Mathematics
Turning Points
In mathematics, a turning point refers to a point on a graph where the direction of the curve changes, such as from increasing to decreasing or vice versa. It is also where the derivative of a function is equal to zero, indicating a local maximum or minimum. Turning points are important in analyzing the behavior and characteristics of functions and their graphs.
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3 Key excerpts on "Turning Points"
eBook - PDF- J Daniels, N Solomon, J Daniels, N Solomon(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
= 0 f ''( x ) = 0 and this does not satisfy the definition of a maximum or minimum turning point. 136 Mathematics: Hands-On Training The point (0; −8) is not a maximum or minimum turning point, but is called the point of inflection. –3 –2 –1 6 4 2 –2 –4 –6 –8 –10 –12 1 2 3 4 –3 • • point of inflection (0; –8) y = x 3 – 8 x y Assessment activity 2.19 1. Calculate the minimum and maximum Turning Points, as well as the nature of the Turning Points of the following functions. a) f ( x ) = − x 2 + 6 x − 8 b) f ( x ) = 2 x 3 – 9 x 2 – 24 x c) f ( x ) = x 3 – 8 x 2 + 5 x + 14 2. Given: y = x 3 – 6 x 2 + 11 x − 6 a) Calculate the Turning Points. b) Prove that the Turning Points are maximum or minimum Turning Points. c) Sketch the graph. 3. a) Determine the coordinates of the maximum and minimum Turning Points of y = (4 – x )(1 – x ) 2 . b) Prove that the Turning Points are maximum or minimum Turning Points. Point of inflection In this section you will learn how to determine the point of inflection of cubic graphs by using second order derivatives. If, at a stationary point, the curve of f ( x ) is neither a maximum nor a minimum, then this point is the point of inflection : d y dx 2 2 = f ( x ) = 0 Note The point of inflection will be dealt with in LO 2.4.8. 2.4.8 Note The second order derivative = 0. 137 Chapter 2 Functions and algebra Consider the following graphs of the cubic function: Stationary point of inflection Q Q x x • • y = ax 3 y = – ax 3 or Figure A Figure B If the cubic function has only one stationary point, as in figure A and B, this point will be the point of inflection (point Q). • y x • • A Q B Figure C We see that a stationary point is where the tangent to the curve crosses the curve. In figure C, the tangent crosses the curve somewhere between point A and B. At this point, Q, d y dx 2 2 = 0. At point Q the curve changes direction from convex to concave (in figure C) and the sign of d y dx 2 2 changes as the tangent moves through this point.
eBook - PDF- Gregory Hine, Robyn Reaburn, Judy Anderson, Linda Galligan, Colin Carmichael, Michael Cavanagh, Bing Ngu, Bruce White(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
Therefore, in functions whose formulas are known, finding these stationary points simply requires finding the values for x where the derivative is equal to zero. Students may, however, forget that the stationary points found this way are often only local maxima or minima and are not the global maximum or minimum of the function. Students also need to be aware that the derivative may also be equal to zero at a point of inflexion. It was mentioned in a previous section that finding limits can be effective if a dynamic view of the limit is used, which uses covariational reasoning. This is also a useful strategy when thinking about what happens to the val- ues of functions as their x-values take on very large positive or negative val- ues. Here, students need to think about dominant terms. For example, for the function f(x) = x 2 + 4x + 1, once the x-value is greater than two the value of x 2 will always be greater than the value of 4x, so it is the x-squared term that needs to be thought about as the student considers what happens as x → ∞ or Chapter 12: Calculus 371 x → –∞. However, this becomes more complicated when rational functions are considered. Graphing of rational functions The National Curriculum document for Mathematics Specialised states that students should ‘sketch the graph of simple rational functions where the numerator and denominator are polynomials of low degree’ (ACMSM 100) (ACARA, 2015). Rational functions are in the form of f x g x h x ( ) = ( ) ( ) A function in this form is more complicated to graph than a simple polynomial as the function is undefined when h(x) = 0. In addition, a function in this form will have asymptotes. An asymptote is a line that the graph of the function f(x) approaches as the x-values approach infinity. Vertical asymptotes are found at the values where f(x) is undefined because h(x) equals zero, but not all values of x that lead to h(x) having a value of zero will have a vertical asymptote.
eBook - PDFAnton's Calculus
Early Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
178 Chapter 4 / The Derivative in Graphing and Applications INFLECTION POINTS IN APPLICATIONS Inflection points of a function f are those points on the graph of y = f (x) where the slopes of Give an argument to show that the function f(x) = x 4 graphed in Figure 4.1.13 is concave up on the interval (−∞, +∞). the tangent lines change from increasing to decreasing or vice versa (Figure 4.1.14). Since the slope of the tangent line at a point on the graph of y = f (x) can be interpreted as the rate of change of y with respect to x at that point, we can interpret inflection points in the following way: x y y = f (x) Slope decreasing Slope increasing x y y = f (x) Slope increasing Slope decreasing x 0 x 0 Figure 4.1.14 Inflection points mark the places on the curve y = f (x) where the rate of change of y with respect to x changes from increasing to decreasing, or vice versa. This is a subtle idea, since we are dealing with a change in a rate of change. It can help with your understanding of this idea to realize that inflection points may have interpretations in more familiar contexts. For example, consider the statement “Oil prices rose sharply during the first half of the year but have since begun to level off.” If the price of oil is plotted as a function of time of year, this statement suggests the existence of an inflection point on the graph near the end of June. (Why?) To give a more visual example, consider the flask shown in Figure 4.1.15. Suppose that water is added to the flask so that the volume increases at a constant rate with respect to the time t, and let us examine the rate at which the water level y rises with respect to t. Initially, the level y will rise at a slow rate because of the wide base. However, as the diameter of the flask narrows, the rate at which the level y rises will increase until the level is at the narrow point in the neck. From that point on the rate at which the level rises will decrease as the diameter gets wider and wider.
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