Are you sure the above is right? If x = 4, then t = +/-1, but when t = 1, y = 4 and when t = -1, y = 0.Hobold said:Homework Statement
This is a very basic problem, though it did confuse me a little:
Find the tangent equations to the curve [tex]x=3t^2+1 \ , \ y = 2t^3+2[/tex] which intercepts the point (4,3).
Hobold said:Homework Equations
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The Attempt at a Solution
I took [tex]\frac{dy}{dx} = t = \frac{y-y_0}{x-x_0} = \frac{y-3}{x-4} \rightarrow y=t(x-4)+3[/tex]
What to do next? I don't think the equation works for all t.
A tangent line to a curve is a straight line that touches the curve at one point, but does not intersect with the curve at any other point. It represents the slope of the curve at that specific point.
The tangent line is important because it helps us understand the behavior of the curve at a particular point. It can also be used to approximate the curve and calculate its slope or rate of change at that point.
The equation of the tangent line can be found using the derivative of the curve at the specific point. The slope of the tangent line is equal to the derivative of the curve at that point. Then, using the point-slope formula, the equation of the tangent line can be written in the form y = mx + b, where m is the slope and b is the y-intercept.
No, a curve can only have one tangent line at a point. This is because the tangent line represents the slope of the curve at that point, and the slope of a curve can only have one value at a specific point.
A tangent line touches the curve at one point, while a secant line intersects the curve at two points. The secant line can be thought of as an approximation of the curve between those two points, while the tangent line represents the exact slope of the curve at a single point.