Suppose \( x \) and \( y \) are varying together at a constant rate of change and we know that the point \( (5,3) \) or the values \( x=5 \) and \( y=3 \) are in the relationship. Note: You can interact with the applet by moving the purple " \( X^{\prime \prime} \) on the horizontal axis to represent varying the value of \( x \). Before moving on to the next question, do the following. In the applet above, click on "Show Hypotenuse," and then vary the value of \( x \) from one value to another--do this several times and pay attention to how \( \Delta y \) and \( \Delta x \) are related as \( x \) and \( y \) vary in tandem. a. Determine if the following statements are true or false. i. Select an answer \( \boldsymbol{V} \) The slanted line (hypotenuse) in the applet represents values of \( x \) and \( y \) as they change together. ii. Select an answer \( \boldsymbol{V} \) The values \( x=2 \) and \( y=7 \), or point \( (2,7) \) is in the relationship represented by the above graph (the hypotenuse in the above applet). iii. Select an answer \( \boldsymbol{V} \) If the values of \( x \) and \( y \) are changing together at a constant rate of change of 2 , then \( y \) is always 2 times as large as \( x \). iv. Select an answer \( \boldsymbol{V} \) If the values of \( x \) and \( y \) vary together at a constant rate of change of 2 , then the value of \( \Delta y=2 \Delta x \) for any \( \Delta x \) away from a point \( \left(x_{1}, y_{1}\right) \) on the graph. b. What is the \( y \)-intercept of the above graph? \( y \)-intercept: Preview The value of the \( y \)-intercept represents the value of the Select an answer \( \boldsymbol{V} \) quantity when the Select an answer \( \boldsymbol{V} \) quantity has a value of 0 . c. Define the formula that represents \( y \) in terms of \( x \) for the graph displayed above. Preview
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In this exercise we compare three functions: $$y=2^{x} \quad y=e^{x} \quad y=3^{x}$$ (a) Begin by graphing all three functions in the standard viewing rectangle. The picture confirms three facts that you know from the text: on the positive $x$ -axis, the functions increase very rapidly; on the negative $x$ -axis, the graphs approach the asymptote (which is the $x$ -axis) as you move to the left; the $y$ -intercept in each case is 1. (b) To compare the functions for positive values of $x,$ use a viewing rectangle in which $x$ extends from 0 to 3 and $y$ extends from 0 to $10 .$ Note that the graph of $e^{x}$ is bounded between the graphs of $2^{x}$ and $3^{x}$, just as the number $e$ is between 2 and $3 .$ In particular, the picture that you obtain demonstrates the following fact: For positive values of $x$ $$2^{x}<e^{x}<3^{x}$$. (c) To see the graphs more clearly when $x$ is negative, change the viewing rectangle so that $x$ extends from -3 to 0 and $y$ extends from 0 to $1 .$ Again, note that the graph of $e^{x}$ is bounded between the graphs of $2^{x}$ and $3^{x}$, but now the graph of $3^{x}$ is the bottom (rather than the top) curve in the picture. This demonstrates the following fact: For negative values of $x$ $$3^{x}<e^{x}<2^{x}$$. (d) Explain how the result in part (c) follows from the result in part (b).
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