True or False---Parametrization

Question

1. True or False: The parametric curve x=(3t+4)^2 , y = 5(3t+4)^2 -9 , for 0 <= t <= 3 is a line segment.
2. True or False: A parametrization of the graph of y=ln(x) for x>0 is given by x = e^t , y =t for -infinity < t < infinity.
3. True or False: The line parametrized by x = 7, y = 5t, z = 6 +t is parallel to the x-axis.

Answer 1

1.

There are two questions

a. Is the curve x = (3t+4)², y = 5(3t+4)² - 9 a line?

b. Does the restriction 0 <= t <= 3 yield a curve segment.

a.

It is a line if we can show that its derivative is the same everywhere.

dy/dx = (dy/dt) / (dx/dt)

= (30(3t+4)) / (6(3t+4))

= 5

The derivative is 5, independent of t, therefore the curve is a line.

b.

The restriction does yield a line segment because it is a closed interval,

which projects on the line as a closed contiguous subset of the line;

i.e., a line segment.

2.

There are two questions

a. Is the curve x = e^t, y = t the logarithmic curve?

b. Does it define y for all x>0?

a.

t = ln(x), therefore it is the logarithmic curve

b.

As -∞ < t < ∞ maps on 0 < x < ∞,

y is defined for all x>0.

3.

A line parallel to the z-axis would have constant y and z coordinates, which is not the case here.

The x-coordinate is constant which makes the line perpendicular to the x-axis.