This is a problem set I do not undestand
Let y1 = a^x, y2 = NDER y1, y3 = (y2/y1), and y4 = e^(y3)
a) Descirbe the grapgh of y4 for a = 2,3,4,5. Generalize your description to an arbitrary a > 1.
b) Descibe the graph of y3 for a = 2,3,4,5. Compare a table of values for y3 for a = 2,3,4,5 with ln (a). Generalize your description to an a > 1.
c)Explain how parts (a) and (b) support the statement
(d/dx)a^x = a^x if and only if a = e
d) Show algebraically that y1 = y2 if and only if a = e
To understand this problem set, let's break it down step by step:
a) In this part, you are asked to describe the graph of y4 for different values of a.
To do this, you need to evaluate y4 using the given equation and plot the graph for each value of a. Here's how you can find the graph for a particular value of a:
- Substitute the given value of a into the equation: y1 = a^x.
- Find the derivative of y1 with respect to x to obtain y2.
- Calculate y3 by dividing y2 by y1.
- Evaluate y4 by raising Euler's number e to the power of y3.
- Plot the graph of y4 using a graphing tool or by calculating several points.
Repeat this process for each value of a (2, 3, 4, and 5). Note any patterns or general characteristics of the graphs. Then, generalize your observations for any value of a greater than 1.
b) In this part, you are asked to describe the graph of y3 for different values of a.
Similarly to part a, start by substituting each value of a into the equation.
- Substitute each value of a into the equation: y1 = a^x.
- Find the derivative of y1 with respect to x to obtain y2.
- Calculate y3 by dividing y2 by y1.
- Observe and describe the graphs of y3 for each value of a.
- Compare each value of y3 with ln(a) by constructing a table of values for y3 and comparing them with ln(a).
- Generalize your observations for any value of a greater than 1.
c) This part asks you to explain how parts (a) and (b) support the statement: (d/dx) a^x = a^x if and only if a = e.
To answer this, you need to analyze the observations from parts (a) and (b).
- Based on the graphs of y4 and y3, observe any instances where the derivative of y1 (y2) is equal to y1 itself (a^x).
- Look for values of a where this occurs. Determine if there is a consistent pattern.
- Consider the relationship between a and e in these instances. Do you see a connection between a and e that supports the given statement?
- Use your findings to explain how parts (a) and (b) provide evidence for the statement.
d) This part asks you to show algebraically that y1 = y2 if and only if a = e.
To demonstrate this:
- Write the equation for y1 and y2 explicitly using the given expressions.
- Set y1 equal to y2 and solve for a.
- Show that the equation is only true when a = e.
- Use logical reasoning to explain the process and support the claim made in this part.
Remember, it's important to provide detailed explanations and evidence based on your calculations and observations.