let f(x) = (x)^(1/2)
Compute the difference quotient for f(x)at a=37 and h=33
Just plug and chug:
(f(x+h) - f(x)) / h
(sqrt(37+33)-sqrt(37))/33
(sqrt(70) - sqrt(33))/33
(8.366-5.744)/33 = 0.079
To compute the difference quotient for the function f(x) = (x)^(1/2) with a value of a = 37 and h = 33, we first need to understand what the difference quotient represents.
The difference quotient measures the average rate of change of a function over a small interval. It provides an approximation to the derivative of the function at a particular point.
The formula for the difference quotient is:
f'(a) = (f(a + h) - f(a)) / h
where f'(a) represents the derivative of f at the point a, f(a) is the value of f at a, f(a + h) is the value of f at a + h, and h is the small interval.
Now, let's substitute the values into the formula:
f'(37) = (f(37 + 33) - f(37)) / 33
We need to find f(37 + 33) and f(37) to compute the difference quotient.
To find f(37 + 33), substitute x = 37 + 33 into the function:
f(37 + 33) = (37 + 33)^(1/2) = 70^(1/2) = √70
To find f(37), substitute x = 37 into the function:
f(37) = 37^(1/2) = √37
Now, we can substitute these values back into the difference quotient formula:
f'(37) = (√70 - √37) / 33
So, the difference quotient for f(x) at a = 37 and h = 33 is (√70 - √37) / 33.