Some PreCalculus Questions, Thanks.?

Question

Find a possible formular for the ln(5(e)^(.3466x). Show your work.
Solve your formula for y.
Write your answer in the form y=ae^(kx) showing all steps.

For functions 2^x and 3^x, it seems that as delta x gets smaller, the average rate of change approaches a particular decimal. One of the decimals is greater than 1 and the other is less than 1. Using trial and error try to find an exponential function whose average rate of chance, using the intervals of 0.1,0.01,0.001, 0.0001, approaches 1. You should present at least 3 other exponential functions and their tables with the interval and average rate of change

-Not sure what the last question really means or how to do it.

Thanks

Answer 1

I can't make much sense out of the first bit, I'm afraid.

For the second bit, although it doesn't say it here you must be considering the average rate of change near the point x = 0. Start with the function y = 2^x and look at the rate of change at x = 0, using the decreasing sequence given for Δx:
e.g. for Δx = 0.1 we have rate of change
(f(0.1) - f(0)) / (0.1 - 0) = (2^0.1 - 1) / 0.1 = 0.7177 (4 d.p.)
Similarly, we get this table:
2^x
Δx . Rate of change
0.10000 0.7177
0.01000 0.6956
0.00100 0.6934
0.00010 0.69317
0.00001 0.69315

Now do the same for 3^x. We get a similar table (the first entry is computed as (3^(0.1) - 1) / (0.1)):
3^x
Δx . Rate of change
0.10000 1.1612
0.01000 1.1047
0.00100 1.0992
0.00010 1.09867
0.00001 1.09862

You can see that each tables is heading towards a specific figure. The value for 2^x is less than 1, the value for 3^x is more than 1. The question is asking you to construct some more tables like this to try and find a value k where the table for k^x generates the value 1.

You can see that the value for 3^x is closer to 1 than the value for 2^x, so you might start at 2.7:
2.7^x
Δx . Rate of change
0.10000 1.0442
0.01000 0.9982
0.00100 0.9937
0.00010 0.99330
0.00001 0.99326

So it's more than 2.7, try 2.8:
2.8^x
Δx . Rate of change
0.10000 1.0845
0.01000 1.0349
0.00100 1.0301
0.00010 1.02967
0.00001 1.02962

Now you can see 2.7 is about four times closer than 2.8, so try something like 2.72 for the next guess:
2.72^x
Δx . Rate of change
0.10000 1.0524
0.01000 1.0057
0.00100 1.0011
0.00010 1.00068
0.00001 1.00064

So the answer is between 2.7 and 2.72, and it should be about 11 times closer to 2.72 than to 2.7. Use that to pick another value to try, and keep going until you're happy with your accuracy. Another one or two well-chosen values would probably be sufficient.