What is the best estimate of the instantaneous rate of hange for the function f(x)= square root 8x at the point of x = 8?
1. 2square root 2/ squareroot8
2. 1
3. 2square root 2
4. 1/2
d/dx [ a b] = adb/dx + b da/dx
d/dx (8^n * x^n) = 8^n * n x^(n-1) + x^n * 0
so
d/dx (8 x)^.5 = 8^.5 * .5 x^-.5
= 8^.5 * .5 / 8^.5
= .5
To find the instantaneous rate of change at a point, we need to find the derivative of the function and evaluate it at that point.
Let's start by finding the derivative of f(x) = √(8x).
We can use the power rule for differentiation, which states that if we have a function of the form f(x) = x^n, then the derivative is given by f'(x) = nx^(n-1).
In this case, n = 1/2 and x = 8, so the derivative of f(x) = √(8x) is:
f'(x) = (1/2)(8)(8^(-1/2)) = 4/√8
Now, we can find the instantaneous rate of change by evaluating f'(x) at x = 8:
f'(8) = 4/√8 = 4/(√(4*2)) = 4/(2√2) = 2√2/2 = √2
Therefore, the best estimate of the instantaneous rate of change for the function f(x) = √(8x) at the point x = 8 is √2.
So, the correct answer is option 3. 2√2
To find the instantaneous rate of change of a function at a specific point, we need to calculate the derivative of the function and evaluate it at that point. In this case, we are given the function f(x) = √(8x) and asked for the instantaneous rate of change at x = 8.
To find the derivative of f(x), we can use the power rule. The power rule states that if we have a function of the form g(x) = ax^n, where a is a constant and n is any real number, then the derivative of g(x) is dg(x)/dx = nax^(n-1).
Applying the power rule to our function f(x) = √(8x), we can rewrite it as f(x) = (8x)^(1/2). Using the power rule, we differentiate f(x) with respect to x:
f'(x) = (1/2)(8x)^(1/2 - 1) * 8
Simplifying this expression, we get:
f'(x) = (8/2)(8x)^(-1/2) * 8
= 4(8x)^(-1/2) * 8
= 32(8x)^(-1/2)
= 32 / √(8x)
Now that we have the derivative f'(x), we can evaluate it at x = 8 to find the instantaneous rate of change. Plugging in x = 8, we get:
f'(8) = 32 / √(8 * 8)
= 32 / √64
= 32 / 8
= 4
Therefore, the correct answer is 4.