Calculus Limit and Continuity Cheat Sheet

Unit 1: Limit and Continuity 1. lim𝑓 (𝑥) = lim! 𝑓 (𝑥) = lim" 𝑓 (𝑥) =𝐿 !→# !→# !→# a. If f(x) is continuous of [a,b] and c is on this interval, then. lim𝑓 ( 𝑥 ) =𝑓(𝑐) !→$ 2. Intermediate Value Theorem – If f(x) is continuous on a closed interval [a,b] and k is any number between f(a) and f(b), inclusive, then there is at least one number x in the interval [a,b] such that f(x) = k. 3. Squeeze Theorem a. If 𝑓( 𝑥 ) ≤𝑔(𝑥) - ≤ℎ(𝑥) on the interval [a,b] and lim𝑓(𝑥)=lim ℎ(𝑥)=𝐿 , then !→$ !→$ lim𝑔 (𝑥) =𝐿 !→$ Units 2 and 3: Differentiation ) ( !+, ).)(!) 1. 𝑚#%&'#(& = , 2. 𝑚/012#02#0&341 = - lim ,→5 3. or 𝑚#%&'#(& = ) ( !+, ).)(!) 6 [𝑓 ( 𝑥) ]=𝑓 7 ( 𝑥) = - lim 6! , ) ( !+, ).)(!) ) (!# ).)(!$ ) !# .!$ or 𝑚/012#02#0&341 = lim ! →! # ) (!# ).)(!$ ) $ !# .!$ , ,→5 4. If a function f is differentiable at 𝑥5 , then f is continuous at 𝑥5 . 6 6 6 5. 6! [𝑓(𝑥) ±𝑔 (𝑥)] = - 6! [ 𝑓(𝑥)] ± 6![𝑔 ( 𝑥)] 6. 6 6! 6 [𝑐𝑓( 𝑥)] = 𝑐∗ 6! [𝑓 (𝑥)] 6 7. Power Rule: 6! [𝑥 0 ] =𝑛𝑥 0.8 6 6 6 8. Product Rule: 6! [ 𝑓 ( 𝑥 ) 𝑔( 𝑥)] =𝑓 ( 𝑥 ) 6! [ 𝑔( 𝑥 )] +𝑔 ( 𝑥 ) 6![𝑓 ( 𝑥) ] 6 )(!) 9. Quotient Rule: 6! 6((!) 7 = 6 % % ( (! ) [) (! )] .)(!) %& %& [( ( ! )] [( (! ) ]' ) 7( ) 10. Chain Rule: 6! 8𝑓 9𝑔 (𝑥 ):; = ( 𝑓 ∘𝑔 𝑥 =𝑓 7 9𝑔( 𝑥 ): 𝑔′(𝑥) 11. Implicit Differentiation: Example for 𝑥 ; +𝑦 ; =1 6 6< 6 6< .! a. 6! [𝑥 ; +𝑦 ; ] = - 6![1] ; 2𝑥 +2𝑦 6! =0 ; 6! = < 12. 6 6! 6) [𝑓 .8 ( 𝑥)] = - 8 ) ( () "# (! ) ) 13. 6! ) [ 𝑓( 𝑥)] =𝑓 (0 ) ( 𝑥) Unit 4: Contextual Applications of Differentiation 1. If s(t), v(t), and a(t) are the position, velocity, and acceleration functions of a particle, 6 6' 6 then [ 𝑠(𝑡 )] =𝑣 (𝑡) ;- ' [ 𝑠(𝑡 )] = - [𝑣 (𝑡 )] =𝑎 (𝑡 ) 6! 6! 6! 2. Related Rates: Example a. Given the rate of change for the radius of a circle, find the rate of change of the 6= 6' Area. 𝐴=𝜋𝑟 ; ;- 62 =2𝜋𝑟 62 3. Local Linear Approximation: 𝑓 (𝑥 ) ≈𝑓 ( 𝑥5 ) +𝑓′(𝑥5 )(𝑥 −𝑥5 ) 4. L’Hospital’s Rule: if the following limit is an indeterminate form of type 0/0, then )(!) lim ((!)= lim !→# ) ( (!) ( !