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Estimate the slope of the curve y = x ^ 3 + 2 at x = 1
Comment on the slope as x gets closer to 1.
Comment on the slope as x gets closer to 1.
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GPT 3.5
To estimate the slope of the curve y = x^3 + 2 at x = 1, we can use the concept of the derivative. The derivative of a function represents its instantaneous rate of change or slope at a specific point.
First, let's find the derivative of the given function. Taking the derivative of y = x^3 + 2 with respect to x, we get:
y' = 3x^2
Now, let's find the slope at x = 1. Plugging x = 1 into the derivative equation, we have:
y' = 3(1)^2 = 3
Therefore, the slope of the curve y = x^3 + 2 at x = 1 is 3.
As x gets closer to 1, the slope remains constant at 3. This implies that the curve becomes steeper as x approaches 1 from both sides, but the rate of increase in steepness remains the same.
First, let's find the derivative of the given function. Taking the derivative of y = x^3 + 2 with respect to x, we get:
y' = 3x^2
Now, let's find the slope at x = 1. Plugging x = 1 into the derivative equation, we have:
y' = 3(1)^2 = 3
Therefore, the slope of the curve y = x^3 + 2 at x = 1 is 3.
As x gets closer to 1, the slope remains constant at 3. This implies that the curve becomes steeper as x approaches 1 from both sides, but the rate of increase in steepness remains the same.
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