You might think that the graph of y = x^2 +cosx would look like a parabola with some waves on it. Use a computer to sketch the actual graph of
y=x^2+cosx and explain what you see by considering y' and y''.
Here is a graph of y = x^2 + cosx:
As you can see, the graph does indeed resemble a parabola with some waves on it. However, there are several interesting features to note.
First, notice that the waves in the graph seem to be "mini parabolas" themselves. This is because the first derivative of y with respect to x (y') is equal to 2x - sinx. So wherever sinx is positive, the slope of the curve will be steeper than the slope of the parabola alone, and wherever sinx is negative, the slope will be shallower. This results in a "wavy" curve that follows the general shape of the parabola but with some added variation.
Second, notice that there are several points where the graph intersects the x-axis. These correspond to points where y = 0, meaning that x^2 = -cosx. These points occur when cosx is negative, and since the cosine function oscillates between 1 and -1, there are infinitely many such points.
Finally, notice that the waves seem to be getting smaller as x increases. This is because the second derivative of y with respect to x (y'') is equal to 2 - cosx, which is always positive. So the concavity of the curve is always upward, meaning that the curve is "smiling" rather than "frowning." However, because the value of cosx oscillates between 1 and -1, the degree of "smiling" varies somewhat. At points where cosx is close to 1, the curve will be more parabolic, and at points where cosx is close to -1, the curve will be more wavy. But overall, the "smiling" shape of the curve causes the waves to get smaller as x increases.
To sketch the graph of y = x^2 + cos(x), we can use a computer or graphing software. Here is the sketch of the actual graph:
Explanation:
1. First, let's consider y' (the first derivative of y) to analyze the slope of the function: y' = 2x - sin(x).
- The term 2x represents the slope of the quadratic term, x^2. It is an increasing function that gets steeper as x increases.
- The term -sin(x) affects the slope with periodic oscillations. It ranges from -1 to 1, depending on the value of x.
2. Now let's consider y'' (the second derivative of y) to understand the concavity:
- Taking the second derivative of y, we have y'' = 2 - cos(x).
- The term 2 is a constant, indicating a positive concavity for the quadratic term.
- The term -cos(x) introduces periodic changes to the concavity, ranging from -1 to 1, depending on x.
Based on the analysis of y' and y'', we can observe the following characteristics in the graph:
- The quadratic term, x^2, dominates the overall shape, creating a parabolic curve.
- The cosine term, cos(x), introduces waves (oscillations) on the graph.
- The graph has positive concavity when y'' > 0 (where 2 - cos(x) > 0), forming a cup-like shape.
- The graph has negative concavity when y'' < 0 (where 2 - cos(x) < 0), forming an inverted cup-like shape.
Overall, the graph of y = x^2 + cos(x) combines the characteristics of a parabola and sinusoidal wave-like behavior.