Determine the derivative at the point (2,−43) on the curve given by f(x)=7−7x−9x^2.
I know that the answer is -43, but I was wondering if it was just a coincidence that the derivative at the point (2,−43) is -43, or is there a reason why -43 is the same as the y-value of the original point.
suppose you try another point, say (1, -9)
then
f ' (x) = -7 - 18x
f ' (1) = -7 - 9 = -16
Yup, just a coincidence.
Question: is there another point where this happens,
that is,
solve f(x) = f'(x)
7 - 7x - 9x^2 = -7 - 18x
-9x^2 + 11x +14 = 0
9x^2 - 11x - 14 = 0
(x - 2)(9x + 7) = 0
x = 2 or x = -7/9
we know about the x=2
when x = -7/9 , f(-7/9) = 7 - 7(-7/9) - 9(49/81)
= 7
and f'(-7/9) = -7 - 18(-7/9) = 7
yup, two cases where it happens.
this is weird...but thanks!
noooooooooo
To determine the derivative at the point (2, -43) on the curve given by f(x) = 7 - 7x - 9x^2, we need to find the derivative of the function and evaluate it at x = 2.
The derivative of a function represents the rate at which the function is changing at a particular point. In this case, we are looking at the rate of change of f(x) with respect to x. To find the derivative, we can use the power rule and the constant rule.
First, let's apply the power rule to find the derivative of the function f(x) = 7 - 7x - 9x^2:
f'(x) = -7 - 18x
Now, let's evaluate the derivative at x = 2:
f'(2) = -7 - 18(2)
= -7 - 36
= -43
So, the derivative at x = 2 is -43. Now let's understand why the y-value of the original point (2, -43) matches the derivative value at that point.
In general, the derivative of a function at a given point represents the slope of the tangent line to the curve at that point. The slope of the tangent line is the rate of change of the function at that point.
So, when the derivative at a specific point has a value equal to the y-value of the original point, it means that the tangent line at that point has a slope of -43. This tells us that the function is changing at a constant rate of -43 at that point, which in turn means that the y-value of the function is decreasing by 43 units per unit of x.
In other words, the coincidence you observed is not just random but is a result of the function's rate of change at the point (2, -43). The value of -43 for the derivative corresponds to the decrease in the y-value of the function at that point.