Let f(x)= (1/b)^x for b>1, and left g(x) = (f'(x))/(f(x))
a) Predict the shape of the graph of g(x).
it will be a straight line, since ln(1/b) will be a constant
b) Test your prediction by exploring two specific cases.
Values of (1/2) and (1/10)
c) Summarize your findings, using words and diagrams.
repeat part a) expect in words and graphs
d) For what value of b will g(x)= -1
it would be undefined, since there is no value of b, and you prove it by taking natural log on both sides and showing that ln -1 is a non real answer
please check the above answers.
(a) Correct.
Also, note that since the function is a constant, it does not contain x as a variable, therefore it is a horizontal line y=-log(b).
(d)For what value of b will g(x)= -1
g(x)=-ln(b)
so
-ln(b)=-1
ln(b)=1
raise to the power of e:
e^(ln(b)) = e^1
b=e (=2.718...)
a) The graph of g(x) will indeed be a straight line. This is because when we differentiate f(x) = (1/b)^x using the power rule, we obtain f'(x) = ln(1/b) * (1/b)^x. The factor ln(1/b) is a constant, so the derivative of f(x) is proportional to f(x), which means that the graph of g(x) will be a straight line.
b) Let's explore two specific cases, (1/2) and (1/10), to test our prediction.
For f(x) = (1/2)^x:
Taking the derivative, we have f'(x) = ln(1/2) * (1/2)^x.
Now, let's calculate g(x) = (f'(x))/(f(x)):
g(x) = (ln(1/2) * (1/2)^x) / ((1/2)^x) = ln(1/2) ≈ -0.6931
For f(x) = (1/10)^x:
Taking the derivative, we have f'(x) = ln(1/10) * (1/10)^x.
Now, let's calculate g(x) = (f'(x))/(f(x)):
g(x) = (ln(1/10) * (1/10)^x) / ((1/10)^x) = ln(1/10) ≈ -2.3026
c) From our calculations, we can see that g(x) is a constant -0.6931 for b = 1/2 and -2.3026 for b = 1/10. This confirms our prediction that g(x) is a straight line, as the value of g(x) does not depend on x.
Graphically, the graph of g(x) will be a horizontal line at the corresponding value of ln(1/b).
d) To find the value of b such that g(x) = -1, we need to solve the equation -1 = ln(1/b).
By taking the natural log on both sides, we have ln(e^-1) = ln(1/b).
Using the property ln(e^x) = x, we simplify it to -1 = ln(b^-1).
Applying the inverse property of ln, we get b^-1 = e^-1.
Taking the reciprocal of both sides, we have b = e.
Therefore, for g(x) = -1, b must be e.