In today's video, we discuss what it means for a function to be increasing or decreasing, provide some contrasting examples and relate these ideas to the sign of the derivative. That is, whether the derivative is positive or negative over a given interval. We say that a function f given by rule y equals f of x is increasing if the outputs y gets bigger as the inputs x get bigger. By getting bigger, we mean moving in the positive direction, to the right along the x axis and upwards along the y axis. The simplest example would be the function y equals x, whose curve is a straight line sloping upwards. Naturally, as x gets bigger, y equal to x also gets bigger. A more precise definition involves, the less than relation. If a and b are two numbers and a is less than b, then b is bigger than a. Thus, y equals f of x is the rule for an increasing function if a less than b implies f of a less than f of b for inputs a and b. This makes precise the idea that as x gets bigger, going from a to b, then f of x also gets bigger going from f of a to f of b. Often, the contexts in which the rule of the function is being used restricts the inputs. Typically, a and b will come from some interval on the real line that is of particular relevance for a given problem. This will become important later when we interpret sign diagrams and sketch curves. We say that a function f given by the rule y equals f of x is decreasing if the outputs y gets "smaller" as inputs x get "bigger". By "getting smaller", we mean moving in the negative direction to the left along the x axis and downwards along the y axis. A simple example would be the function y equals minus x, whose curve is a straight line sloping downwards. As x gets "bigger" in the sense of moving in the positive direction y equal to minus x gets "smaller" in the sense of moving in the negative direction. More precisely, y equals f of x is the rule for a decreasing function if a less than b implies f of a is greater than f of b for inputs a and b. Note how the inequality less than for inputs turns around into greater than for outputs. Functions whose graphs are lines fall into three classes. They're increasing if the slope of the line is positive, decreasing if the slope is negative or neither increasing nor decreasing if the slope is zero, that is, if the function is constant. Consider the simplest quadratic y equals x squared, whose graph is a parabola with lowest point at the origin. The function is increasing, if inputs are restricted to being greater than or equal to zero, but decreasing if inputs are restricted to being less than or equal to zero. If inputs are allowed to be taken from anywhere on the real line, then the function is neither increasing nor decreasing. Consider the simplest cubic polynomial function y equals x cubed, whose graph looks like this with an inflection at the origin where the tangent line passes through the curve. This function is increasing on all of the real line even though it seems to flatten out momentarily at the origin. It's always the case that if a is less than b, then a cubed is less than b cubed. Here are the graphs, the natural logarithm and exponential functions on the same diagram, they form a natural pair of increasing functions obtained from each other by reflecting of the line y equals x. The exponential function is increasing on all of the real line without any restriction on inputs. The natural logarithm function however, is increasing only over the positive reals. Remember, you aren't allowed to take the logarithm of zero or of a negative number. The fact that both are increasing on their respective domains it's not an accident, it's a nice fact, elaborated on a bit later, that is f is increasing then the inverse function f to the minus one is also increasing. The function associated with exponential decay y equals e to the minus x is by contrast decreasing on all of the real line. This is to be expected because it's the result of reflecting the graph of the increasing function y equals e to the x in the y axis. Reflecting graphs in the y axis interchanges increasing and decreasing functions. The circular functions y equals sine x and y equals cos x undulate forever backwards and forwards. So, neither increasing nor decreasing over the real line. If we restrict the domain of the sine curve to the interval from minus Pi on two to Pi on two, we get a fragment of the curve which now represents an increasing function. This is the standard restriction that's used to invert the sine function. If we restrict the domain of the cosine curve to the integral from zero to Pi we get a fragment of the curve which now represents a decreasing function. This is the standard restriction used to invert the cosine function. It's a general fact that increasing and decreasing functions are invertible because they satisfy the horizontal line test and these restrictions of the sine and cosine functions are two such examples. The natural exponential and logarithm functions are also examples that we've remarked upon earlier. It's a general fact explaining the notes that the inverse of an increasing function is increasing and the inverse of a decreasing function is decreasing. You can see how this works geometrically with these restrictions of sine and cosine. For example, take the restriction of the sine curve which is increasing and the line y equals x reflect and you produce the inverse sine function which is also increasing. You should take the restriction of the cosine curve which is decreasing and the line y equals x, make be a bit more room in the vertical direction, reflect and you produce the inverse cosine function, which is also decreasing. Here's the graph of the tan function which is neither increasing nor decreasing over its entire domain. Though it's comprised of these infinitely repeating pieces sandwiched between and separated by vertical asymptotes and each of these pieces represents an increasing function. We've highlighted one of them here, the result of restricting the tan function to the interval strictly between negative Pi on two and Pi on two. This is the standard restriction to make the tan function invertible and the result is an increasing function. Here's the graph of the inverse tan function which is also increasing. What about a physical example drawn from real life? Here's a curve we looked at in an earlier video that describes the distance function of a car trip from Sydney to Melbourne. While it's generally sloped upwards as you move from left to right, there are two places where the curve is flat corresponding to rest periods in the trip with the car isn't moving. So, overall the function is not increasing over the entire time interval. But there are many intervals of time on that trip when the car is moving all the time such as this section from Coolac to North Gundagai and over that time period, the function is increasing. If you're inside the car, but not looking outside, you could be sure the car is actually moving because you could always say a positive velocity on the speedometer, even though it might be fluctuating from moment to moment. These positive velocities correspond to slopes of tangent lines to the curve which are the derivatives of the function. It's clear then that if all the derivatives are positive, then the car is moving at positive velocities, moving closer to North Gundagai from Coolac, which just means, in our mathematics parlance that the associated function is increasing. This is a physical demonstration of the following mathematical fact. If the derivative's positive on an interval, then the function is increasing. We have correspondingly, if the derivative is negative on an interval then the function is decreasing. You can think of this physically in terms of a car moving backwards in the reverse direction with negative velocities. Consider the function with rule f of x equals x cubed minus x, then the derivative is 3x squared minus one. Observe that if 3x squared is greater than one then the derivative is positive. This occurs when x squared is greater than a third, which is precisely when x is greater than one over the square root of three or less than negative one over the square root of three. Thus, the curve for this rule will be increasing on the intervals from one over root three to infinity and from negative infinity to negative one over root three. Observe also, that if 3x squared is less than one, then the derivative is negative and this occurs when x lies between negative and positive one over root three. Thus, the curve will be decreasing over this interval. Notice that the rule of f factorizes as x times x plus one times x minus one, which tells us that the curve for this rule must cross the x axis at zero minus one and one. We're also aware of the importance of plus or minus one over root three. So, here's the curve crossing the x axis at those points we noted before, increasing for x less than negative one over root three and for x greater than one over root three and decreasing for x in between. What we've just completed in fact, are some of the main steps in the technique of curve sketching which is the topic of a future video. Notice the importance of the sign of the derivative, whether it's positive or negative and that can be captured in something called a sign diagram for the derivative in the form of a table, with horizontal line represents the real line and we are note the important points plus and minus one over the root three where the derivative is zero and the changing sides of the derivatives from positive to negative to positive which correspond to the curve being increasing, decreasing and then increasing again. That's a good point to close the discussion for today, leading into a more general discussion about sign diagrams in the next video. In today's video, we discussed what it means for a function to be increasing or decreasing and saw many familiar examples interpreted with this new terminology and noted that if the derivative's positive on an interval then the function is increasing. Or, if the derivative is negative, then the function is decreasing. Please read the notes and when you're ready, please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
