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Calculus 2 UNIT 0: Pre-requisite (Review of Antiderivative and Integral) 1 LEARNING OUTCOMES: After this unit, students are expected to: • Students will learn how to navigate through the course • Identify course requirements • Review important Calculus concepts, such as differentiation and integration Resources: • Big Picture: Derivatives (MIT OpenCourseware) https://www.youtube.com/watch?v=T_I-CUOc_bk&list=PLBE9407EA64E2C318&index=3 • Big Picture: Integrals (MIT OpenCourseware) https://www.youtube.com/watch?v=2qxY859dzzQ&list=PLBE9407EA64E2C318&index=6 • Antiderivatives and indefinite integrals (Khan Academy) I. REVIEW OF DERIVATIVES The derivative of a function 𝑓 at a number 𝑎, denoted by 𝑓’(𝑎) is 𝑓′(𝑎)= limℎ→0𝑓(𝑎+ ℎ)− 𝑓(𝑎)ℎ 1) Given is a function: 𝑓(𝑥)= −12𝑥2+ 2 a) Use the definition of derivative to find 𝑓’(𝑥). b) Sketch 𝑓’(𝑥) and f(x) on the same Cartesian Coordinate.Calculus 2 UNIT 0: Pre-requisite (Review of Antiderivative and Integral) 2 2) Given that: lim𝜃→0sin𝜃𝜃= 1 and lim𝜃→0cos𝜃− 1𝜃= 0 and sin(∝ +𝛽)= sin𝛼cos𝛽+ cos𝛼sin𝛽 Use the definition of derivative to show that 𝑑(sin𝜃)𝑑𝜃= cos𝜃 3) Differentiate the functions. a) 𝑓(𝑥)= 𝑎𝑥2+ 𝑏𝑥 + 𝑐 b) 𝑓(𝑥)= (𝑥+ 𝑥−1)3 c) 𝑓(𝑥)= √𝑥5+ 4√𝑥5 d) 𝑦 =𝑥21 + 2𝑥Calculus 2 UNIT 0: Pre-requisite (Review of Antiderivative and Integral) 3 Derivatives of Trigonometric Functions 𝑑𝑑𝑥(sin𝑥)=cos𝑥 𝑑𝑑𝑥(csc𝑥)= −csc𝑥 cot𝑥 𝑑𝑑𝑥(cos𝑥)= −sin𝑥 𝑑𝑑𝑥(sec 𝑥)= sec𝑥 tan𝑥 𝑑𝑑𝑥(tan𝑥)= sec2𝑥 𝑑𝑑𝑥(cot 𝑥)= −csc2𝑥 4) Find the derivatives of the following functions a) 𝑓(𝑥)= 1 + sin𝑥𝑥+ cos𝑥 b) 𝑔(𝑥)= 1 − sec𝑥tan𝑥 5) Given is an equation: 𝑥2+ 𝑦2= 25 a) Find: 𝑑𝑦𝑑𝑥 b) Find an equation of the tangent to the circle 𝑥2+ 𝑦2= 25 at the point (3,4).Calculus 2 UNIT 0: Pre-requisite (Review of Antiderivative and Integral) 4 Derivatives of Inverse Trigonometric Functions 𝑑𝑑𝑥(sin−1𝑥)=1√1 −𝑥2 𝑑𝑑𝑥(csc−1𝑥)=−1𝑥√𝑥2− 1 𝑑𝑑𝑥(cos−1𝑥)= −1√1 −𝑥2 𝑑𝑑𝑥(sec−1𝑥)=1𝑥√𝑥2− 1 𝑑𝑑𝑥(tan−1𝑥)=11 +𝑥2 𝑑𝑑𝑥(cot−1𝑥)= −11 + 𝑥2 6) Find the derivative of the function. Simplify where possible. a) 𝑓(𝑥)= tan−1√𝑥 b) 𝑔(𝑥)= sin−1(2𝑥+ 1) Derivatives of exponential and logarithmic Functions 𝑑𝑑𝑥(log𝑎𝑥)=1𝑥ln 𝑎 𝑑𝑑𝑥(ln|𝑥|)=1𝑥 𝑑𝑑𝑥(𝑒𝑥)=𝑒𝑥 𝑑𝑑𝑥(𝑎𝑥)=(ln𝑎)𝑎𝑥 7) Differentiate the following functions: a) 𝑓(𝑥)=ln(sin𝑥) b) 𝑓(𝑥)= log(2 + sin𝑥) c) 𝑓(𝑥)= 23𝑥 d) 𝑓(𝑥)= 𝑥𝑒−2𝑥Calculus 2 UNIT 0: Pre-requisite (Review of Antiderivative and Integral) 5 II. REVIEW OF ANTIDERIVATIVES Definition of Antiderivative A function 𝐹 is called an antiderivative of 𝑓 on an interval