Want to graph a function? A calculator will basically plot a hundred points and connect the dots. There's of course no nefarious intent on their part - but calculators often try to answer our questions using techniques that are not quite up to the task. I know what you are thinking - couldn't we just graph the function on our calculator to discover the same thing? Interestingly, the answer to that question is "No"!Ĭalculators can quickly calculate things, but sometimes they lie. For example, if we know that as we approach some $x$-value from the left, the slopes of the corresponding tangent lines increase without bound, then we can conclude there is a vertical asymptote at that $x$-value. Knowing the slopes of tangent lines at various points on the graph of a function can help one better understand the graph of the overall function. Often, finding places where the tangent line has horizontal slope reveals maximum (and minimimum) values. Notice how the slope of the tangent line at the highest point on the curve below is horizontal. One naturally would be interested in knowing how many widgets must one sell to maximize profit. Suppose the function in question gave the expected profit earned upon selling $x$ widgets at some company. Let's consider just a couple of these applications: The first such problem is this: Given a point on the graph of some given function, what is the slope of the tangent line to the function at that point?Īs simply stated as this problem is, its applications are huge and far-reaching. Hence, the graph of y = a e x + bx has NO horizontal tangent line if -a/b <= 0ġ) Find all points on the graph of y = x 3 - 3 x where the tangent line is parallel to the line whose equation is given by y = 9 x + 4.Ģ) Find a and b so that the line y = - 2 is tangent to the graph of y = a x 2 + b x at x = 1.ģ) Find conditions on a, b and c so that the graph of y = a x 3 + b x 2 + c x has ONE tangent line parallel to the x axis (horizontal tangent).An initial study of calculus can be miraculously distilled down to just a couple of carefully stated general problems.
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