k that satisfies the regression line? a) 1 b) 2 c) 3 d) 0 14) In curve fitting, what does the equation y= mx + b represent? a) Derivative of a function b) Regression line c) Equation for interpolation d) Second-order polynomial 15) GIVEN: f(x) = e-x - x  According to the presentation, what is the approximate root of f(x) = e-1 -x, using the bisection method? a) 0.3156 b) 0.5671 c) 1.2310 d) 0.8042

Numerical Methods

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