Write a difference quotient that can be used to estimate the instantaneous rate of change of y=x^2 at x=-3.
Please explain in detail please and thank you.
Review the section in your text. The difference quotient is
(f(-3+h) - f(-3))/h
= ((-3+h)^2 - (-3)^2)/h
= (9-6h+h^2 - 9)/h
= -6+h
To find the instantaneous rate of change of a function at a specific point, we can use the difference quotient. The difference quotient is a mathematical expression that estimates the slope of a curve at a given point.
The difference quotient for a function y = f(x) at a point x = a is given by:
(f(a + h) - f(a)) / h
In this case, we want to estimate the instantaneous rate of change of the function y = x^2 at x = -3. So, we substitute a = -3 into the difference quotient:
(f(-3 + h) - f(-3)) / h
Now, let's substitute the function y = x^2 into the difference quotient:
((-3 + h)^2 - (-3^2)) / h
Simplifying further,
((-3 + h)^2 - 9) / h
Now, we have the difference quotient that can be used to estimate the instantaneous rate of change of y = x^2 at x = -3.