Category: Calculus

Earlier we plotted the polynomial function whose graph is reported again for convenience in Fig. 2.23. The stationarity points can be found by setting its derivative to zero: Using numerical methods, we find the following roots of f′(x): which are indeed the points at which f is stationary. Observing the graph, we see that x1 is the global minimum, x2 is a local…

What we have learned so far about function derivatives suggests that in order to optimize a function, assuming that it is differentiable, a good starting point is to set its first-order derivative to zero. However, we know that this first-order, stationarity condition may not be enough, as it does not even discriminate between a maximum…

Reading on, you will notice that a large part of deals with uncertainty. Uncertainty comes in many forms: One way to deal with uncertainty is to rely on the tools of probability theory and statistics. The main limitation of these tools is that they may require a lot of past data to characterize uncertainty, assuming…

The derivative tells us something about the rate at which a function f increases or decreases at some point x. This rate is the slope of the tangent line to the graph of f at x. So, the derivative tells us something about the “linear” behavior of a function. However, this does not tell us anything about its curvature. To visualize…

The derivative is the slope of the tangent line to the graph of a function. Hence, the sign of the derivative at a point tells us whether the function is increasing or decreasing there and how rapidly. We can use this to figure out essential features of a function and to sketch its graph. Example…

The rules of previous section do not help us in finding derivatives of functions like the square root or, given the derivative of logarithm, in finding the derivative of the exponential. We need a rule to deal with the derivative of an inverse function. THEOREM 2.9 (Derivative of an inverse function) Let x = g(y) be the…

Given two functions g and h, we may build a new function by composition, namely, g o h. It would be nice to have a way of finding the derivative of the composite function by decomposing the task and exploiting knowledge about the derivatives of g and h. THEOREM 2.8 (Chain rule) Given functions g and h, we obtain the derivative of their composition as…

Given two functions f and g, there are a few easy ways to build other functions by ordinary arithmetic operations such as sum, multiplication, and division. If we are able to find the derivative of f and g, the following theorem shows how to find the derivative of functions defined by the mechanisms above. THEOREM 2.7 Let f and g be functions…

The direct application of the definition to find the derivative of a function is typically a rather difficult and cumbersome procedure, possibly requiring some intuition. Example 2.19 (Derivative of logarithm and exponential function) One of the most useful results concerning derivatives is that the derivative of the exponential is just the exponential itself: As a first…

If the derivative of function f at point Xo exists, then we say that the function is differentiable at point x0; if this holds for all points on an interval or domain, the function is differentiable on that interval or domain. If the derivative f′(x) exists at all points x on an interval and the derivative is a continuous function, we say that…