Fundamental Theorem of Calculus. Help?

Question

So on a quiz, I was asked to find the derivative of the integral from 0 to ln(7) of e^(2x) dx.
A. e^2ln(7) - e^0
B. 0
C. ln(7)
D. e^2ln(7)

When I was taught the first part of the fundamental theorem of calculus, I sort of thought of it as "if you see a zero in the derivative of an integral, f*** that zero and go to the non-zero number. Otherwise, split into two integrals and f*** the two zeroes." With that logic, I put down D. It was somehow wrong.

I am stumped as to how in the world I was wrong. I looked online using a derivative calculator and it said the answer was 48, which is equal to A. But why would I care about e^0 when I thought I was supposed to move on from the zero whenever a zero popped up in a derivative of an integral equation.

If you're one of those people who knows the answer and is willing to share with explanation, feel free to share. If you're one of those who will help me get a better understanding of FTOC (which I have a hard time understanding) and lead me to, but not directly tell me, the correct answer, that's also fine.

ted s · Answer

THE ANSWER TO YOUR QUERY IS " 0 " SINCE YOU HAVE A DEFINITE INTEGRAL...{ sorry : cap lock was on } and thus a number

Alan · Answer

The integral calculators and I were wrong.  
the results   of 
d/dx (  ∫  f(x)dx | 0 to x ) = d/dx ( F(x) - F(0) ) 
since the zero value is a constant , you can forget about 
However, in the case about , you do have to worry 
about the term that still has an x  in x.

integral is    F(x) - F(0)  , but F(0) is a constant 
so when so when you take the 
derivative of a constant it is zero.  
d (F(x))/dx and it returns   f(x) the original function.

But in your case, you are doing this
d/dx ( ∫ f(x)dx | 0 to ln(7) ) = d/dx ( F(ln(7)) - F(0) ) 
both F(ln(7) ) and F(0) are constants

both values are constant 
you are asking for the derivative 
d ( e^(2*ln(7) - e^(0) ) /dx  
Notice there is no x term in your equation 
both e^(2*ln(7)) and -e^(0) are constants 
derivative of a constant is 0 
so as your other answer suggest , the answer is 0

The calculators are taking step by step 
and making a mistake somewhere.

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Fundamental Theorem of Calculus. Help?

2 Answers