# Calculus and Ordinary Differential Equations (Modular by David Pearson

By David Pearson

Professor Pearson's e-book starts off with an advent to the world and an evidence of the main favourite capabilities. It then strikes on via differentiation, exact features, derivatives, integrals and onto complete differential equations.

As with different books within the sequence the emphasis is on utilizing labored examples and tutorial-based challenge fixing to realize the arrogance of scholars.

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Extra resources for Calculus and Ordinary Differential Equations (Modular Mathematics Series)

Example text

Later we shall see the useful role of flow diagrams in clarifying the process of differentiation for more functions in combination. FURTHER EXERCISES 1. For which values of x is it true that (1 + x)/(l + 2x) ~ O? Draw a flow diagram for the function J(l + x)/(l + 2x). What are the domain and range of this function? 2. Using only the basic operations 'square' and 'add one', show how to evaluate the function x 2 + 2x + 4. If f == 'square' and g == 'add one', use the notation for composition of functions to "express the function x 2 + 2x + 4 in terms of compositions of'jwith g.

If f(x) == sin x (with x in radians), use your calculator to evaluate [f(X2) - f(XI)]/(X2 - Xl) for various pairs Xl, X2 with Xl close to X2, say IX2 - xII of the order 10- 1,10-2,10-3 , etc. Verify that your results are consistent withf'(x) == cos x. Carry out a similar investigation for the function u(() == (3 + 11(. 3. Explain why any even differentiable function has 1'(0) == O. If 1 is an even function, what is the relation between f'(x) and f'( -x)? How are f'(x) and f'( -x) related if f is an odd function?

Paying careful attention to signs, show that cos/sin"! x) == sin(cos- I x) == Vl- x 2 (-1 ~ x ~ 1). Show that cosftan" x) == 1/(Vl +x2 ) , for all x E JR. Summary Starting from a small collection of basic functions, such as powers, trigonometric functions, the modulus function, later to be supplemented by the exponential, logarithm, and related functions, we can combine them in various ways to make new functions, and thereby very considerably enlarge our collection. For this reason, anyone with a scientific or programmable calculator, or computer, will have access numerically to a huge variety of different functions.