2009 S Test1 solution

34 views4 pages
10 Apr 2012
School
Department
Course
Professor
MAT137: Term Test 1 Answers
1. Calculate tan π
12 exactly.
Double angle formulas are,
sin2θ=1cos(2θ)
2cos2θ=1 + cos(2θ)
2
Use θ=π
12 to get,
tan2π
12=1cos π
6
1 + cos π
6=13
2
1 + 3
2
=23
2 + 3=1
2 + 32
Since π
12 is in quandrant I, tan π
12 >0. Implies tan π
12 =1
2+3.
2. Let f(x) = x2+ 5 with domain(f)=(−∞,+), g(x) = 1xwith domain(g)=(−∞,1]
and h(x) = 2 cos(x) with domain(h) = [0,2π]. Give the formula and domain of fgh(x).
For h(x) to be in domain(g), need 2 cos(x)1cos(x)1
2. Note cos π
3=1
2. By
inspection of the unit circle, cos(x)1
2for exactly,
xhπ
3,2ππ
3i.
For g(x) to be in domain(f), do not need any further restriction.
fgh(x) = f1h(x) = (1 h(x)) + 5, so we have,
fgh(x) = 6 2 cos(x) with domain xhπ
3,2ππ
3i.
3. Prove that for all n1,
1
1·2+1
2·3+1
3·4+··· +1
n(n+ 1) =n
n+ 1.
Let S(n)denote the previous equation.
Base 1
1(1+1) =1
2=1
1+1 . Therefore S(1) is true.
Hypo Let statement S(n)be true for some n1.
Unlock document

This preview shows page 1 of the document.
Unlock all 4 pages and 3 million more documents.

Already have an account? Log in

Get access

Grade+
$10 USD/m
Billed $120 USD annually
Homework Help
Class Notes
Textbook Notes
40 Verified Answers
Study Guides
1 Booster Class