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Homework Statement
If x = cos(3t) and y = (sin(3t))² find dy/dx and d²y/dx².
Homework Equations
d/dx cosx = -sinx
d/dx sinx = cosx
The Attempt at a Solution
dx/dt = -3sin3t
dy/dt = 6sin3tcos3t
Not sure what to do now.
The 1st derivative, also known as the slope of a curve, represents the rate of change of a function at a specific point. It is calculated as the change in output divided by the change in input.
The 1st derivative is used in various fields such as physics, economics, and engineering to analyze and predict changes in quantities over time. For example, it can be used to calculate the velocity of an object or the rate of change of a company's profits.
The 2nd derivative is the rate of change of the 1st derivative. In other words, it measures how quickly the slope of a curve is changing. The 2nd derivative can provide information about the concavity of a curve and whether it is increasing or decreasing.
Critical points are points on a function where the derivative is equal to zero or does not exist. To find these points, we use the 2nd derivative test, which involves looking at the sign of the 2nd derivative at a given point. If the 2nd derivative is positive, the point is a minimum, and if it is negative, the point is a maximum.
Yes, the 1st and 2nd derivative can be used to find the inflection point of a curve. The inflection point is where the concavity of a curve changes, and it can be found by setting the 2nd derivative equal to zero and solving for the input value.