I if 𝐹′(𝑥)=𝑓(𝑥) for all 𝑥 in 𝐼. If 𝐹 is an antiderivative of 𝑓 on an interval 𝐼, then the most general antiderivative of 𝑓 on 𝐼 is 𝐹(𝑥) + 𝐶 where 𝐶 is 1) Determine whether each statement is TRUE or FALSE. Statement True False Reason a) 𝐹(𝑥) = 2𝑥3+ 𝐶 is the antiderivative of 𝑓(𝑥)= −6𝑥4 b) 𝐹(𝑥)= 𝑥1/3 is the antiderivative of 𝑓(𝑥)= 34𝑥4/3 c) 𝐹(𝑥)= 𝑥𝑒𝑥− 𝑒𝑥 + 𝐶 is the antiderivative of 𝑓(𝑥)= 𝑥𝑒𝑥 Definition of Indefinite Integral The collection of all antiderivatives of 𝑓 is called the indefinite integral of 𝑓 with respect to 𝑥, and is denoted by ∫𝑓(𝑥)𝑑𝑥. Table of Antiderivative formulas, k a nonzero constant Function General Antiderivatives Function General Antiderivatives 𝑥𝑛 1𝑛𝑥𝑛+1+ 𝐶 𝑒𝑘𝑥 1𝑘𝑒𝑘𝑥+ 𝐶 sin𝑘𝑥 −1𝑘cos𝑘𝑥 + 𝐶 1𝑥 ln|𝑥|+ 𝐶,𝑥 ≠ 0 cos𝑘𝑥 1𝑘sin𝑘𝑥+ 𝐶 1√1 −𝑘2𝑥2 1𝑘sin−1𝑘𝑥 + 𝐶 sec2𝑘𝑥 1𝑘tan𝑘𝑥+ 𝐶 11 + 𝑘2𝑥2 1𝑘tan−1𝑘𝑥 +𝐶 csc2𝑘𝑥 −1𝑘cot𝑘𝑥+ 𝐶 1𝑥√𝑘2𝑥2− 1 sec−1𝑘𝑥 + 𝐶 sec𝑘𝑥tan𝑘𝑥 1𝑘sec𝑘𝑥+ 𝐶 𝑎𝑘𝑥 (1𝑘ln𝑎)𝑎𝑘𝑥+ 𝐶,𝑎 > 0,𝑎 ≠ 1 csc𝑘𝑥 cot𝑘𝑥 −1𝑘csc𝑘𝑥+ 𝐶Calculus 2 UNIT 0: Pre-requisite (Review of Antiderivative and Integral) 6 2) Find the indefinite integral and check the result by differentiation. a) ∫(4𝑡2+ 3)2𝑑𝑡 b) ∫(√𝑥34+ 1)𝑑𝑥 c) ∫𝑥+ 6√𝑥𝑑𝑥 d) ∫3𝑥7𝑑𝑥 e) ∫(2sin𝑥− 5𝑒𝑥)𝑑𝑥 f) ∫(tan2𝑦 + 1) 𝑑𝑦 g) ∫(2𝑥− 4𝑥)𝑑𝑥 h) ∫(𝑥−5𝑥)𝑑𝑥Calculus 2 UNIT 0: Pre-requisite (Review of Antiderivative and Integral) 7 3) An evergreen nursery usually sells a certain type of shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by: 𝑑ℎ𝑑𝑡= 1.5𝑡 + 5 Where 𝑡 is the time in years and ℎ is the hight in centimeters. The seedlings are 12 centimeters tall when planted (𝑡 = 0) a) Find the height after 𝑡 years. b) How tall are the shrubs when they are sold? (Larson & Edwards, 2015, p.288) 4) Given is a differential equation: 𝑑𝑦𝑑𝑥= 𝑥− 1 The curve on the right, is the solution curves of the equation. Find an equation of the curve through the labelled point.Calculus 2 UNIT 0: Pre-requisite (Review of Antiderivative and Integral) 8 III. UNIT 0 REVIEW 1) Find an equation of the tangent line to the graph of at the given point. (𝑥+ 2)(𝑥2+ 5) ,(−1 ,6) 2) Is the following statement TRUE? Write down your reason. ∫−15(𝑥+3)2(𝑥− 2)4𝑑𝑥 = (𝑥+ 3𝑥− 2)3+ 𝐶 3) Solve the initial value problems: a) 𝑑𝑦𝑑𝑥= 9𝑥2− 4𝑥+5 , 𝑦(−1)= 0 b) 𝑑𝑆𝑑𝑡= cos𝑡+ sin𝑡 , 𝑠(𝜋)= 1 c) 𝑑2𝑟𝑑𝑡2=2𝑡3 , 𝑑𝑟𝑑𝑡|𝑡=1= 1 , 𝑟(1)= 1 4) A ball is thrown upward with an initial velocity of 64 feet per second from an initial height of 80 feet. Assume that the acceleration of gravity is 32 ft𝑠2⁄. a) Find the position function

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