sheldonrocks97 said:Homework Statement
Find dy/dx and d^y/dx^2
x = cos 2t, y = cos t, 0 < t < ∏
For which values of t are concave upward? (write your answer in interval notation).
Homework Equations
The Attempt at a Solution
I used the formula to find d^2y/dx^2.
d^2y/dx^2= e^(-3t)*(2t-3)
I solved it and got t>3/2, but the computer told me it was wrong. What am I doing wrong here?
The OP edited his post, so the parametric equations now have exponential formLCKurtz said:Where did the exponential come from?
Yes. Show your work for dx/dt and dy/dt and so on.LCKurtz said:After you fix that, please show your work so we can follow it.
The concavity of parametric equations refers to the shape or curvature of a graph created by plotting the x and y coordinates of a set of parametric equations. It can be described as either concave up (curving upward) or concave down (curving downward).
The concavity of parametric equations is determined by calculating the second derivative of the equations with respect to the independent variable (usually t). If the second derivative is positive, the graph is concave up, and if it is negative, the graph is concave down.
The concavity of parametric equations can provide important information about the behavior and characteristics of the graph. For example, concave up graphs tend to have a minimum point, while concave down graphs tend to have a maximum point. Concavity can also affect the rates of change and the direction of the graph.
Yes, parametric equations can have changing concavity. This occurs when the second derivative changes sign, resulting in a change in the direction of the curvature. This can be observed on the graph as a point where the graph changes from being concave up to concave down, or vice versa.
The concept of concavity in parametric equations can be applied in various fields such as physics, engineering, and economics. For example, it can be used to analyze the trajectory of a projectile, the shape of a bridge, or the cost and revenue functions of a business. Understanding the concavity can help predict the behavior and make important decisions based on the graph.