Question 1

Given the parametric equations 
$$ x = f(t) = t^3$ - 4t^2$ $$
$$ y = g(t) = t^2$ $$
find the value of $$t$$ where the tangent line to the curve is **vertical**.

Accepted Answer

$$ t = 2 $$

Question 2

For the parametric curve 
$$ x = t^3$ - 4t^2$ $$
$$ y = t^2$ $$
find the equation of the tangent line at $$ t = -2 .$$

Accepted Answer

$$ y + 4 = -\frac{1}{4}(x + 24) $$

Question 3

Given the parametric equations 
$$ x = t^3$ - 4t^2$ $$
$$ y = t^2$ $$
find the value of $$t$$ where the tangent line is **horizontal**.

Accepted Answer

$$ t = 0 $$

Question 4

For the parametric curve 
$$ x = t^3$ - 4t^2$ $$
$$ y = t^2$ $$
what is the slope of the tangent line at $$ t = 2 $$?

Accepted Answer

$$ \infty $$

Question 5

Given the parametric equations 
$$ x = f(t) = t^3$ - 4t^2$ $$
$$ y = g(t) = t^2$ $$
find the coordinates of the point where the tangent is horizontal.

Accepted Answer

$$ (0, 0) $$

Question 6

For the parametric curve 
$$ x = t^3$ - 4t^2$ $$
$$ y = t^2$ $$
what is the equation of the tangent line at $$ t = 0 $$?

Accepted Answer

$$ y = 0 $$

Question 7

Given the parametric equations 
$$ x = f(t) = t^3$ - 4t^2$ $$
$$ y = g(t) = t^2$ $$
find $$ \frac{dy}{dx}  in $$terms of $$t. $$

Accepted Answer

$$ \frac{2t}{3t^2$ - 8t} $$

Question 8

For the parametric curve 
$$ x = t^3$ - 4t^2$ $$
$$ y = t^2$ $$
what is the geometric significance of the points where $$ \frac{dx}{dt} = 0 $$?

Accepted Answer

The tangent line is vertical.

Question 9

Given the parametric equations 
$$ x = f(t) = t^3$ - 4t^2$ $$
$$ y = g(t) = t^2$ $$
find the coordinates of the point where the tangent is vertical.

Accepted Answer

$$ (-8, 4) $$

Question 10

For the parametric curve 
$$ x = t^3$ - 4t^2$ $$
$$ y = t^2$ $$
what is the equation of the tangent line at $$ t = 2 $$?

Accepted Answer

$$ x = -8 $$

Question 11

Given the parametric equations 
$$ x = f(t) = t^3$ - 4t^2$ $$
$$ y = g(t) = t^2$ $$
find the coordinates of the point at $$ t = -2 .$$

Accepted Answer

$$ (24, 4) $$

Question 12

For the parametric curve 
$$ x = t^3$ - 4t^2$ $$
$$ y = t^2$ $$
what is the slope of the tangent line at $$ t = -2 $$?

Accepted Answer

$$ -\frac{1}{4} $$

Question 13

Given the parametric equations 
$$ x = f(t) = t^3$ - 4t^2$ $$
$$ y = g(t) = t^2$ $$
find the value(s) of $$t$$ where the curve intersects the x-axis.

Accepted Answer

$$ t = 0 $$

Question 14

For the parametric curve 
$$ x = t^3$ - 4t^2$ $$
$$ y = t^2$ $$
what is the geometric meaning of the points where $$ \frac{dy}{dt} = 0 $$?

Accepted Answer

The tangent line is horizontal.

Question 15

Given the parametric equations 
$$ x = f(t) = t^3$ - 4t^2$ $$
$$ y = g(t) = t^2$ $$
find the area enclosed by the curve and the x-axis for $$ t $$ from $$ 0  to$$ $$ 2 .$$

Accepted Answer

$$ \$$int_0$$^2 (t^3$ - 4t^2$) 2t dt $$

Question 16

For the parametric curve 
$$ x = t^3$ - 4t^2$ $$
$$ y = t^2$ $$
what is the value of $$ \frac{dy}{dx}  at$$ $$ t = 1 $$?

Accepted Answer

$$ -2 $$

10.2 Homework

Study Guide - Practice Questions

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