Discuss the curve using the guidelines of Section 3.5.
To discuss: The curve using the guidelines of Section 3.5.
The curve is .
Even function: The function f is said to be even if , for all x in D.
Odd function: The function f is said to be odd if , for all x in D.
Periodic function: The function f is said to be periodic if , for all x in D, where p is a positive constant.
Horizontal Asymptote: If either , then the line is a horizontal asymptote of the curve .
Vertical Asymptote: The line is a vertical if at least one of the following statements is true.
Concave upward and concave downward:
(a) If for all x in I, then the graph of f is concave upward on I.
(b) If for all x in I, then the graph of f is concave downward on I.
A point P on a curve under the function is called an inflection point if f is continuous and curve changes from concave upward to concave downward or from concave downward to concave upward at P.
The given function , defined on .
Thus, the domain will be .
The y intercept is given by .
That is, the y intercept is .
The x intercept is determined by .
That is, implies that .
Notice that the exponential function is never zero.
Thus, the only possibility is .
That is, .
Thus, the x intercepts are .
Check the symmetry of as follows.
Since and , there is no symmetry.
The asymptotes are found as follows.
The value of is not a finite value.
Therefore, by the definition, there is no horizontal asymptote.
Similarly, there is no value for such that at least one of the following statements is true.
Therefore, by the definition, there is no vertical asymptote.
Obtain the derivative of .
That is, . (1)
If , then .
Since the exponential function is always positive, .
This happens for the interval,
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