Estimate the slope of the curve y = x ^ 3 + 2 at x = 1
Comment on the slope as x gets closer to 1.
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.
To estimate the slope of the curve at x = 1, we need to find the derivative of the function y = x^3 + 2.
The derivative will give us the rate of change of the function at any given point. In this case, we are interested in finding the rate of change at x = 1.
Step 1: Find the derivative of the function y = x^3 + 2.
To find the derivative, we can use the power rule for differentiation. The power rule states that the derivative of x^n is equal to n*x^(n-1).
Using this rule, we differentiate each term of the function separately:
d/dx (x^3) = 3x^2
d/dx (2) = 0 (since 2 is a constant)
Therefore, the derivative of y = x^3 + 2 is dy/dx = 3x^2.
Step 2: Substitute x = 1 into the derivative.
To find the slope at x = 1, we substitute x = 1 into the derivative we found in step 1:
dy/dx = 3(1)^2
dy/dx = 3
So the slope of the curve at x = 1 is 3.
As x gets closer to 1, the value of the slope remains constant at 3. This means that the rate of change of the function at x = 1 is consistently increasing by 3 units for every unit increase in x.
To estimate the slope of the curve y = x^3 + 2 at x = 1, we can use the concept of derivatives. The derivative of a function tells us the rate at which the function is changing at any given point. In this case, we are interested in finding the derivative (slope) of the curve at x = 1.
To find the slope at x = 1, we'll need to take the derivative of the given curve with respect to x. Let's differentiate y = x^3 + 2 with respect to x.
dy/dx = 3x^2
Now that we have the derivative, we can substitute x = 1 into dy/dx to find the slope at x = 1:
Slope at x = 1 = dy/dx (x = 1)
= 3(1)^2
= 3
Therefore, the slope of the curve y = x^3 + 2 at x = 1 is 3.
Now, let's comment on the slope as x gets closer to 1.
As x gets closer to 1, the slope of the curve y = x^3 + 2 also gets closer to 3. This is because the derivative measures the instantaneous rate of change, or in other words, the slope of the tangent line to the curve at a specific point. So, as x approaches 1 from both smaller and larger values, the slope of the curve at those points will approach 3.