Dick said:If the lines are perpendicular to y=(1/4)*x then what should the slopes of the tangent lines be? The slopes of the tangent lines should also be f'(x), right?
fghtffyrdmns said:The slope would be -4, no?
I've made the two equations of y = -4x + 17/2 and y = -4x -17/2.
Dick said:Yes, the slope should be -4. But how did you get those two line equations?
fghtffyrdmns said:I equated y and f(x) to see when they intercept. I got 2 and -2. I put these two into f(x) to get the y cordinate so I can make the equation as I have the slope.
Dick said:I'm still not totally clear what you are doing. But if you know the slope is -4, then f'(x) should be -4, right? What are the possibilities for x?
Dick said:Yes! If you know x=1/2 or -1/2, then you know y. So now you know x and y and the slope. Pretty easy, right?
To find the slope of a line perpendicular to y=0.25x, you need to take the negative reciprocal of the slope of y=0.25x. In this case, the slope of y=0.25x is 0.25, so the slope of the perpendicular line would be -4.
The equation of a line that is perpendicular to y=0.25x and passes through a given point (x1, y1) can be found using the point-slope formula. The slope of the perpendicular line would be -4, so the equation would be y - y1 = -4(x - x1).
To prove that two lines are perpendicular, you need to show that their slopes are negative reciprocals of each other. In this case, the slope of y=0.25x is 0.25, and the slope of the perpendicular line would be -4, which are negative reciprocals of each other.
To find the equation of a line that is tangent to f(x)=1/x, you need to find the derivative of f(x) and evaluate it at the point where the tangent line touches the curve. This will give you the slope of the tangent line, and you can use the point-slope formula to find the equation.
Perpendicular lines and tangent lines are both types of lines that intersect with other lines or curves. Perpendicular lines intersect at a 90-degree angle, while tangent lines intersect at a single point on a curve. In this case, the perpendicular line to y=0.25x will intersect the curve f(x)=1/x at a 90-degree angle, making it a tangent line.