Question

What is the slope of the tangent line to the parametric curve x = t^2 + 2t, y = t^2 + 1 at t = 1? (4 points)

Answer 1

ANSWER

$\frac{1}{2}$

EXPLANATION

The given parametric curve has equation:

$x = {t}^{2} + 2t$

and

$y = {t}^{2} + 1$

Let us differentiate each equationtion with respect to t.

$\frac{dx}{dt} = 2t + 2$

and

$\frac{dy}{dt} = 2t$

The slope function is given by:

$\frac{dy}{dx} = \frac{ \frac{dy}{dt} }{ \frac{dx}{dt} }$

This implies that:

$\frac{dy}{dx} = \frac{2t}{2t + 2}$

At t=1,

$\frac{dy}{dx} = \frac{2(1)}{2(1)+ 2}$

$\frac{dy}{dx} = \frac{2}{4}$

$\frac{dy}{dx} = \frac{1}{2}$

The correct choice is A.