(a)
To find: The equation of the tangent line to the given equation at the point.
(a)
Answer to Problem 30E
The equation of the tangent line to the equation
Explanation of Solution
Given:
The curve with equation
The point is
Derivative rules:
(1) Chain rule: If
(2) Product rule: If
Formula used:
The equation of the tangent line at
Here, m is the slope of the tangent line at
Calculation:
Obtain the equation of tangent line to the given point.
Differentiate the above equation implicitly with respect to x,
Apply the chain rule (1) and simplify the terms,
Therefore, the derivative of the equation is
The slope of the tangent line at
Thus, the slope of the tangent line at
Substitute
Therefore, the equation of the tangent line to the equation
(b)
To find: The points if the curve has horizontal tangent.
(b)
Answer to Problem 30E
The curve has horizontal tangent at
Explanation of Solution
Given:
The curve with equation
The derivative of the equation
Calculation:
Obtain the point if the curve has horizontal tangent.
Note that, the curve horizontal tangent if
Suppose that
Rewrite the given equation as follows,
Substitute
The derivative of the equation does not exist at
Substitute
Therefore, the curve has horizontal tangent at
(c)
To sketch: The curve and tangent line to the given point.
(c)
Explanation of Solution
Graph:
Using online graphic calculator to draw the curve and the tangent line as shown below in Figure 1,
From Figure 1, it is observed that the line
Chapter 3 Solutions
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