Determine the tangent equation for y=2-3x^2+4x at x=2.
To determine the tangent equation at x = 2, we first need to find the derivative of the function y = 2 - 3x^2 + 4x.
Taking the derivative with respect to x, we get:
dy/dx = -6x + 4
Now, we can substitute the value x = 2 into the derivative to find the slope of the tangent line:
dy/dx = -6(2) + 4
= -12 + 4
= -8
Therefore, the slope of the tangent line at x = 2 is -8.
To find the equation, we need to use the point-slope form of a linear equation, where the equation is given by y - y1 = m(x - x1), with (x1, y1) being a point on the line and m being the slope.
Using the point (2, y), where y = 2 - 3(2)^2 + 4(2) = 2 - 12 + 8 = -2, the equation becomes:
y - (-2) = -8(x - 2)
Simplifying further:
y + 2 = -8x + 16
Rearranging to slope-intercept form, we get:
y = -8x + 14
Therefore, the tangent equation for y = 2 - 3x^2 + 4x at x = 2 is y = -8x + 14.
To determine the tangent equation at a specific point, we need to find the derivative and evaluate it at that point. Let's go through the steps:
Step 1: Find the derivative of the function.
The given function is y = 2 - 3x^2 + 4x. To find its derivative, we differentiate each term with respect to x.
The derivative of 2 with respect to x is 0 since it's a constant.
The derivative of -3x^2 with respect to x is -6x.
The derivative of 4x with respect to x is 4.
So the derivative of the function is dy/dx = -6x + 4.
Step 2: Evaluate the derivative at x=2.
Substitute x=2 into the derived equation to find the slope of the tangent at that point:
dy/dx = -6(2) + 4
dy/dx = -12 + 4
dy/dx = -8.
Step 3: Write the tangent equation using the slope and the given point.
The tangent equation is given by the point-slope form, which is:
y - y1 = m(x - x1),
where (x1, y1) is the given point and m is the slope.
We know that the given point is (2, 2 - 3(2)^2 + 4(2)), which simplifies to (2, -6).
Substituting the values, we have:
y - (-6) = -8(x - 2),
which simplifies to:
y + 6 = -8x + 16.
Rearranging the equation will give us the tangent equation in slope-intercept form:
y = -8x + 10.
So, the tangent equation for y = 2 - 3x^2 + 4x at x = 2 is y = -8x + 10.