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''.

Answer 1

Unfortunately, as an AI language model, I cannot create a graph. However, I can explain what I would observe with the use of calculus.

Firstly, the graph of y = x^2 is a parabola that opens upwards. As for cosx, it is a periodic function that oscillates between 1 and -1.

When we add the two functions together, we get y = x^2 + cosx. This means that each point on the resulting graph will have a y-coordinate that is equal to the sum of the corresponding y-coordinates on the individual graphs.

The derivative of y (y') is given by:

y' = 2x - sinx

This means that when x = 0, y' = 0. This is the lowest point on the graph, where the slope is 0. We can also see that y' = 0 again where sinx = 2x.

The second derivative of y (y'') is given by:

y'' = 2 - cosx

This means that y'' is always positive (greater than 0) since -1 ≤ cosx ≤ 1, so 1 ≤ 2 - cosx ≤ 2. This indicates that the graph is always curving upwards or concave up.

Overall, the graph of y = x^2 + cosx will resemble a parabola with oscillations on top of it. The troughs of the cosine function will dip lower than the parabola, while the peaks will rise higher than the parabola. The entire graph will curve upwards, with the slope being equal to 0 at two points.

Answer 2

y = x^2 + cosx is a parabola with waves on it
y' = 2x - sinx is a straight sloping line with waves on it
y" = 2 - cosx is a horizontal line with waves on it

Answer 3

That is correct.