Consider f(x)=3*2^x find the rate of change between X=1 and X=4
To find the rate of change between x = 1 and x = 4 for the function f(x) = 3 * 2^x, we need to calculate the slope of the function over that interval.
The rate of change, or slope, is determined by the derivative of the function. Let's find the derivative of f(x) first.
Step 1: Compute the derivative of the function f(x) = 3 * 2^x with respect to x.
To find the derivative, we can use the power rule. For a function of the form f(x) = a * b^x, the derivative is given by f'(x) = ln(b) * a * b^x.
Applying the power rule to f(x) = 3 * 2^x, we have:
f'(x) = ln(2) * 3 * 2^x
Step 2: Evaluate the derivative at the given values of x.
We want to find the rate of change between x = 1 and x = 4, so we need to evaluate the derivative at these points.
At x = 1:
f'(1) = ln(2) * 3 * 2^1
At x = 4:
f'(4) = ln(2) * 3 * 2^4
Step 3: Calculate the rate of change.
The rate of change between x = 1 and x = 4 is the difference in values of the derivative at these points.
Rate of change = f'(4) - f'(1) = ln(2) * 3 * 2^4 - ln(2) * 3 * 2^1
We can simplify this expression to get the numerical value of the rate of change.