To sketch: The parabolas
Explanation of Solution
Derivative rules:
(1) Power Rule:
(2) Constant multiple rule:
(3)
(4)
Result used:
The equation of the tangent line at
where, m is the slope of the tangent line at
Graph:
The graph of two parabolas
From Figure 1, it is observed that there may be a line that is tangent to both the parabolas.
It is required to find the equation of the tangent line to the parabolas.
Calculation:
Consider the parabolas
Choose the point P
Suppose the slope of the required tangent line passes through the points P
The derivative of parabola
Apply the power rule (1) and simplify the terms,
Thus, the derivative of
Therefore, the slope of the tangent to
The derivative of parabola
Apply the derivative rules (1), (2), (3) and (4),
Thus, the derivative of
Therefore, the slope of the tangent to
Since the required equation of the tangent is linear from
From equations (2) and (3),
From equations (3) and (4),
Substitute
Add 2 on both sides and obtain the value of b.
Substitute the value
For
Substitute
Therefore, the equation of the tangent line to the parabolas is
Chapter 3 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition