UltimateCalculusGuide

UltimateCalculusGuide - T HE U LTIMATE C ALCULUS S TUDY G...

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Unformatted text preview: T HE U LTIMATE C ALCULUS S TUDY G UIDE V 2 ROBERT SMITH FUNCTIONS LINEAR FUNCTIONS A function of the form ݕ ൌ ݉ݔ ൅ ܾ , with slope ݉ and ݕ ‐ intercept ܾ . POWER FUNCTIONS A function of the form ݕ ൌ ݇ݔ ௣ . POLYNOMIAL FUNCTIONS Any linear combination of power functions with integer powers ݌ ൒ 0 . A function of the form ݕ ൌ ∑ ܽ ௞ ݔ ௞ ௡ ௞ୀ଴ with coefficients ܽ ௞ . RATIONAL FUNCTIONS A function of the form ܴሺݔሻ ൌ ܲሺݔሻ/ܳሺݔሻ where ܲ and ܳ are polynomial functions. EXPONENTIAL FUNCTIONS A function of the form ݕ ൌ ܲ ଴ ܽ ௫ or ݕ ൌ ܲ ଴ ݁ ௞௧ . ܲ ଴ is the starting value to an exponential growth or decay model. LOGARITHMIC FUNCTIONS A function of the form ݕ ൌ log ௕ ݔ with logarithmic base ܾ . The equation has the equivalent representation ܾ ௬ ൌ ݔ . ( ln ݔ ൌ log ௘ ݔ ) Logarithm Properties log ௕ ݔ ൌ ln ݔ ln ܾ log ௕ ݉݊ ൌ log ௕ ݉ ൅ log ௕ ݊ log ௕ ݉ ݊ ൌ log ௕ ݉ െ log ௕ ݊ log ௕ ݔ ௣ ൌ ݌ log ௕ ݔ TRIGONOMETRIC FUNCTIONS If we have a unit circle, any point on the circle is defined by the coordinates ሺcos ݔ , sin ݔሻ . We have the following: sin ݔ csc ݔ ൌ 1 sin ݔ cos ݔ sec ݔ ൌ 1 cos ݔ tan ݔ ൌ sin ݔ cos ݔ cot ݔ ൌ cos ݔ sin ݔ INVERSE TRIGONOMETRIC FUNCTIONS Function Inverse sin ݔ sin ିଵ ݔ ൌ arcsin ݔ cos ݔ cos ିଵ ݔ ൌ arccos ݔ tan ݔ tan ିଵ ݔ ൌ arctan ݔ Common Trigonometric Identities sin ଶ ݔ ൅ cos ଶ ݔ ൌ 1 sinሺ2ݔሻ ൌ 2 sin ݔ cos ݔ cosሺ2ݔሻ ൌ cos ଶ ݔ െ sin ଶ ݔ ൌ 2 cos ଶ ݔ െ 1 ൌ 1 െ 2 sin ଶ ݔ cosሺܽ േ ܾሻ ൌ cos ܽ cos ܾ ט sin ܽ sin ܾ sinሺܽ േ ܾሻ ൌ sin ܽ cos ܾ േ cos ܽ sin ܾ HYPERBOLIC FUNCTIONS cosh ݔ ൌ 1 2 ሺ݁ ௫ ൅ ݁ ି௫ ሻ sinh ݔ ൌ 1 2 ሺ݁ ௫ െ ݁ ି௫ ሻ SYMMETRY A function is odd if ݂ሺെݔሻ ൌ െ݂ሺݔሻ holds true. This is a 180 degree rotation about the origin. A function is even if ݂ሺെݔሻ ൌ ݂ሺݔሻ holds true. This is a reflection about the ݕ ‐ axis. LIMITS IDEA OF A LIMIT There is a number ܮ such that as ݔ approaches some number ܿ , ݂ሺݔሻ approaches that ܮ . Mathematically, lim ௫՜௖ ݂ሺݔሻ ൌ ܮ . FORMAL DEFINITION OF A LIMIT If there is a number ܮ such that for any ߳ ൐ 0 , there exists a ߜ ൐ 0 such that if |ݔ െ ܿ| ൏ ߜ , and ݔ ് ܿ , then |݂ሺݔሻ െ ܮ| ൏ ߳ , then lim ௫՜௖ ݂ሺݔሻ ൌ ܮ . LIMIT PROPERTIES Let ݂ and ݃ be functions of ݔ and let ݇ and ܿ be a constant. lim ௫՜௖ ݂݇ ൌ ݇ lim ௫՜௖ ݂ lim ௫՜௖ ሺ݂ േ ݃ሻ ൌ lim ௫՜௖ ݂ േ lim ௫՜௖ ݃ lim ௫՜௖ ݂݃ ൌ ቀlim ௫՜௖ ݂ቁ ቀlim ௫՜௖ ݃ቁ lim ௫՜௖ ݂ ݃ ൌ lim ௫՜௖ ݂ lim ௫՜௖ ݃ lim ௫՜௖ ݇ ൌ ݇ lim ௫՜௖ ݔ ൌ ܿ For indeterminate forms 0/0 or ∞/∞ , we may use L’Hôpital’s rule....
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This note was uploaded on 04/02/2008 for the course 640 162 taught by Professor Scheffer during the Spring '08 term at Rutgers.

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UltimateCalculusGuide - T HE U LTIMATE C ALCULUS S TUDY G...

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