Math1210-001 Lab 8
Spring 2016; due March 31 at beginning of lab
Name : _______________________________ uid number: _________________
To be filled in by the TA:
Total Raw Score:
/170
Total Percent Score:
Your lab score this week will be your percent score plus your 25 points for attendance.
This lab contains sections 3.7-3.9.
n
5
6
7
3
4
1
2
1. Use the Bisection Method to approximate the real root of the given equations on the given interval.
Fill in the given table with all your work/results. Report the final answer correct to two decimal places.
(a) (10 points) f ( x )= x 3 + 4x − 6 = 0 on the interval [1, 2] Final answer: _______________ a n b n m n f  a n
 f  b n
 f ( m n
) n
5
6
7
3
4
1
2
(b) (10 points) f ( x )= 2cos x − sin ( 2x )= 0 on the interval [1, 2]
Final answer: _________________ a n b n m n f  a n
 f  b n
 f ( m n
)
1

2. Use Newton's Method to approximate the indicated root of the given equation, accurate to five decimal places. Fill in the table and answer all questions to show your work. Additionally, show (in some way that does NOT use technology) why you think your initial guess is reasonable.
3 (a) (10 points) f ( x )= 7x + x − 5 = 0
Newton's method formula for x n for this particular problem: x n
= ________________________________________________ n
0
3
4
1
2
5 x n
(initial guess)
Final answer: _________________
Show why you chose your initial guess (without technology).
(b) (10 points) cos ( x )= 2x (You're going to have to figure out what f(x) is here.)
Newton's method formula for x n for this particular problem: x n
= ________________________________________________ n
2
3
0
1
4
5 x n
(initial guess)
Final answer: _________________
Show why you chose your initial guess (without technology).
2

3. (a) (2 points)Sketch the graph of f ( x )= x 1 / 3 .
(b) (1 point) What is the only zero of this function? _____________________
(c) (8 points)Use Newton's Method to approximate this root. I've provided two tables. Fill in both of them, using two different non-zero numbers for the initial guess.
Newton's method formula for x n for this particular problem: x n
= ________________________________________________ n
3
4
5
0
1
2 x n n
3
4
5
0
1
2 x n
(d) (3 points) Did you find the zero you reported in (b)? If not, explain what's happening here.
3

4. Evaluate each of these antiderivatives (a.k.a. indefinite integrals).
(a) (8 points) ∫ ( 3x 2 +
√
π) dx
(b) (10 points)
∫
( 32 x 7 − 24 x 5 + 15 x 3 +
Answer: _______________________________________
4 x 2
) dx
Answer: _______________________________________
(c) (10 points)
∫ 2x 6 x
− x
3 dx
(d) (10 points)
∫
( 1
3 x − 5 / 4 +
2x 3 − 7 x 3
) dx
Answer: _______________________________________
4
(e) (10 points) ∫ ( x 2 − 2 ) 2 dx
Answer: _______________________________________
Answer: _______________________________________

(f) (8 points) ∫ ( sin x − cos x ) dx
(g) (10 points)
∫
( 4x 2 + 1 )( 4x 3 + 3x − 1 ) 4 dx
Answer: _______________________________________
(h) (10 points)
∫ cos x ( 1 − 2sin x ) 3 / 2 dx
Answer: _______________________________________
Answer: _______________________________________
5. (10 points) Find f ( x ) by anti-differentiating as many times as you need to. Pay close attention to what happens with the integration constants.
f ' ' ' ( x )= 3x + 1
5 f ( x ) = ______________________________________________

6. Solve the differential equations subject to the indicated condition.
(a) (10 points) dy dx
=
x y
; y (0) = 2
(b) (10 points) dy dx
= y 3 sin x y + 1
; y (0) = 1
Answer: _______________________________________
Answer: _______________________________________
7. (10 points) From what height above the earth must a ball be dropped in order to strike the ground with velocity -160 feet per second? (Assume that g = 32 feet per second per second and neglect air resistance.)
Answer: _______________________________________
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