How to find instantaneous rate of change for #x^3 +2x^2 + x# at x from 1 to 2?

Answer 1

The instantaneous rate of change is the same as taking the derivative. This is defined at a point.

We can easily find that function via power rule: #f(x) = x^3 + 2x^2 + x implies f'(x) = 3x^2 + 4x + 1 #
I hope it is clear here that the concept of "instantaneous rate of change" from one point to another doesn't make any sense. However, you can think about the average rate of change very easily. We just do that in the way we'd think: the change in y divided by the change in x. This gives the following: #(f(2) - f(1))/(2 - 1) = 14/1 = 14#
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Answer 2

To find the instantaneous rate of change of a function (f(x)) at a specific point (x = a), you can use the derivative (f'(a)). In this case, the function is (f(x) = x^3 + 2x^2 + x), and we want to find the instantaneous rate of change at (x = 1) and (x = 2).

To find the derivative of (f(x)), we first differentiate the function with respect to (x): [f'(x) = 3x^2 + 4x + 1]

Now, we can find the instantaneous rate of change at (x = 1) and (x = 2) by evaluating (f'(x)) at these points: [f'(1) = 3(1)^2 + 4(1) + 1 = 3 + 4 + 1 = 8] [f'(2) = 3(2)^2 + 4(2) + 1 = 3(4) + 8 + 1 = 12 + 8 + 1 = 21]

So, the instantaneous rate of change of (x^3 + 2x^2 + x) at (x = 1) is 8, and at (x = 2) is 21.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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