Calculus: An integrated approach to functions and their by Robin J. Gottlieb

By Robin J. Gottlieb

An important criticism of professors instructing calculus is that scholars would not have the precise historical past to paintings in the course of the calculus path effectively. this article is concentrated at once at this underprepared viewers. this can be a single-variable (2-semester) calculus textual content that comes with a conceptual re-introduction to key precalculus rules through the exposition as applicable. this can be the fitting source for these faculties facing poorly ready scholars or for colleges introducing a slower paced, built-in precalculus/calculus path.

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Write a formula for this function, where s is the distance between the man and the lamppost. 42. Assume that f is a function with domain (−∞, ∞). Which of the following statements is true for every such function f and all p, w, and z in the domain of f ? If a statement is not true for every function, find a function for which it is false. 3 Representations of Functions 43 43. , an equation. Then comment on the validity of the statement, making qualifications if necessary. You will need to define your variables.

They will also serve as a review of some geometry including similar triangles and the Pythagorean Theorem. ) REMARK Examples in a mathematics text are meant to be read actively, with a pencil and paper. A solution will have more impact, and stay with you longer, if you have spent a bit of time tackling the problem yourself. Read the problems that follow and try each one on your own before reading the solutions. The problem-solving strategies highlighted below should help you out. Think of them as a way of coaching yourself through a problem.

The functions f and g are equal if: f and g have the same domain, and f (x) = g(x) for every x in the domain. 2 For example, the functions f (x) = x x−x and g(x) = x − 1 are not equal; the domain of g is all real numbers, while the domain of f does not include x = 0. 5 SOLUTION √ x+1 . Find the natural domain of f . Suppose f (x) = x−2 √ In order for x + 1 to be defined, we need x + 1 ≥ 0. Therefore, x ≥ −1. 7 In fact, it was not until the nineteenth century that the mathematician Lejeune Dirichlet formulated the modern definition of a function as an input-output mapping such as we gave in the previous section.

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