Q. 2(a) - Calculus

Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 1)$$ that trace the unit circle exactly once clockwise, starting at the point $$(1, 0)$$.

(b) Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 2π)$$ that trace the circle centered at $$(2, −3)$$ with radius 5 exactly once counterclockwise, starting at the point $$(7, −3)$$.

(c) Parametric equations $$x = f(t), y = g(t)$$ whose graph is not the graph of a function $$y = f(x)$$.

Step-by-Step Solution

Answer

(a) $$(cost, -sint)$$ 

(b) $$(2+5cost, 5sint-3)$$

(c) $$(rcost, rsint)$$ 

Step 1. Given Information

Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 1)$$ that trace the unit circle exactly once clockwise, starting at the point $$(1, 0)$$.

Step 2. Explanation

The constructed example can be given as, $$(cost, -sint)$$

Step 3. Given Information

Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 2π)$$ that trace the circle centered at $$(2, −3)$$ with radius 5 exactly once counterclockwise, starting at the point $$(7, −3)$$. 

Step 4. Explanation

The constructed example can be given as, $$(2+5cost, 5sint-3)$$ 

Step 5. Given Information

Parametric equations $$x = f(t), y = g(t)$$ whose graph is not the graph of a function $$y = f(x)$$. 

Step 6. Explanation

The constructed example can be given as, $$(rcost, rsint)$$