Determine the instantaneous rate of change at x=2 for the function defined by m(x) = -x^2 +2x
If you know calculus
dm/dx = -2 x + 2 = -4 +2 = -2
a minus times a minus = a positive
-x^2 +2x = 4 + 4 = 8
??
-x^2 ≠ (-x)^2
If with algebra
m(x+dx) = -(x+dx)^2 + 2 x + 2 dx
= -x^2 -2 x dx -dx^2 + 2 x + 2 dx
m(x) = -x^2 +2x
subtract
change in m = -2 x dx -dx^2 + 2 dx
slope = ( -2 x dx -dx^2 + 2 dx)/dx
= - 2 x - dx + 2
for instantaneous slope find limit as dx --->0
-2x + 2
when x = 2
-4 + 2 = -2 (but of course we knew that)
To determine the instantaneous rate of change at x = 2 for the function m(x) = -x^2 + 2x, we need to find the derivative of the function and evaluate it at x = 2.
Step 1: Find the derivative of m(x) using the power rule of derivatives. The power rule states that the derivative of x^n is n*x^(n-1).
m'(x) = d/dx (-x^2 + 2x)
= -2x + 2
Step 2: Evaluate the derivative at x = 2.
m'(2) = -2(2) + 2
= -4 + 2
= -2
Therefore, the instantaneous rate of change at x = 2 for the given function is -2.
Explanation:
To find the instantaneous rate of change, we need to find the derivative. The derivative of a function tells us the rate at which the function is changing at any given point. In this case, we are interested in finding the rate of change at x = 2.
To find the derivative of the given function, we use the power rule, which states that the derivative of x^n is n*x^(n-1). Applying this rule, we differentiate each term of the function.
After finding the derivative, we have the derivative function m'(x) = -2x + 2. This represents the rate of change of the original function at any given point.
To find the instantaneous rate of change at x = 2, we evaluate the derivative function at that specific value by substituting x = 2. The result is -2, indicating that the function is changing at a rate of -2 at x = 2.