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course Phy 121
10/16 3
I tried to pull this simulation up, but it says,
A concave up parabola type graph f(x) is drawn using the freeform tool, click the smooth option the graphs smoothness is improved. The graph is a bit crooked, and so is the differential and the integral crookedness. The differential function f’(x) is the function of the slope of the function f(x). The f’(x) graph is a curved graph almost like a straight line. Since the function f(x) is concave up, thus the first half of the slope function has negative values and the second half has positive values. The slope for the function f(x) is continuously increasing and thus the slope function is also continuously increasing. Since f(x) is almost parabolic, f’(x) is almost a straight line. The concave up f(x) has all negative values and so does the integral function g(x). g(x) is a curve function, the slope of which is negative at the start and then decreases and then again increases to the same value at which it was at the start.
Next we need to construct a graph of f’(x) which is concave up. I thought of constructing a concave up f’(x) graph for which the slope has positive values, it starts with 0 slope and then increases at an faster rate to reach the maximum slope, thus making the graph a concave up. To construct such derivative function we need a function f(x) which has 0 slope initially and then there is an increase in the slope at an increasing rate. Thus the function of f(x) is like the right hand part of a concave up parabola when divided along the line of symmetry. Integral function for the same f(x) function is also a positive function. The integral function is again a curve function which is concave up and the slope continuously increases.
Finally it is to construct a graph of f’(x) which is concave down. I thought of constructing a concave down f’(x) graph for which the slope has negative values, it starts with 0 slope and the decreases at an increasing rate to reach the least value of the slope, thus making the graph a concave down. To construct such a derivative function we need a function f(x) which has 0 slope initially and then there is a decrease in the slope at an increasing rate. Thus the function f(x) is like a right hand part of a concave down parabola when divided along the line of symmetry. Integral function for the same f(x) is also a negative function, that is the values of the function at x values is negative. The integral function is again a curve function with negative slopes decreasing at an increasing rate.
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Excellent.
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