The derivative of a function at one point. 1. Secant lines and tangents. 2. The tangent problem

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1 1. Secant lines and tangents The derivative of a function at one point A secant line (or just secant ) is a line passing through two points of a curve. As the two points are brought together (or, more precisely, as one is brought towards the other), the secant line tends to a tangent line. A tangent line to a curve is a line that only touches the curve at one point, without intersecting it. A more formal definition is that it is the limit of the slopes of the secant lines. 2. The tangent problem Consider the problem of trying to find an equation of the tangent line t to a curve with equation y = f(x) at a given point P with x-coordinate a and y-coordinate f(a) (see Figures 1 and 2 below). Since the tangent line goes through P, all we need in order to determine the equation of the tangent line is the slope m of the line. Then from the slope and the point we will be able to determine the equation of this tangent line. But how can we find the slope if we only know one point? To get around the problem, we first find an approximation to m by taking a nearby point Q on the curve and computing the slope mpq of the secant line PQ. From Figure 2 below we see that : m!" = f x f(a) x a Fig. 1: Tangent line at P Fig. 2: Secant line PQ Fig. 3: Secant lines approaching tangent line Now imagine that Q moves along the curve toward P as in Figure 3. You can see that the secant line rotates and approaches the tangent line as its limiting position. This means that the slope mpq of the secant line becomes closer and closer to the slope m of the tangent line.

2 We write: m = lim!! m!" and we say that m is the limit of mpq as Q approaches P along the curve. Since x approaches a as Q approaches P, we could also use this notation: f x f(a) m = lim!! x a This is very similar to what we have been doing in physics to find the instantaneous velocity from the slope of the tangent line to a position-time graph, except that we were closing in on one point from the outside on both sides (as opposed to keeping one fixed point and bringing the second closer and closer). 3. The derivative of a function at one point The value of the slope of the tangent line at point P is called the derivative of f(x) at point P. In general, the derivative of function f(x) at the point with x-coordinate a is defined as follows: f a = lim!! f x f(a) x a We use f and an apostrophe, pronounced f prime, to denote the derivative of function f. It is painful, but feasible, to find the derivatives of functions using the formula above. But in the near future we will learn general algebraic rules to compute derivatives fast. Example: a) Find the derivative of the function f x =!! (x 3)! + 1 at x = 5, x = 3, and x = 0.

3 b) Find the derivative of the function f x =!! (x 3)! + 1 at x = 3. c) Find the derivative of the function f x =!! (x 3)! + 1 at x = 0.

4 4. The derivative of a function (as a function of x) If we never substitute a by a specific value, we find a generalized equation for the derivative of the original function for any value of the x-coordinate. This is algebraically even a bit more painful to achieve than before, but I m sure you are ready for the challenge! As an example, let s look at the same parabolic function as before: f x =!! (x 3)! The limit L of a function We have seen how limits arise when we want to find the equation of the tangent line to a curve or the slope of that tangent line. In that case we were writing the expression for the slope of a secant line, and then we were bringing one of the two points of intersection between the curve and the secant closer and closer to the first point, to obtain the slope of the tangent to the curve at that first point. This is how we introduced the concept of a derivative. It turns out that the limit of a function (not the limit of the slope of a secant line to a function) also has many uses for the study of functions. In that case, the limit of the function as x approaches a specific x-coordinate a is defined as: lim f x = L!! We say the limit of f(x), as x approaches a, equals L. We will be looking at some of these applications, but not today.

5 Finding the derivative function of a parabola using Geogebra In this activity, we are going to use Geogebra to geometrically determine the derivatives of parabolic functions at different points. Then we will try to write some rules for finding the generalized derivative function of parabolic functions (we will use them a lot in physics!). Open the software Geogebra. Input the parabolic function f x =!! (x 3)! + 1. Does the graph of this parabola make sense? (What does the +1 do? What does the - 3 do? What does the ½ do?) Use your algebraic skills to expand the expression for f(x) so you can write it in its standard form f x = ax! + bx + c. This will make it easier to guess derivative rules. If you are not sure whether you expanded correctly, enter your result in the Input bar with a different name, like g(x), and check whether both functions are coincident. Place a point anywhere on the parabola. Rename this point P. Using the Move tool, move P around to make sure you actually placed it on the line. Point P is going to be the point on the parabola where we are finding the derivative. Using the constructed line menu (4 th starting from the left), construct a tangent to the curve at point P. We are going to name this tangent line t. To do this, double click on the line; a menu should appear which shows Tangent[P, f]. Type t = in front of it so that it reads t = Tangent[P, f]. Press OK. We are now going to name m the slope of the tangent line t to the parabola at point P. To achieve this, type m = slope(t) in the Input bar. Finally we will draw another point A that will have the same x-coordinate as point P but its y-coordinate will be the slope of the tangent line at that point. Basically, point A is giving us the value of the derivative of the parabola at point P. To achieve this, type A= (x(p), m) in the Input bar. You will notice that point A is not on the parabola; does this make sense? Move point P around. The position of point A should move accordingly, since the value of the derivative at different points on the parabola is different (the slope of the tangent line is changing). You will notice as point P goes through the vertex of the parabola that point A is on the x-axis. What does this mean? Does it make sense? Keep on moving point P around and try to follow the position of point A. What kind of graph does point A move on? Is there a pattern, like a specific curved or straight line? Right-finger click (or two-finger click on Mac) on point A and select Trace on. Now move point P around again. Find the equation (as a function of x) of the the graph formed by point A as it moves. Call this function f (x). f (x) =

6 This function f (x) is the derivative function of the original parabola; for any value of x, the derivative function will give us the value of the slope of the tangent line to the parabola at this location. Pretty cool, uh? Take a snapshot of your Geogebra figure (including the expression for the function), and record your results in the first row of the table below. Then see if you can start to find a pattern about how f (x) is related to f(x). Write your guess in the Notes column. f(x) vertex form f(x) standard form f (x) derivative function Notes about patterns observed Try other parabolas, changing the values of the coefficients (in the standard or vertex form of the function), and refine your guess about how to find the derivative function of any parabola. Record all of your attempts in the table above, and keep on making notes about the patterns you observe.