Question

1.The domain of r(t)=〈sin(t),cos(t),ln(t+1)〉 is

Select one:

a. (−∞,−1]
b. (π,2π)

c. (−∞,−2]
d. (−π,2π)
e. ℝ

f. (−1,∞)

2. The limt→πr(t) where r(t)=〈t2,et,cos(t)〉 is

Select one:

a. 〈π2,eπ,−1〉

b. undefined

c. 〈π2,eπ,1〉
d. π2+eπ
e. 〈π2,eπ,0〉

3.Let v(t)=〈2t,4t,t2〉. Then the length of v′(t) is

Select one:

a. 20‾‾‾√+2t

b. 20+4t2

c. 20+2t‾‾‾‾‾‾‾√

d. 20+4t2‾‾‾‾‾‾‾‾√

e. 6+2t‾‾‾‾‾‾√
f. 1

4. The curvature of a circle centred at the origin with radius 1/3 is

Select one:

a. 1

b. 1/3

c. undefined

d. 3

e. 10

Answer 1

1.

sin(t) is defined for all real values of t.

cos(t) is defined for all real values of t.

ln(t+1) is defined when

Thus, ln(t+1) is defined on the interval (-1, ).

Hence, the domain of r(t) = 〈sin(t),cos(t),ln(t+1)〉 is

2.

Given that

Hence, the limit is

3.

Given that

The length of v′(t) is

4.

The parametrization of the circle centered at the origin with radius 1/3 is

The unit tangent vector T(t) is defined as

Hence, the curvature of the circle centered at the origin with radius 1/3 is