Calculus 2 UNIT 0 Pre requisite Review of Antiderivative and Integral LEARNING OUTCOMES After this unit students are expected to Students will learn how to navigate through the course Review important Calculus concepts such as differentiation and integration Identify course requirements 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 at a number denoted by is 1 Given is a function lim 0 2 2 1 2 a Use the definition of derivative to find b Sketch and f x on the same Cartesian Coordinate 1 Calculus 2 UNIT 0 Pre requisite Review of Antiderivative and Integral 2 Given that sin lim 0 1 and cos 1 lim 0 0 Use the definition of derivative to show that sin cos and sin sin cos cos sin 3 Differentiate the functions a c 2 5 4 5 b d 1 3 2 1 2 2 Calculus 2 UNIT 0 Pre requisite Review of Antiderivative and Integral Derivatives of Trigonometric Functions sin cos cos sin tan sec2 csc csc cot sec sec tan cot csc2 4 Find the derivatives of the following functions a 1 sin cos b 1 sec tan 5 Given is an equation 2 2 25 a Find b Find an equation of the tangent to the circle at the point 3 4 2 2 25 3 Calculus 2 UNIT 0 Pre requisite Review of Antiderivative and Integral Derivatives of Inverse Trigonometric Functions sin 1 cos 1 tan 1 1 1 2 1 1 2 1 1 2 csc 1 sec 1 cot 1 1 2 1 1 2 1 1 1 2 a 6 Find the derivative of the function Simplify where possible tan 1 b sin 1 2 1 Derivatives of exponential and logarithmic Functions 1 ln log ln ln 1 a c 7 Differentiate the following functions ln sin 23 b d log 2 sin 2 4 Calculus 2 UNIT 0 Pre requisite Review of Antiderivative and Integral II REVIEW OF ANTIDERIVATIVES Definition of Antiderivative A function is called an antiderivative of on an interval I if for all in If is an antiderivative of on an interval then the most general antiderivative of on is where is 1 Determine whether each statement is TRUE or FALSE True False Reason a Statement 2 3 is the antiderivative of 6 4 1 3 b is the antiderivative of 4 3 3 4 c is the antiderivative of Definition of Indefinite Integral The collection of all antiderivatives of is called the indefinite integral of with respect to and is denoted by Table of Antiderivative formulas k a nonzero constant General Antiderivatives Function sin cos sec2 csc2 sec tan csc cot 1 cos sin tan cot sec csc 1 1 1 1 1 1 1 Function General Antiderivatives 1 1 1 2 2 1 1 2 2 1 2 2 1 1 ln 0 sin 1 1 1 tan 1 sec 1 1 ln 0 1 5 Calculus 2 UNIT 0 Pre requisite Review of Antiderivative and Integral 2 Find the indefinite integral and check the result by differentiation a c e g 4 2 3 2 6 2 sin 5 2 4 b d f h 4 3 1 3 7 tan2 1 5 6 Calculus 2 UNIT 0 Pre requisite Review of Antiderivative and Integral 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 1 5 5 Where is the time in years and is the hight in centimeters The seedlings are 12 centimeters tall when planted 0 a Find the height after years b How tall are the shrubs when they are sold Larson Edwards 2015 p 288 4 Given is a differential equation 1 The curve on the right is the solution curves of the equation Find an equation of the curve through the labelled point 7 Calculus 2 UNIT 0 Pre requisite Review of Antiderivative and Integral III UNIT 0 REVIEW 1 Find an equation of the tangent line to the graph of at the given point 2 2 5 1 6 Is the following statement TRUE Write down your reason 15 3 2 2 4 3 2 3 Solve the initial value problems 2 3 a b c 9 2 4 5 1 0 cos sin 1 2 3 2 2 1 1 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 ft 2 a Find the position function giving the height as a function of the time b When does the ball hit the ground c What is the domain of d Sketch the graph of 8