→# ( (!) Note: This can be repeated until it is no longer an indeterminate form Unit 5: Analytical Applications of Differentiation ) (> ).)(#) 1. Mean-Value Theorem: 𝑓 7( 𝑐 ) = - >.# 2. Extreme-Value Theorem: If a function f is continuous on a finite closed interval [a,b], then f has both an absolute maximum and an absolute minimum on [a,b]. 3. Critical Points: values of x where 𝑓 7(𝑥) =0 or 𝑓 is non-differentiable 4. Inflection Points: values of x where 𝑓 77(𝑥) =0 5. Increasing/Decreasing Intervals a. If 𝑓 7( 𝑥 ) >0 on (a,b), then 𝑓(𝑥) is increasing on [a,b] b. If 𝑓 7(𝑥) <0 on (a,b), then 𝑓(𝑥) is decreasing on [a,b] c. If 𝑓 7(𝑥) =0 on (a,b), then 𝑓(𝑥) is constant on [a,b] 6. Concavity a. If 𝑓 7 ′(𝑥 ) >0 on (a,b), then 𝑓(𝑥) is concave up on [a,b] b. If 𝑓 77(𝑥 ) <0 on (a,b), then 𝑓(𝑥) is concave down on [a,b] 7. First Derivative Test; Let 𝑥5 be a critical point of 𝑓(𝑥) a. If 𝑓 7(𝑥) >0 to the left of 𝑥5 and 𝑓 7( 𝑥) <0 to the right of 𝑥5 , then 𝑥5 is a relative maximum. b. If 𝑓 7(𝑥) <0 to the left of 𝑥5 and 𝑓 7( 𝑥) >0 to the right of 𝑥5 , then 𝑥5 is a relative minimum. c. If 𝑓 7( 𝑥 ) has the same sign on both sides of 𝑥5 , then 𝑥5 is not a relative extremum. 8. Second Derivative Test; Let 𝑥5 be a critical point of 𝑓(𝑥) a. If 𝑓 7(𝑥5 ) =0 and 𝑓 77(𝑥5 ) >0 , then 𝑥5 is a relative minimum. b. If 𝑓 7(𝑥5 ) =0 and 𝑓 77(𝑥5 ) <0 , then 𝑥5 is a relative maximum. c. If 𝑓 7(𝑥5 ) =0 and 𝑓 77(𝑥5 ) =0 , then it is inconclusive. Unit 6: Integration and Accumulation 1. Reimann Sum: 𝐴= lim ∑0@B5 𝑓(𝑥@ ∗ )(∆𝑥) 0→? > 2. 𝐴𝑟𝑒𝑎= ∫ # 𝑓( 𝑥)𝑑𝑥 3. Fundamental Theorem of Calculus > a. ∫ # 𝑓(𝑥)𝑑𝑥=𝐹 (𝑏) −𝐹(𝑎) b. 6 6! ! 8∫# 𝑓( 𝑡)𝑑𝑡; =𝑓(𝑥) > 4. Mean-Value Theorem for Integrals: ∫ # 𝑓( 𝑥) 𝑑𝑥=𝑓(𝑥 ∗ )(𝑏 −𝑎) 5. Integration by Parts: ∫ 𝑢𝑑𝑣= 𝑢𝑣− ∫ 𝑣𝑑𝑢 Unit 7: Differential Equations 1. Logistic Growth Differential Equation, where L is the carrying capacity and k is just a constant < 6< a. 62 = 𝑘(1− C )𝑦 2. Euler’s Method (𝑥5 ) = - 𝑦5 a. Given 𝑦 7 =𝑓 (𝑥, 𝑦 ),------𝑦 b. 𝑥0 =𝑥0.8+∆𝑥 where n is the number of steps taken c. 𝑦0 =𝑦0.8+𝑓(𝑥5 , 𝑦5 )∆𝑥 Unit 8: Applications of Integrals 1. Average Value of a function 8 > a. 