This is difficult to read. Use * to indicate products, like so:cummings15 said:Homework Statement
Verify that (1,0) is on the following curve and find the tangent line and normal line to the curve at the point.
y=pisin(pix-y)
cummings15 said:The Attempt at a Solution
y prime = picos(u)(du)
y prime = picos(pix-y)(pi-y prime)
cummings15 said:i think i got it is y '
{-1/pi*cos(pi*x-y)} + pi
If you expand the right side, the equation becomescummings15 said:i got this y ' = pi * cos(pi * x - y) * (pi - y')
and then i solved for y '
cummings15 said:so the slope would = pi
The tangent line is a straight line that touches a curve at one point, known as the point of tangency. It represents the instantaneous rate of change or slope of the curve at that specific point.
To find the tangent line at a specific point on a curve, you can use the slope formula, which involves finding the derivative of the function at that point. Alternatively, you can also use the geometric method of drawing a line that touches the curve at the point of interest and has the same slope as the curve at that point.
The tangent line is useful in many areas of mathematics and science, such as in calculus, physics, and engineering. It allows us to approximate the behavior of a curve at a specific point and make predictions about its behavior in the surrounding area.
Yes, the tangent line can be found at any point on a curve as long as the curve is continuous and differentiable at that point. In simpler terms, this means that the curve is smooth and has a defined slope at that point.
Yes, finding the tangent line has numerous real-world applications. For example, in physics, the tangent line can be used to determine the velocity of an object at a specific point in time. In economics, the tangent line can be used to estimate the marginal cost or revenue of a product. In engineering, the tangent line can be used to design efficient and stable structures.