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
No idea.
We do not see the figure. We do not know the extent of the x-axis.
The slope of a given function f(x) can be calculated by finding the derivative, f'(x), if you have done calculus. Otherwise you can estimate the slope by placing a ruler that follows and rides on the curve.
In addition, we do not know what help you need.
To add a sketch of the graph of dy/dx against x, we need to find the derivative of the function y(x) = x^2cos(3x). Let's calculate the derivative using the product rule and the chain rule:
1. Apply the product rule:
d/dx (x^2cos(3x)) = 2xcos(3x) + x^2(-3sin(3x))
2. Apply the chain rule:
d/dx (x^2cos(3x)) = 2xcos(3x) - 3x^2sin(3x)
Now that we have the derivative dy/dx, we can sketch its graph.
The main features of the graph of dy/dx are as follows:
1. Local maxima: These occur when dy/dx changes from positive to negative. Look for points where the graph reaches its highest values and the slope transition occurs.
2. Local minima: These occur when dy/dx changes from negative to positive. Look for points where the graph reaches its lowest values and the slope transition occurs.
3. Points of inflection: These occur where dy/dx changes concavity. Look for points where the graph changes from being concave up to concave down, or vice versa.
To find the local maxima, minima, and points of inflection, we can do the following:
1. Set dy/dx = 0 and solve for x to find possible critical points.
2. Calculate the second derivative (d²y/dx²) to determine the concavity at each critical point.
3. Analyze the behavior of the second derivative around each critical point to identify the type of point.
I hope this explanation helps you understand the steps needed to sketch the graph and find the local maxima, minima, and points of inflection.
To address this question, you need to understand the concept of derivative and its relation to the original function. Let's break down the steps to find the graph of dy/dx against x and identify the local maxima, minima, and points of inflection.
1. Find the derivative of y(x):
The derivative of a function represents the rate of change of that function at each point. In this case, you need to find dy/dx, the derivative of y(x), which will give the slope of the graph of y(x).
To find the derivative of y(x) = x^2cos(3x), you can use the product rule and the chain rule of differentiation.
dy/dx = (2x * cos(3x)) + (x^2 * (-sin(3x)) * 3)
Simplifying the expression, dy/dx = 2x * cos(3x) - 3x^2 * sin(3x)
2. Sketch the graph of dy/dx against x:
To create a rough sketch of the graph of dy/dx against x, you will need to plot some key points. Start by identifying critical points where dy/dx = 0 or is undefined. Critical points occur when the derivative changes sign or when it does not exist.
Set dy/dx = 0 and solve for x to find the x-values of the critical points.
2x * cos(3x) - 3x^2 * sin(3x) = 0
At this point, you will likely need to use numerical approximation methods or a graphing calculator to find the exact values of the critical points.
3. Analyze the critical points:
Once you have the critical points, evaluate the second derivative (d^2y/dx^2) at these points to determine the concavity and classify each critical point as a local maximum, minimum, or point of inflection.
If the second derivative is positive, the graph is concave up, indicating a local minimum. If the second derivative is negative, the graph is concave down, indicating a local maximum. If the second derivative is zero or undefined, it suggests a point of inflection.
To find the second derivative, take the derivative of dy/dx, which will give you d^2y/dx^2, and then evaluate it at the critical points.
4. Add graphical arguments pointing out the main features:
Include graphical arguments, such as arrows or annotations, on the sketch you made earlier to demonstrate the main features of the graph. This might involve highlighting the critical points, indicating whether they are local maxima, minima, or points of inflection, and showing the concavity changes.
Remember that the sketch is expected to be rough, but it should provide visible main features of the graph of dy/dx against x that illustrate how it represents the slope of the graph of y(x) against x.
By following these steps, you should be able to find the graph of dy/dx against x and identify the local maxima, minima, and points of inflection for the given function y(x) = x^2cos(3x).