I'm after understanding more than an answer but an answer would be greatly appreciated.
The question reads as
"On the answer sheet for this question is shown an x-y graph of the function [y(x)=x^2cos(3x)]
Add to that same picture a rough sketch of the graph of dy/dx against x, adding graphical arguments pointing out the main features of this graph which convince you that it represents the slope of the graph of y(x) against x. Find all local max, min and points of inflexion."
any idea?
thanks
To answer this question, you need to know how to find the derivative of y(x) with respect to x, which will give you dy/dx. Once you have dy/dx, you can plot it on the same x-y graph that shows the function y(x)=x^2cos(3x).
Here's how you can find dy/dx:
1. Start with the function y(x)=x^2cos(3x).
2. To find the derivative, use the product rule and the chain rule.
- Apply the derivative of the first factor (x^2) using the power rule, which states that d/dx (x^n) = n*x^(n-1).
- Multiply it by the derivative of the second factor (cos(3x)) using the chain rule. The derivative of cos(x) is -sin(x), so the derivative of cos(3x) would be -3sin(3x), as you need to apply the chain rule.
- Add these two derivatives together to get the final derivative.
3. Simplify the derivative expression as much as possible.
Once you have the derivative dy/dx, you can plot it on the same graph as the function y(x)=x^2cos(3x). Here are some key features to look for:
1. Local Maximum: These are points where dy/dx changes from positive to negative. They correspond to peaks in the graph of y(x).
2. Local Minimum: These are points where dy/dx changes from negative to positive. They correspond to valleys in the graph of y(x).
3. Points of Inflexion: These are points where dy/dx changes its concavity. When dy/dx changes from increasing to decreasing, or vice versa, it indicates a point of inflexion in the graph of y(x).
To find the local max, min, and points of inflexion, you need to identify where dy/dx is zero or undetermined (i.e., where the derivative does not exist). These points will help you determine the critical points of the graph.
To summarize, to answer the given question, you need to find the derivative dy/dx of the function y(x)=x^2cos(3x). Plot dy/dx on the same x-y graph, observing its main features such as local maxima, minima, and points of inflexion. To find these features, identify where dy/dx is zero or undetermined.