Differential Calculus – Definitions, Rules and Theorems Sarah Brewer, Alabama School of Math and Science Precalculus Review Functions, Domain and Range : a function from to assigns to each a unique the domain of is the set - the set of all real numbers for which the function is defined 푓 푋 → 푌 푓 푋 푌 푥 ∈ 푋 푦 ∈ 푌 is the image of under , denoted ( ) 푓 푋 the range of is the subset of consisting of all images of numbers in 푌 푥 푓 푓 푥 is the independent variable, is the dependent variable 푓 푌 푋 is one-to-one if to each -value in the range there corresponds exactly one -value in the domain 푥 푦 is onto if its range consists of all of 푓 푦 푥 푓Implicit v. Explicit Form 푌 3 + 4 = 8 implicitly defines in terms of = + 2 explicit form 푥 푦 푦 푥 3 푦Graphs− 4 o푥f Functions A function must pass the vertical line test A one-to-one function must also pass the horizontal line test Given the graph of a basic function = ( ), the graph of the transformed function, = ( + ) + = + + 푦 푓 푥, can be found using the following rules: 푐 - vertical stretch (mult. -values by ) 푦 푎푓 푏푥 푐 푑 푎푓 �푏 �푥 푏�� 푑 - horizontal stretch (divide -values by ) 푎 푦 푎 – horizontal shift (shift left if > 0, right if < 0) 푏푐 푥 푐 푏 푐 > 0 < 0 푏 - vertical shift (shift up if 푏 , down if 푏 ) You푑 should know the basic 푑graphs of: 푑 = , = , = , = | |, = , = , = , 3 1 = sin , =2 cos , 3 = tan , = csc , = sec , = cot 푦 푥 푦 푥 푦 푥 푦 푥 푦 √푥 푦 √푥 푦 푥 For푦 further푥 discussion푦 푥 of 푦graphing,푥 see푦 Precal 푥and푦 Trig Guides푥 푦 to Graphing푥 Elementary Functions Algebraic functions (polynomial, radical, rational) - functions that can be expressed as a finite number of sums, differences, multiples, quotients, & radicals involving Functions that are not algebraic are trancendental (eg. trigonometric,푛 exponential, and logarithmic 푥 functions) Polynomial Functions ( ) = + +. + + + = leading푛 coefficient푛− 1 2 푓 푥 푎푛푥 푎푛−1푥 푎2푥 푎1푥 푎0 = lead term 푎푛 = 푛degree of polynomial (largest exponent) 푎푛푥 = constant term (term with no ) 푛 푎Lead0 term test for end behavior: even푥 functions behave like = , odd functions behave like = 2 3 Odd and Even Functions 푦 푥 푦 푥 = ( ) is even if ( ) = ( ) = ( ) is odd if ( ) = ( ) 푦 푓 푥 푓 −푥 푓 푥 푦Rational푓 푥 Functions푓 −푥 −푓 푥 ( ) ( ) = , , , ( ) 0 ( ) 푝 푥 푓 푥 푝 푞 − 푝표푙푦푛표푚푖푎푙푠 푞 푥 ≠ Zeros of푞 a 푥function Solutions to the equation ( ) = 0; When written in the form ( , 0) are called the x-intercepts of the function 푓 푥 Y-intercepts 푥 Found by evaluating (0) (plugging 0 in for x) Vertical Asymptotes 푓 Found by setting the denominator equal to 0 and solving for x. Note that any factors found in both numerator and denominator will result in holes rather than vertical asymptotes. The graph will approach but never cross a