![]() ![]() ![]() For example, if x = 100,000,000, which is approximately the value of e, accurate to seven decimal places. To visually verify that the limit is accurate, use your graphing calculator to calculate for a very large value of x. That small difference becomes magnified when raised to the x power, and the result is e. Justification: Using the first special limit rule, Technically, is a number very close to, but not quite, one. Consider The denominator is growing larger more quickly than in the previous example, while 7 remains constant. This limit rule works the same way for other eligible n values. The denominator will eventually get so large, in fact, that no matter how large the numerator is, the fraction has an extremely small value so close to zero that the difference is negligible. Remember that the denominator is approaching infinity, so it is getting very large, while the numerator is remaining fixed. Justification: In the simplest case (c = 1, n =1), you are considering You should know the graph of 1/x by heart, and its height clearly approaches zero as x approaches infinity. if c is a nonzero constant and n is a positive integer (BC students have still another topic to cover concerning limits-L’Hopital’s Rule-but that occurs in Chapter 5.) I call these “special” limits because we accept them without formal proof and because of the special way they make you feel all tingly inside.ġ. Before you are completely proficient at limits, however, there are four limits you need to be able to recognize on sight. Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.You have a number of techniques available to you now to evaluate limits and to interpret the continuity that is dependent on those limits.Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.Use technology to solve problems, experiment, interpret results, and support conclusions.Model a written description of a physical situation with a function, a differential equation, or an integral.Communicate mathematics and explain solutions to problems verbally and in writing.Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.Understand the meaning of the definite integral as a limit of Riemann sums and as the net accumulation of change and use integrals to solve problems.Understand the meaning of the derivative in terms of a rate of change and local linear approximation and use derivatives to solve problems. ![]() They should understand the connections among these representations. Work with functions represented in multiple ways: graphical, numerical, analytical, or verbal.Students who are enrolled in AP Calculus BC are expected to: Students learn how to use technology to help solve problems,Įxperiment, interpret results, and support conclusions. Graphically, numerically, analytically, and verbally, and to makeĬonnections amongst these representations. The course teaches students to approachĬalculus concepts and problems when they are represented Limits, derivatives, definite integrals, the Fundamental Theorem The AP course covers topics inĭifferential and integral calculus, including concepts and skills of Learned in AB to different types of equations and introduces the Semester college calculus courses and extends the content ![]() AP Calculus BC is roughly equivalent to both first and second ![]()
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