China
Africa
Explore chapters and articles related to this topic
Solving quadratic equations
Published in John Bird, Basic Engineering Mathematics, 2017
Quadratic equations have many applications in engineering and science; they are used in describing the trajectory of a ball, determining the height of a throw, and in the concept of acceleration, velocity, ballistics and stopping power. In addition, the quadratic equation has been found to be widely evident in a number of natural processes; some of these include the processes by which light is reflected off a lens, water flows down a rocky stream or even the manner in which fur, spots or stripes develop on wild animals. When traffic police arrive at the scene of a road accident, they measure the length of the skid marks and assess the road conditions. They can then use a quadratic equation to calculate the speed of the vehicles and hence reconstruct exactly what happened. The U-shape of a parabola can describe the trajectories of water jets in a fountain and a bouncing ball, or be incorporated into structures like the parabolic reflectors that form the base of satellite dishes and car headlights. Quadratic functions can help plot the course of moving objects and assist in determining minimum and maximum values. Most of the objects we use every day, from cars to clocks, would not exist if someone somewhere hadn’t applied quadratic functions to their design. Solving quadratic equations is an important skill required in all aspects of engineering.
Extrema and Variational Calculus
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
This is easily recognizable as a quadratic function in x whose graph is a parabola. Thus, we have a quadratic approximation of f (x). If x = a is a critical point of f (x), then this approximation leads to f(x)=f(a)+12!f″(a)(x−a)2.
Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative
Published in International Journal of Mathematical Education in Science and Technology, 2018
Annie Burns-Childers, Draga Vidakovic
Not only is the quadratic function an important concept in a college algebra or other introductory mathematics courses, but it is also important in calculus courses. In calculus courses, usually a first year calculus course, students are taught how the derivative function can be used to find critical points, c, of a function. Critical points for f are the interior points c of the domain of f for which or does not exist [13]. Once the critical point is found, algebraic calculations can be used to find f(c), the corresponding y-value. For quadratic functions, (c, f(c)) form the coordinates of the vertex. The vertex can be identified as a local maximum or local minimum by the application of what is often referred to as the first-derivative test (see Figure 1).
Intra-mathematical connections made by high school students in performing Calculus tasks
Published in International Journal of Mathematical Education in Science and Technology, 2018
Javier García-García, Crisólogo Dolores-Flores
This was identified in seven students (28%). The connection indicates that these students consider a function as a rule of correspondence, that is to say, there is an initial value x (x ∈ Df) that is transformed by f(x) to generate a value for y (belongs to the image of the function). In this way, they assume that a function f(x) is a formula that transforms values of the domain and they declare the uniqueness of the results obtained for y as values that belong to the domain of the function (see excerpt from E7). Interviewer: Here you can see an expression f(x) equal to something (interviewer shows the expression f (x) = 3x2); for you, what does this expression represent and what are the elements that are part of it?E7: It is a function f that is equal to y; it is assumed that the value you add to x will give you a value in y. It is a relation, a function, and can be both derived, or given values to create a graph.Graph of a quadratic function is a parabola