Math 170B HW #0 Partial Solutions
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1a. f(x) = x^5 - x^4 - 2x^3 + x^2 - 1.

1b. f(x) = xsinx - 1.

1c. f(x) = x^2 - sinx - xe^x.


2. f_1 = x + e^x - cosx. 
   f_2 = cosx - x - e^x.


3a. No because f(0)*f(pi/2) > 0.

3b. Yes because f(-1)*f(0) < 0 and f is continuous on [-1,0].

3c. No because f is not continuous on [-1,1].


4a. p_1 = 2.5, p_2 = 2.75, p_3 = 2.875, p_4 = 2.8125.

4b. The bound of the absolute error of the final approximation 
    (without knowing the exact root) is 0.0625.  The exact absolute 
    error is 0.015927125.


5. The bisection method approximates x = -2 using the starting interval [-3,2].


6a. |p_n - p| <= (b-a)/2^n = 3/2^30.

6b. We need 39 approximations in order to get the error less than 10^-11.


7a. c_1 = pi/8, c_2 = 7pi/32, c_3 = 37pi/128, c_4 = 175pi/512.

7b. The bound on the absolute error of the 4th approximation without knowing the 
    exact root is 81pi/512. (Hint: Use |p_n - p| <= (3/4)^n (b-a). How did I get 
    that?)

7c. We need 50 approximations to achieve an absolute error less than 10^-6.

8a. True.

8b. False.  See Exercise 5.

8c. False.  Change the interval in Exercise 5 to [-3, 4] and see what happens.

8d. False.  |p_n - p| <= (b-a)/2^n.