Let f(x)=e^x. Compute the average rate of change over the interval [ln3, ln4].
average f'
f'=f(final)-f(initial) / (ln4-ln3)
avg f'= (4-3)/(ln4-ln3)
luckily, f' = f
To compute the average rate of change of a function over an interval, we need to use the formula:
Average rate of change = (f(x2) - f(x1))/(x2 - x1)
In this case, the function is f(x) = e^x, and the interval is [ln3, ln4].
Step 1: Find the values of f(x2) and f(x1)
Let's find the values of f(x2) and f(x1) using the given function.
f(x2) = f(ln4) = e^(ln4)
= 4
f(x1) = f(ln3) = e^(ln3)
= 3
Step 2: Find the values of x2 and x1
The interval is [ln3, ln4], so x2 = ln4 and x1 = ln3.
Step 3: Compute the average rate of change
Now, we can substitute the values we found into the average rate of change formula.
Average rate of change = (f(x2) - f(x1))/(x2 - x1)
= (4 - 3)/(ln4 - ln3)
Step 4: Simplify the expression
The natural logarithm of a number corresponds to the exponent e raised to that value. So, ln4 = ln(2 * 2) = ln(2) + ln(2) = 2ln2. Similarly, ln3 = ln(3).
Therefore, the average rate of change is:
Average rate of change = (4 - 3)/(2ln2 - ln3)
Simplify the denominator:
= (4 - 3)/(ln2^2 - ln3)
= 1/(2ln2 - ln3)
So, the average rate of change over the interval [ln3, ln4] is 1/(2ln2 - ln3).