vertical asymptote. Horizontal/Oblique Asymptotes If degree of the numerator is smaller than the degree of denominator, there will be a horizontal asymptote at = 0. If the degree of the numerator is the same as the degree of the denominator, there will be a horizontal 푦 asymptote at = , where is the constant ratio of the leading coefficients. If the degree of the numerator is one higher than the degree of the denominator, there will be an 푦 푐 푐 oblique asymptote (one degree higher - linear, 2 degrees higher, quadratic, etc), found by long division. Limits Informal Definition of a Limit If ( ) becomes arbitrarily close to a single number as approaches from either side, the limit of ( ), as approaches , is . 푓 푥 퐿 푥 푐 푓Note:푥 the푥 existence or nonexistence푐 퐿 of ( ) at = has no bearing on the existence of the limit of ( ) as approaches . 푓 푥 푥 푐 Formal푓 푥 Definition푥 of a푐 Limit Let be a function defined on an open interval containing (except possibly at ) and let be a real number. The statement lim ( ) = means that for each > 0, there exists a > 0 such that if 푓 푐 푐 퐿 0 < | | < , then | ( ) | < . 푥→푐 푓 푥 퐿 휀 훿 Translation:푥 − 푐 If given훿 any푓 arbitrarily푥 − 퐿 small휀 positive number , there exists another small positive number such that ( ) is -close to whenever is -close to , then the limit of as approaches exists 휀 훿 and is equal to . 푓 푥 휀 퐿 푥 훿 푐 푓 푥 푐 Note: if the limit퐿 of a function exists, then it is unique. Example proof of a limit using the definition Problem: Prove that lim 2 + 5 = 1 휺 − 휹 Proof: We want to show that for each > 0 there exists a > 0 such that |(2 + 5) ( 1)| < 푥→−3 푥 − whenever | ( 3)| < . Since we can rewrite |2 + 5 + 1| = |2 + 6| = 2| + 3|, if we take = , 휀 훿 푥 − − 휀 휀 | ( 3)| < |(2 + 5) ( 1)| = 2| + 3| < 2 = then whenever푥 − − 훿 , we have 푥 푥 푥 . Since we have훿 2 휀 found a that works for any , we have proved that the limit of the function is 1. 푥 − − 훿 푥 − − 푥 ∙ 2 휀 Evaluating훿 Limits Analytically휀 − Basic Limits , , > 0 , , , lim ( ) = , lim ( ) = 1. Constant lim = 퐿푒푡 푏 푐 ∈ ℝ 푛 푎푛 푖푛푡푒푔푒푟 푓 푔 − 푓푢푛푐푡푖표푛푠 푥→푐 푓 푥 퐿 푥→푐 푔 푥 퐾 2. Identity lim = 푥→푐 푏 푏 3. Polynomial lim = 푥→푐 푥 푐 4. Scalar Multiple lim [ 푛( )] 푛= 푥→푐 푥 푐 5. Sum or Difference lim [ ( ) ± ( )] = ± 푥→푐 푏푓 푥 푏퐿 lim [ ( ) ( )] = 6. Product 푥→푐 푓 (푥) 푔 푥 퐿 퐾 7. Quotient lim푥→푐 = , 0 푓 (푥) 푔 푥 퐿퐾 푓 푥 퐿 [ ] 8. Power lim푥→푐 �푔(푥 )� =퐾 퐾 ≠ 푛 푛 푥→푐 Note: If substitution yields , an푓 indeterminate푥 퐿 form, the expression must be rewritten in order to 0 evaluate the limit. 0 Examples: ( )( ) lim = lim = lim (2 3) = 5 1. 2 2푥 −푥−3 푥+1 2푥−3 2. lim푥→−1 푥+1 = lim 푥→−1 푥+1 =푥→lim−1 푥 − − = lim = √2+푥−√2 √2+푥−√2 √2+푥+√2 2+푥−2 1 1 ( ) ( ) lim푥→0 푥 = lim푥→0 푥 ∙ √2+푥+=√2lim 푥→0 푥�√2+=푥+lim√2� 푥→0 √=2+lim푥+√(22 +2√2) = 2 3. 