Calculus 220 section 2 1 Describing Graphs of Functions notes by Tim Pilachowski Reminder You will not be able to use a graphing calculator on tests First a quick scan of what we know so far The slope of a curve at a point slope of line tangent to the curve at that point instantaneous rate of change of the curve at that point first derivative evaluated at that point dy Notations are f x and dx To find the first derivative of a given function we have the power rule both general and specific constant multiple rule and sum rule We also have that the derivative of the first derivative is the second derivative 2 d d d y with notations f x and f dx dx dx 2 Chapter 2 applies all this in various circumstances and explores the meaning of it all Example A Consider the graph of y x 2 pictured to the left Reading from left to right From forever left to x 0 the curve is going down graph is falling the slope of the curve is negative The graph bottoms out at the vertex 0 0 where the slope of the curve 0 From x 0 onward to forever right the curve is going up the graph is rising the slope of the curve is positive In technical terms the function f x x 2 is decreasing on the interval x 0 has a minimum at 0 0 and is increasing on the interval 0 x We can say that the minimum value of f is equal to 0 because there are no lower values in the range of f In other words the minimum here is an absolute minimum Let s go back to f x x 3 8 x 2 and take a closer look at the curve pictured to the left below increasing decreasing relative maximum relative minimum The function f x x 3 8 x 2 has no absolute maximum or minimum the range is x Vocabulary to know relative extrema plural and relative extremum singular The function f x 25 x 2 has a limited domain 5 x 5 and range 0 y 5 increasing absolute maximum decreasing The minimum value of the function is 0 Because the minimum occurs at the endpoints of the domain it is called an endpoint extreme value or endpoint extremum The maximum value of the function is 5 Compare the graphs of y x2 and y x on the interval 0 x Neither has an absolute maximum Also both are increasing on this domain but in different ways y x2 y x Going back to f x x 3 8 x 2 concave down 0 2 concave up point of inflection Now consider the graph of y x 3 concave down 0 0 concave up point of inflection Note also that the curve levels out at this spot and that at x 0 the slope of the curve is 0 We ll say that this function is increasing for all values of x in the domain x Not all functions have unbounded domains For example f x 25 x 2 is not defined for values of x less than 5 or greater than 5 The domain of y x is 0 x Rational functions will usually have limited domains they are not defined for values of x that would put a 0 in the denominator 3x 1 For example as x approaches 2 from the left the function f x x 2 approaches As x approaches 2 from the right the function approaches vertical asymptote As x approaches both and the value of f approaches 3 horizontal asymptote Both x intercepts and y intercepts will also be of great interest and use to us in determining an accurate shape for the graph of a function Now its time to put all of this terminology to work in describing the graph of 3 a new function f x x x whose graph is to the right Numbers used e correspond to the attributes listed in your text 1 increasing decreasing 2 maximum minimum 3 points of inflection 4 x and y intercepts 5 domain undefined values 6 vertical horizontal asymptotes