𝑓#%& = >.# ∫ # 𝑓 ( 𝑥) 𝑑𝑥 2. If s(t), v(t), and a(t) are the position, velocity, and acceleration functions of a particle, then ∫ 𝑣( 𝑡)𝑑𝑡=𝑠 (𝑡) ;-∫ 𝑎(𝑡 )𝑑𝑡=𝑣 (𝑡) 3. Area between two curves (given 𝑓(𝑥)≥𝑔(𝑥) on [a,b] and ℎ(𝑦)≥𝑗(𝑦) on [c,d]) > a. 𝐴= ∫# [ 𝑓 (𝑥) −𝑔 (𝑥)] 𝑑𝑥 6 b. 𝐴= ∫$ [ 𝑓( 𝑦) −𝑔 ( 𝑦)] 𝑑𝑦 4. Volume with Cross Sections > a. 𝑉= ∫# 𝐴(𝑥)𝑑𝑥 where 𝐴(𝑥) is the cross-sectional area in terms of x 6 b. 𝑉= ∫$ 𝐴( 𝑦)𝑑𝑦 where 𝐴(𝑦) is the cross-sectional area in terms of y 5. Disk Method (Washer Method is just one disk minus another) > a. 𝑉= ∫# 𝜋[𝑓(𝑥) ]; 𝑑𝑥 6 b. 𝑉= ∫$ 𝜋[𝑔(𝑦)]; 𝑑𝑦 6. Cylindrical Shells Method > a. 𝑉= ∫# 2𝜋𝑥𝑓(𝑥)𝑑𝑥 6 b. 𝑉= ∫$ 2𝜋𝑦𝑔(𝑦)𝑑𝑥 7. Arc Length of a curve > a. 𝐿= ∫# [ 1+ [𝑓 7(! ) ] ; 𝑑𝑥 6 b. 𝐿= ∫$ [ 1+[𝑔′ ( 𝑦) ]; 𝑑𝑦 Unit 10: Infinite Sequences and Series # 1. ∑ ?@B5𝑎𝑟 @ = 8.' 8 2. Harmonic Series: ∑?@B8 @ 3. P-Series: 8 a. ∑ ?@B8@ *; converges if 𝑝>1 and diverges if 0<𝑝≤1 4. Alternating Series Test for Convergence a. An alternating series converges if the following conditions are satisfied: i. 𝑎8 ≥𝑎; ≥𝑎D ≥⋯≥𝑎@ ≥⋯ ii. lim 𝑎@ =0 @→? 5. Ratio Test #+!# a. 𝜌= lim @→? # + i. Converges if 𝜌<1 ii. Diverges if 𝜌>1-𝑜𝑟-𝜌 =+∞ iii. Test inconclusive if 𝜌=1 6. Taylor Polynomial for 𝑓 about 𝑥 =𝑥5 ) () ) ( ! ) $ (𝑥 −𝑥5 )0 a. 𝑇0 ( 𝑥) =𝑓 ( 𝑥5 ) +𝑓 7 (𝑥5 )(𝑥 −𝑥5 ) +⋯ + 0! 7. Taylor Polynomial Remainder Estimation Theorem a. |𝑅0 (𝑥) | ≤ FGH .)/012/3)024567 I ) ()!# ) (!) I (0+8 ) ! | 𝑥 −𝑥5 | 0+8 List of Common Limits and Derivatives Limits: JKL! 1. lim =1 ! !→5 2. 3. lim !→5 8.MNJ! ! lim d1+ 0→? =0 0 8 0 e =𝑒 Differentiation: 6 1. [sin(𝑥 )] = cos (𝑥 ) 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 6! 6 6! 6 6! 6 6! 6 6! 6 6! 6 6! 6 6! 6 6! 6 6! 6 6! 6 6! 6 6! 6 6! [cos ( x)] = - − sin ( 𝑥) [tan ( 𝑥)] = - (sec 𝑥); [csc(𝑥)] = - − csc ( 𝑥) tan-(𝑥) [sec ( x)] = sec ( 𝑥 ) tan-(𝑥) [cot (𝑥)] = - −(csc 𝑥); [𝑒 ! ] = - 𝑒 ! [ln(𝑥)] = - 8 ! [sin.8 (𝑥 )] = [cos.8 (x)] = - − sin ( 𝑥) [tan.8 ( 𝑥)] = - (sec 𝑥); [csc.8 ( 𝑥 )] = - − csc ( 𝑥 ) tan-(𝑥) [sec.8 (x)] = sec (𝑥) tan-(𝑥) [cot.8 ( 𝑥 )] = - −(csc 𝑥);