2 2 2 2 2 2 푥+ℎ −푥 푥 +2푥ℎ+ℎ −푥 2푥ℎ+ℎ ℎ 2푥+ℎ ℎ→0 ℎ ℎ→0 ℎ ℎ→0 ℎ ℎ→0 ℎ Squeeze Theorem ℎ→0 푥 ℎ 푥 If ( ) ( ) ( ), that is, a function is bounded above and below by two other functions, and lim ( ) = lim ( ) = , that is, the two upper and lower functions have the same limit, then 푔 푥 ≤ 푓 푥 ≤ ℎ 푥 lim ( ) = , that is, the limit of the function that is “squeezed” between the two functions with 푥→푐 푔 푥 푥→푐 ℎ 푥 퐿 equal limits must have the same limit. 푥→푐 푓 푥 퐿 Example: Prove that lim sin = 0 using the Squeeze Theorem. Solution: 1 sin 1. Multiplying2 both sides of each inequality by yields sin . 푥→0 푥 푥 Now, applying limits, we have lim lim sin lim 2 0 2lim 2 sin 2 0 − ≤ 푥 ≤ 푥 −푥 ≤ 푥 푥 ≤ 푥 Since our limit is bounded above and below2 by 0, by the2 Squeeze theorem,2 we have that 2 푥→0 −푥 ≥ 푥→0 푥 푥 ≥ 푥→0 푥 ↔ ≥ 푥→0 푥 푥 ≥ lim sin = 0. 2 Two푥 →important0 푥 푥 limits that can be proven using the Squeeze Theorem that we will use to find many other limits are: sin 1 cos lim = 1 lim = 0 푥 − 푥 푥→0 푎푛푑 푥→0 Examples: 푥 푥 1. lim = lim = 1 1 = sin 2푥 sin 2푥 3푥 2 2 2 ( ) ( ) 2. lim푥→0 sin 3푥 푥→=0 lim2푥 ∙ sin 3푥 ∙ 3 ∙ ∙ 3 3 = lim = sin 푥+ℎ −sin 푥 sin 푥 cos ℎ+cos 푥 sin ℎ−sin 푥 sin 푥 cos ℎ−1 +cos 푥 sin ℎ limℎ→0 ( sinℎ ) +ℎ→0lim(cos ) ℎ = ( sin ) ℎ0→+0 cos 1 =ℎ cos 1−cos ℎ sin ℎ ℎ→0 � − 푥 ∙ ℎ � �ℎ→0 푥 ∙ ℎ � − 푥 ∙ 푥 ∙ 푥 Continuity If lim ( ) = ( ), we say that the function is continuous at . Discontinuities푥→푐 푓 푥 occur푓 푐 when either 푓 푐 1. ( ) is undefined, 2. lim ( ) does not exist, or 푓 푥 3. lim ( ) ( ) 푥→푐 푓 푥 If a discontinuity푥→푐 푓 푥 ≠ can푓 푐 be removed by inserting a single point, it is called removable. Otherwise, it is nonremovable (e.g. verticals asymptotes and jump discontinuities) One-Sided Limits lim ( ) = lim+ ( ) = 푥→푐 푓 푥 퐿 푙푖푚푖푡 푓푟표푚 푡ℎ푒 푟푖푔ℎ푡 − Examples푥→푐 푓 푥 : 퐿 푙푖푚푖푡 푓푟표푚 푡ℎ푒 푙푒푓푡 | | | | 1, 2 lim = 1 = 1. Note that 1, < 2 푥−2 푥−2 + , 1 푥 ≥ lim푥→2 푥(−2) , ( ) = 푥−2 � 2. 1 , >− 1 푥Since we want the limit from the right, we look at the + 푥 푥 ≤ piece푥→ where1 푓 푥 the푤 functionℎ푒푟푒 푓 푥 is defined� > 1. Plugging 1 into 1 gives us that the limit is 0 . − 푥 푥 Continuity at a point − 푥 A function is continuous at if the following 3 conditions are met: 1. ( ) is defined 푓 푐 2. Limit of ( ) exists when approaches 푓 푐 3. Limit of ( ) when approaches is equal to ( ) 푓 푥 푥 푐 Continuity푓 on푥 an open푥 interval 푐 푓 푐 A function is continuous on an open interval if it is continuous at each point in the interval. A function that is continuous on the entire real line ( , )is everywhere continuous. Continuity on a closed interval −∞ ∞ A function f is continuous on the closed interval [ , ] if it is continuous on the open interval l ( , ) and lim ( ) = ( ) and lim ( ) = ( ).