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

25 min Numerical Methods

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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