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Download PDF by K. O. Friedrichs: Advanced Ordinary Differential

By K. O. Friedrichs

ISBN-10: 0677009607

ISBN-13: 9780677009605

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Example text

The equation of the graph is y ෇ x 2, which represents a parabola (see Appendix C). The domain of t is ‫ޒ‬. The range of t consists of all values of t͑x͒, that is, all numbers of the form x 2. But x 2 ജ 0 for all numbers x and any positive number y is a square. So the range of t is ͕ y y ജ 0͖ ෇ ͓0, ϱ͒. This can also be seen from Figure 8. Խ 1 0 1 x EXAMPLE 3 If f ͑x͒ ෇ 2x 2 Ϫ 5x ϩ 1 and h 0, evaluate f ͑a ϩ h͒ Ϫ f ͑a͒ . h SOLUTION We first evaluate f ͑a ϩ h͒ by replacing x by a ϩ h in the expression for f ͑x͒: FIGURE 8 f ͑a ϩ h͒ ෇ 2͑a ϩ h͒2 Ϫ 5͑a ϩ h͒ ϩ 1 ෇ 2͑a 2 ϩ 2ah ϩ h 2 ͒ Ϫ 5͑a ϩ h͒ ϩ 1 ෇ 2a 2 ϩ 4ah ϩ 2h 2 Ϫ 5a Ϫ 5h ϩ 1 Then we substitute into the given expression and simplify: f ͑a ϩ h͒ Ϫ f ͑a͒ ͑2a 2 ϩ 4ah ϩ 2h 2 Ϫ 5a Ϫ 5h ϩ 1͒ Ϫ ͑2a 2 Ϫ 5a ϩ 1͒ ෇ h h The expression f ͑a ϩ h͒ Ϫ f ͑a͒ h in Example 3 is called a difference quotient and occurs frequently in calculus.

Convert from degrees to radians. (b) Ϫ18Њ (a) 300Њ 2. Convert from radians to degrees. (a) 5␲͞6 (b) 2 3. Find the length of an arc of a circle with radius 12 cm if the arc subtends a central angle of 30Њ. 4. Find the exact values. (a) tan͑␲͞3͒ (b) sin͑7␲͞6͒ (c) sec͑5␲͞3͒ 5. Express the lengths a and b in the figure in terms of ␪. 24 6. If sin x ෇ 3 and sec y ෇ 4 , where x and y lie between 0 and ␲ր 2, evaluate sin͑x ϩ y͒. 1 a 5 7. Prove the identities. ¨ (a) tan ␪ sin ␪ ϩ cos ␪ ෇ sec ␪ b FIGURE FOR PROBLEM 5 (b) 2 tan x ෇ sin 2x 1 ϩ tan 2x 8.

2a 2 ϩ 4ah ϩ 2h 2 Ϫ 5a Ϫ 5h ϩ 1 Ϫ 2a 2 ϩ 5a Ϫ 1 h ෇ 4ah ϩ 2h 2 Ϫ 5h ෇ 4a ϩ 2h Ϫ 5 h Representations of Functions There are four possible ways to represent a function: ■ verbally (by a description in words) ■ numerically (by a table of values) ■ visually (by a graph) ■ algebraically (by an explicit formula) If a single function can be represented in all four ways, it’s often useful to go from one representation to another to gain additional insight into the function. ) But certain functions are described more naturally by one method than by another.

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Advanced Ordinary Differential by K. O. Friedrichs

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