Math 5070(N. Mackey)
Homework 1
Due: 2pm Thurs 15 Sept 2011.
Please attach this page to the front of your work.
Name:
Do not write in the space below.
#
1a
Component
Max Your Score Comments
Reasoning/Justification
4
1b Organization, Clarity
1c
2
3
4a
Correctness
3
Organization, Clarity
3
Correctness
3
Organization, Clarity
3
Correctness
3
Reasoning/Justification
4
Correctness
3
Reasoning/Justification
4
Correctness
3
4b Reasoning/Justification
4c
3
4
Correctness
3
Organization, Clarity
3
Reasoning/Justification
4
Correctness
4
Total
54
Fall 2011
Some Writing Tips:
• Use complete sentences.
• Use “we are given”, “we want to show that” to distinguish clearly between what is
given and what is to be found.
• Do not put an equals sign between unequal quantities, or quantities you have not yet
proved are equal.
• The equals sign should only be used in an equation.
• Use words like “since”, “therefore” to connect statements or use the implication sign
⇒ to connect equations/inequalities, and indicate the chain of reasoning. For example
Since x2 > 9, we know x > 3 or x < −3.
Alternatively,
x2 > 9 ⇒ (x > 3 or x < −3)
• Avoid the word ‘it’ like the plague. For instance, I’ve often read sentences like
Because it is positive, it is increasing.
instead of
Because f ′ is positive, f is increasing.
• Interpret your answers. Write a concluding sentence or paragraph.
• Read your work. Does the logic flow? Do the sentences scan? Expect to revise. It’s
normal.
2
Due: 2pm Thurs 15 Sept 2011.
1. Consider the function f (x) = 5x7 + 2x − 1.
(a) Argue, using tools from calculus that f has exactly one real root.
(b) Approximate this root to 5 decimal places using bisection.
(c) Repeat using inverse linear interpolation with the same initial bracket.
For parts (b) and (c), and any future problems where this is applicable, organize your
results in a neat table. For example:
Bisection
Iteration
0
1
..
.
Left
Right
Mid
..
.
..
.
..
.
Signs (L, M, R)
+, +, −
Don’t forget to test the final approximation, and report the error.
2. Use the bisection method on sin(x) = 0 to approximate π to four decimal places.
3. Problem 13, p.11 of the text.
4. Problem 14, p.11 of the text.
Some notes on this problem:
• ODE stands for Ordinary Differential Equation
IVP stands for Initial Value Problem. An Initial Value Problem is a differential
equation, together with the value of the unknown function at a specific point.
• Part (b) should read:
dx
, x(0) = x0 , for x(t).
dt
• The differential equations in parts (a) and (b) can be solved by the separation of
variables technique.
Solve the ODE IVP v(t) =
• For part (c) careful consideration of the basic features of the ln function would
be useful.
3