I'm supposed to sketch a graph of two functions that have the same derivative of a graph.
The graph having a horizontal line on (0,4)
clearly,
y(0) = 4
y'(0) = 0
So, a parabola with vertex there will work:
y = 2x^2+4
or how about a cosine function?
y = 4cos(x)
To sketch two functions that have the same derivative, we can start by finding the derivative of a function that has a horizontal line at (0,4).
Let's call the function with the horizontal line "f(x)". Since it is a horizontal line, its slope is zero, which means the derivative of f(x) is zero for all values of x.
Therefore, the derivative of f(x) is f'(x) = 0.
Now, to find the two functions that have the same derivative, we can integrate f'(x). Since f'(x) = 0, when we integrate it, we get a constant value. Let's call this constant "C".
So, the two functions that have the same derivative as the horizontal line f(x) are:
g(x) = C
h(x) = C
Both g(x) and h(x) are constant functions because their slopes are zero, just like the slope of the horizontal line f(x).
To sketch the graph of these functions, draw a horizontal line at y = C for both g(x) and h(x). The value of C can be any real number since it represents the constant. This means that both g(x) and h(x) will be parallel horizontal lines on the y-axis.