Training Numerical Methods Placement Test Practice — Numerical Methods
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Placement Test Practice — Numerical Methods

25 min Numerical Methods
This practice test covers bisection and Newton's methods for root finding, Lagrange interpolation, trapezoidal and Simpson's rules for integration, and Euler's and RK4 methods for ODEs. Apply each algorithm step by step, estimate errors, and compare convergence rates.

Placement Test Practice — Numerical Methods

Practice Test — 20 Questions

1. Bisection: how many steps for error $< 0.0001$ on $[0,1]$?
2. Newton's: $f(x)=x^3-2$, $x_0=1$. Find $x_1$.
3. Trapezoidal: $\int_0^1 x\,dx$, $n=2$.
4. Simpson's: $\int_0^1 x\,dx$, $n=2$.
5. Euler: $y'=2y$, $y(0)=1$, $h=0.1$. Find $y_1$.
6. What is quadratic convergence?
7. Linear interpolation: $(0,0)$, $(4,8)$. $f(3) \approx$?
8. Bisection requires $f$ to be ___.
9. Error order of Simpson's rule?
10. RK4 needs how many $k$-evaluations per step?
11. Secant vs. Newton: advantage?
12. Trapezoidal error halves when $n$ is ___.
13. Name two methods for root-finding.
14. What is a divided-difference table?
15. Euler's method is derived from?
16. Simpson's rule approximates $f$ with ___.
17. Can numerical methods give exact answers?
18. Newton's: two steps for $f(x) = x^2-4$, $x_0 = 3$.
19. What is truncation error?
20. Bisection first step: $f(x)=x-\cos x$, $[0,1]$.
Show Answer Key

1. $2^n > 10000$; $n \geq 14$

2. $x_1 = 1-(-1)/3 = 4/3 \approx 1.333$

3. $h=0.5$; $T = 0.25[0+2(0.5)+1] = 0.5$. Exact.

4. $0.5$ (exact for linear functions)

5. $y_1 = 1+0.1(2)=1.2$

6. Error squared each step: digits of accuracy double

7. Slope $= 2$; $f(3) = 6$.

8. Continuous on $[a,b]$ with a sign change

9. $O(h^4)$

10. $4$

11. No derivative needed

12. Doubled (error $\propto 1/n^2$)

13. Bisection, Newton's (also secant, fixed-point)

14. Recursive table of function value differences for Newton interpolation

15. Truncating the Taylor series after the linear term

16. Parabolas (quadratic arcs)

17. Sometimes (e.g., Simpson's for polynomials $\leq$ degree 3)

18. $x_1 = 3-5/6 \approx 2.167$; $x_2 \approx 2.167 - 0.694/4.333 \approx 2.007$

19. Error from approximating an infinite process (series, derivative) by a finite one

20. $f(0)=-1<0$, $f(1)=0.46>0$. $c=0.5$; $f(0.5)=-0.378<0$ → $[0.5,1]$

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