To determine
(a)
How is the number e defined.
Answer
e=limx→∞(1+1n)n.
Explanation
Given: A number e.
The number e can be defined a few ways. e is defined in this book as the number such that
limx→∞en−1n=1.
Later on, one can observe:
e=∑n=0∞1n!=limx→∞(1+1n)n
Conclusion: The number e is defined as e=limx→∞(1+1n)n.
To determine
(b)
Express e as a limit.
Answer
e=limx→∞(1+1n)n.
Explanation
Given: The number is e.
e is defined as a limit.
limx→∞en−1n=1
e=∑n=001n!=limx→∞(1+1n)n
Conclusion: Hence, e as a limit is e=limx→∞(1+1n)n.
To determine
(c)
Why is the natural logarithm function y=ex used, more often in calculus than the other exponential function y=bx.
Answer
The natural exponential function is its own derivative.
Explanation
Given: y=ex,y=bx.
When one use axx however, the derivative would be axlna, which adds another function lna
This make take its integral slightly harder as well.
So y=ex is more used than y=bx.
Conclusion: Hence, the natural exponential function is its own derivative other exponential function are not. So y=ex is more used than y=bx.
To determine
(d)
Why is the natural logarithm function y=lnx used more often in the other logarithmic function y=logx.
Answer
The natural exponential function is its own derivative. Other exponential function is not.
Explanation
Given: y=lnx and y=logbx.
Well the good thing about ex is that the ‘ln’ function is based off of it. So lne=1,lna=??
One would need a calculator for doing that, when one have a calculator problem, such as taking the derivative of ex, it would be itself, ex=lne, which is simply ex because lne=1
Which mean its integral is also ex.
Conclusion: Hence, the natural exponential function is its own derivative, other exponential function is not so y=lnx is more used other than u=logbx.