Question 1

Given the parametric equations 
$$ x = f(t) = t^2$ + 4t^2$ $$
$$ y = g(t) = t^2$ $$
for $$ t = -2 $$, what is the equation of the tangent line at this point?

Accepted Answer

$$ y - 4 = \frac{1}{8}(x - 20) $$

Question 2

For the parametric equations 
$$ x = f(t) = t^2$ + 4t^2$ $$
$$ y = g(t) = t^2$ $$
find the value(s) of $$ t $$ where the tangent line is horizontal.

Accepted Answer

$$ t = 0 $$

Question 3

For the parametric equations 
$$ x = f(t) = t^2$ + 4t^2$ $$
$$ y = g(t) = t^2$ $$
find the value(s) of $$ t $$ where the tangent line is vertical.

Accepted Answer

There are no values of $$ t $$ where the tangent is vertical.

Question 4

Given the parametric equations 
$$ x = f(t) = t^2$ + 4t^2$ $$
$$ y = g(t) = t^2$ $$
for $$ t = -2 $$, what is the slope of the tangent line?

Accepted Answer

$$ \frac{1}{8} $$

Question 5

For the parametric curve $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, what is $$ \frac{dy}{dx}  in $$terms of $$ t $$?

Accepted Answer

$$ \frac{2t}{10t} = \frac{1}{5} $$

Question 6

Given $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, what is the point on the curve when $$ t = -2 $$?

Accepted Answer

$$ (20, 4) $$

Question 7

For the parametric equations $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, what is $$ \frac{dx}{dt} $$?

Accepted Answer

$$ 10t $$

Question 8

For the parametric equations $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, what is $$ \frac{dy}{dt} $$?

Accepted Answer

$$ 2t $$

Question 9

Given $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, what is the equation of the tangent line at $$ t = 0 $$?

Accepted Answer

$$ y = 0 $$

Question 10

For the parametric equations $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, what is the geometric shape of the curve traced out as $$ t $$ varies?

Accepted Answer

A parabola

Question 11

Given $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, eliminate the parameter $$ t  to $$find the Cartesian equation of the curve.

Accepted Answer

$$ x = 5y $$

Question 12

For the parametric equations $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, what is the area enclosed by the curve and the x-axis for $$ t $$ from $$ 0  to$$ $$ 2 $$? (Hint: Use $$ \int y \frac{dx}{dt} dt )$$

Accepted Answer

$$ 32 $$

Question 13

Given $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, what is the value of $$ \frac{dy}{dx}  at$$ $$ t = 1 $$?

Accepted Answer

$$ \frac{1}{5} $$

Question 14

For the parametric equations $$ x = f(t) = t^2$ + 4t^2$ $$, $$ y = g(t) = t^2$ $$, what is the equation of the tangent line at $$ t = 1 $$?

Accepted Answer

$$ y - 1 = \frac{1}{5}(x - 5) $$

Parametric Equations and Tangent Lines: Calculus II Homework Study Notes

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