Calculus 220 section 1 2 The Slope of a Curve at a Point notes by Tim Pilachowski Finding the slope of a line is fairly simple once you get the hang of it because the slope is the same everywhere on the line Another way to say the same thing is that while the value of a linear function such as y 4x 1 may change its rate of change is constant There is only one type of linear function whose value doesn t change What type is that Consider the function f x x 3 8 x 2 pictured to the left To reproduce this on your graphing calculator set your window to 10 10 by 8 12 The curve s slope is far from constant From until someplace between x 2 and x 1 the function is increasing has a positive slope The curve then levels off slope 0 a relative maximum changes direction and is decreasing slope is negative until someplace between x 1 and x 2 On the other side of this relative minimum toward the function is once again increasing The task at hand will be to find a way to describe the rate of change of f x in a way that not only makes sense intuitively but also complements all of the other information we know about functions and their graphs We ll develop a method by traveling along the curve a piece at a time Example A First stop the open interval 2 The series of graphs below zoom in on the point 5 51 You can duplicate this on your graphing calculator Begin with the window in Standard Use the 2 8 TRACE key to get your cursor close to x 2 5 press the ZOOM key then choose Zoom In Each time you press ENTER the calculator will show pictures similar to those below Dimensions of the graph window are given below each picture 6 1 by 2 875 9 875 4 1 by 4 875 7 875 3 2 by 5 875 6 875 2 8 2 2 by 6 075 6 675 2 6 2 4 by 6 275 6 475 2 55 2 45 by 6 325 6 425 We could continue the process but the pictures would change very little Notice what happens as we zoom in It 5 51 is clear in the original graph that at the point the function has a steep positive slope As we zoom in 2 8 the graph appears to have less curve and is more like a line The final picture is reproduced to the left with a grid marked in intervals of 0 01 We can 5 51 2 8 roughly estimate the slope by approximating points and calculating using the formula for the slope of a line The point at the top of the picture is approximately 2 495 6 425 The point at the bottom of the picture is approximately 2 505 6 325 We estimate the 6 325 6 425 0 1 slope to be 10 This estimate is close but is not exact the 2 505 2 495 0 01 slope at this point is actually 10 75 Before we move to the next interval on our curve near the relative maximum we need to be a little more formal in our definitions The slope of a curve at a point P is defined to be the slope of the line tangent to the curve at point P The function f x x 3 8 x 2 has a relative maximum someplace in between x 2 and x 1 The lines tangent to the curve vary greatly see dotted lines in the picture below How can we estimate the slope of the curve near this point Also how can we determine the coordinates of this relative maximum The answer is that we need some more rigorous mathematics of the kind which we ll derive in the next section For now we ll simply state as a fact that for the function f x x 3 8 x 2 the slope of the curve at a point x f x can be found with the formula 3 x 2 8 Example B Find the slope of the curve f x x 3 8 x 2 at x 2 Answer 4 L Example B extended Find the equation of this tangent line Answer y 4x 18 L Both f and its tangent at x 2 have been graphed in the picture to the left Example C Find the x coordinates of the relative maximum and relative minimum of f Answers 2 2 3 x 2 2 is the exact answer You could use the TRACE key and arrows on your graphing calculator but the 3 answer you get would be only a decimal approximation Example D How does the graph of f x x 3 8 x 2 relate to the graph of the slope of the curve equation f x 3 x 2 8 Both are graphed in the picture to the left Note that the parabola has positive values on 2 2 corresponding to a positive slope for f At x 2 the open interval 2 3 3 the parabola is at 0 corresponding to the relative maximum on f On the open interval 2 2 x 2 the parabola has negative values corresponding to a negative slope for f At x 0 the 3 3 2 parabola has a vertex corresponding to what we ll later call a point of inflection on f At x 2 the parabola 3 8 is at 0 corresponding to the relative maximum on f On the open interval x the parabola once again 3 has positive values corresponding to a positive slope for f 2 Example E Many of the practice exercises in the text rely on the fact that the slope of y x 2 y 2x the graph of y x 2 at the point x y 2x Note that this formula fits what we see in the graph When x is negative the function is decreasing and the slope is negative When the curve levels out at x 0 the slope also equals 0 To the right of the vertex the function is increasing and the slope is positive These concepts will become increasingly important as we go along particularly the notion of the slope equaling 0 where a curve levels out Using this idea will provide us the means of determining the precise location of all relative maxima and minima