Im having trouble with this.
Find an equation of the tangent line to the parabola at the given point and find the x-intercept of the line.
x^2=2y (4,8)
y = (1/2) x^2
The slope of the curve is dy/dx = x
At x = 4, the slope is 4.
For a line though (4,8) with slope = 4, the equation is
(y-8) = 4(x-4)
Rewrite that in standard y = mx + b form.
well, from the above equation y = (x^2)/2
to get an equation of a line you need a slope and a point. You have a point and to get the slope we need to get dy/dx
and sub in the value for x in the point given [in this example 4]
Then sub your point (x1,y1) and the slope (m) into
y - y1 = m(x - x1)
and then just tidy up
Hope that helps
To find the tangent line to a parabola at a given point, you need to first find the derivative of the equation of the parabola. The derivative will give you the slope of the tangent line.
Let's start by finding the derivative of the equation x^2 = 2y with respect to x.
To do this, differentiate both sides of the equation using the power rule of differentiation.
d/dx (x^2) = d/dx (2y)
2x = 2(dy/dx)
Simplifying further, we get:
dy/dx = x
Now, we have the derivative dy/dx representing the slope of the tangent line at any given point (x, y) on the parabola.
Next, we need to find the slope of the tangent line at the given point (4, 8).
Substitute the x-coordinate of the given point into the derivative expression:
dy/dx = 4
So, the slope of the tangent line at the point (4, 8) is given by m = 4.
Now, we have the slope (4) and the point (4, 8). We can use the point-slope form of a linear equation to write the equation of the tangent line.
The point-slope form of a linear equation is given by:
y - y₁ = m(x - x₁)
Substituting the values into the equation, we get:
y - 8 = 4(x - 4)
Simplifying this equation gives:
y - 8 = 4x - 16
Rearranging the equation to the standard form (y = mx + b), we have:
y = 4x - 8
Now, to find the x-intercept of the line, we set y = 0 and solve for x:
0 = 4x - 8
Solving this equation for x gives:
4x = 8
x = 8/4
x = 2
Therefore, the x-intercept of the tangent line is 2.