For the function f(x) = 3x2 − 5x − 7.
Find the rate of change of f(x) at x = 3
Find derivative with respect to x:
y'(x)=dy/dx=6x-5
Evaluate derivative at x=3:
y'(3)=6(3)-5=13
To find the rate of change of a function at a given point, we need to find the derivative of the function and evaluate it at that point. Let's find the derivative of the function f(x) = 3x^2 - 5x - 7 first.
To find the derivative of a polynomial function, we can use the power rule. The power rule states that if we have a term of the form ax^n, the derivative is given by d/dx(ax^n) = nax^(n-1).
Applying the power rule to each term of the function f(x) = 3x^2 - 5x - 7:
f'(x) = d/dx(3x^2 - 5x - 7)
= d/dx(3x^2) - d/dx(5x) - d/dx(7)
= 3(2)x^(2-1) - 5(1)x^(1-1) - 0
= 6x - 5
So, the derivative of f(x) is f'(x) = 6x - 5.
To find the rate of change of f(x) at x = 3, we need to evaluate the derivative at x = 3:
f'(3) = 6(3) - 5
= 18 - 5
= 13
Therefore, the rate of change of f(x) at x = 3 is 13.