Concept explainers
(a)
To draw: Thegraph of equations
(a)
Explanation of Solution
Given information:The given parametric equations are
Graph:
Consider
The parametric equation for
The parametric equation for
The parametric equation for
Now, draw the graph of parametric equations in the same viewing window in the interval
Figure (1)
Interpretation: From the graph it can be observed that the graph of parametric equation for different values of a andb is the right half of a hyperbola that lies in Quadrant I and IV. The value of a gives x -intercept, b describes the shape of hyperbola.
As the value of b increases, the curve of hyperbola goes away from the x -axis and becomes shallower.
(b)
To draw: The graph of equations
(b)
Explanation of Solution
Given information:The given parametric equations are
Graph:
Substitute 2 for a in the equation
Substitute 3 for b in the equation
Now, draw the graph of parametric equations in the interval
Figure (2)
Interpretation: From the graph it can be observed that the graph of parametric equations is a left half of the hyperbola in second quadrant and third quadrant.
(c)
To draw: The graph of equations
(c)
Explanation of Solution
Given information:The given parametric equations are
Graph:
Substitute 2 for a in the equation
Substitute 3 for b in the equation
Now, draw the graph of parametric equations in the interval
Figure (3)
Interpretation: From the graph it can be observed that the graph of parametric equations is a hyperbola. A student must be careful while drawing the graph in an interval that includes
(d)
To prove: The equation
(d)
Explanation of Solution
Given information:The given parametric equations are
Proof:
It is known that the standard
Substitute
Hence, it is proved that
(e)
To evaluate: Repeatpart (a), (b), (c) and (d) for parametric equations
(e)
Explanation of Solution
Given information:The given parametric equations are
Part (a): Consider
The parametric equation for
The parametric equation for
The parametric equation for
Now, draw the graph of parametric equations in the same viewing window in the interval
Figure (4)
From the graph it can be observed that the graph of parametric equations is the top half of a hyperbola that lies in first and second Quadrant. The value of b gives y -intercepts and the values ofadescribe the shape of hyperbola.
As the value of a increases, the curve of hyperbola goes away from the y -axis and becomes shallower.
Part (b): Consider
Substitute 2 for a in the equation
Substitute 3 for b in the equation
Now, draw the graph of parametric equations in the interval
Figure (5)
From the graph it can be observed that the graph of parametric equations is the bottom half of the hyperbola that lies in third and fourth quadrant.
Part (d): It is known that the standard trigonometric identity for
Substitute
Chapter 1 Solutions
Calculus: Graphical, Numerical, Algebraic
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