Question

t^2+4t+4 t^2-2t+1 Consider the parametric curve a. Find points on the parametric curve where the tangent lines are vertical. b. Find points on the parametric curve where the tangent lines are horizontal. c. Find the interval(s), in terms of , where the parametric curve is concave upward. d. Find the interval(s), in terms of , where the parametric curve is concave downward.

Answer 1

a)
`t = 1`, b)
`t = -2`, c) Concave upward:
`(-\infty, 1)`. No intervals being concave downward.

Explanation:

a) Tangent line is vertical if slope is undefined, whose points are associated with discontinuities of the function. Rational functions have discontinuities associated with the denominator:

`f(t) = (t^(2)+4\cdot t +4)/(t^(2)-2\cdot t + 1)`

By factorizing each component, the function is re-arranged as:

`f(t) = ((t+2)^(2))/((t-1)^(2))`

There is no avoidable discontinuities. The only point where tangent line is vertical is
`t = 1`.

b) Tangent line is horizontal if slope is equal to zero, whose point is associated to the points that makes function equal to zero. Rational functions have horizontal tangent lines if numerator is equal to zero and denominator is different to zero. Hence, the only point where tangent line is horizontal is
`t = -2`.

c) An interval is concave upward if exist an absolute minimum inside, which can be found by the help of the First and Second Derivative Tests.

`f'(t) = (2\cdot (t+2)\cdot (t-1)^(2)-2\cdot (t-1)\cdot (t+2)^(2))/((t-1)^(4))`

`f'(t) = 2\cdot (t+2)\cdot (t-1)\cdot \left[(t-1-t-2)/((t-1)^(4))\right]`

`f'(t) = -(6\cdot (t+2)\cdot (t-1))/((t-1)^(4))`

`f'(t) = -(6\cdot (t+2))/((t-1)^(3))`

The only critical point is:

`-6\cdot (t+2) = 0`

`t = -2` (which coincides with the result of point b)

`f''(t) = -6\cdot \left[((t-1)^(3)-3\cdot (t-1)^(2)\cdot (t+2))/((t-1)^(6))\right]`

`f''(t) = -6\cdot \left[(1)/((t-1)^(3))-(3\cdot (t+2))/((t-1)^(4)) \right]`

The value associated with the critical point is:

`f''(-2) = (2)/(9)`

Which means that critical point is an absolute minimum, and, consequently, the interval that is concave upward is
`(-\infty, 1)`. There is no absolute maximums and, therefore, there is no interval that is concave downward.