Final answer:
To find the equation of the tangent line to a curve at a given point, you can use methods such as the point-slope form, the normal vector, the parametric form, and the chain rule.
Step-by-step explanation:
The equation of the tangent line to a curve at a given point can be found using different methods:
A) To use the point-slope form, you need to know the slope of the tangent line at the given point. This can be found by finding the derivative of the function and plugging in the x-coordinate of the given point. The point-slope form of the equation can then be used to find the equation of the tangent line.
B) Applying the normal vector involves finding the derivative of the function and plugging in the x-coordinate of the given point to find the slope of the tangent line. The normal vector to the tangent line can be found by taking the negative reciprocal of the slope. The equation of the tangent line can then be determined using the point-slope form.
C) Employing the parametric form is useful when dealing with curves defined by parametric equations. Differentiate the parametric equations to find the derivatives and the slopes of the tangent line. You can then use the point-slope form to find the equation of the tangent line.
D) Applying the chain rule is used when dealing with composite functions. Find the derivative of the composite function using the chain rule, and then plug in the x-coordinate of the given point to find the slope of the tangent line. The point-slope form can be used to determine the equation of the tangent line.