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Is the first derivative of a function is a constant then its graph is?
A line. The derivative of a function is its slope. If the slope is a constant then the graph is a line.
What are the uses of derivatives?
A derivative of a function represents that equation's slope at any given point on its graph.
How do you graph function g?
use y = g(x) make a table of y values for several x values Find max/min values using derivative. graph the ordered pairs.
How do you find critical value for a total revenue function?
If it is a differentiable function, you find the value at which its derivative is 0. But in general, you can plot it as a line graph and see where it peaks.
What is concavity of a function?
Just as the slope of the tangent line to the graph of f at the point (x, f(x)) describes the behavior of the function, concavity describes the behavior of the slope. As x increases (graph goes from left to right), one of the following is true:Concavity is positive, so the slope slowly increases.Concavity is negative, so the slope slowly decreases.Concavity is equal to zero, so the slope is constant.Again, remember that concavity directly affects the slope, NOT the function itself. I mean this in the sense that concavity affects slope affects function.Mathematically speaking, you can find the concavity at a certain point by taking the derivative of the derivative of the function (accurately called the second derivative, f''). So, when you take the derivative of a function, you get the first derivative, f' (describing slope), and the derivative of that is the second derivative (describing the concavity).Last but not least, here is a handy way to find the concavity of a function by looking at its graph:Concavity is positive when the graph turns up, like a smiling emoticon (look at a graph of f(x) = x2 for an example).First observe that f'(x) = 2x.We see that f' < 0 when x < 0 and f' > 0 when x > 0. So that the graph is decreasing on the negative real axis and the graph is increasing on the positive real axis.Next observe that f''(x) = 2.Thus, f'' > 0 at all points. Thus the graph is concave up everywhere.Finally observe that the graph passes through the origin.Concavity is negative when the graph turns down, like a frowning emoticon (look at a graph of f(x) = -x2 for an example).First observe that f'(x) = -2x.We see that f' > 0 when x < 0 and f' < 0 when x > 0. So that the graph is increasing on the negative real axis and the graph is decreasing on the positive real axis.Next observe that f''(x) = -2.Thus, f'' < 0 at all points. Thus the graph is concave down everywhere.Finally observe that the graph passes through the origin.Look at the graph of f(x) = x3First observe that f'(x) = 3x2.Thus, f' ≥ 0 everywhere. The function is always increasing.Next observe that f''(x) = 6x.Thus, f'' < 0 when x < 0 and f'' > 0 when x > 0. So the graph is concave down on the negative real axis and concave up on the positive real axis.Finally observe that the graph passes through the origin.Concavity is zero when the graph is linear OR at a point where it stops turning up and starts turning down, and vice versa.
