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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 25

Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 25. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | 6.1 A Profile of Quadratics from a Calculus Perspective 221 Notice that we have drawn families of parabolas. This is because shifting a graph vertically does not change its slope at a given point. The derivative determines a function only up to a constant. We have indicated by a dotted line the parabolas corresponding to those given in Example 6.2. Although the derivative gives us information about the shape of the parabola it doesn t give us any information about the vertical positioning of the parabola. The derivative will not help us pick out any one member of the family of parabolas drawn. The Graph of a Quadratic Function The derivative of the quadratic function x ax2 fex c is the linear function x 2ax fe. The linear function is not horizontal because a 0.1 Any nonhorizontal linear function has exactly one x-intercept at this intercept the line cuts the x-axis and changes sign. This corresponds to a turning point of the quadratic function either changes from increasing to decreasing or vice versa. The turning point of a parabola is called the vertex of the parabola. The derivative gives us easy access to the shape of the parabola and the x-coordinate of its vertex we could have used derivatives to solve the problem in Example 6.1 efficiently and exactly. Question How can we find the x -coordinate of the vertex of the parabola Answer The x-coordinate of the vertex is the zero of the derivative function so x fe. There is no need to memorize this. In practice simply differentiate and find the zero of . Question How can we determine whether the parabola opens upward or downward Answer If the graph off has a positive slope the parabola looks like If the graph off has a negative slope the parabola looks like If x ax2 fex c then the slope of the graph of is given by a. Therefore we know that if a 0 then the parabola opens upward if a 0 then the parabola opens downward. Definition The slope of is written read double prime and is called the second derivative of . In .