AP Calculus AB Ch. 2 Derivatives (Part I) Intro to Derivatives: Definition of the Derivative and the Tangent Line 9/15/14

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1 AP Calculus AB Ch. Derivatives (Part I) Name Intro to Derivatives: Deinition o the Derivative an the Tangent Line 9/15/1 A linear unction has the same slope at all o its points, but non-linear equations will have ierent slopes at ierent points. Tangent Line: The slope o the tangent line is the same as the slope o the unction at the point o tangenc. To in the slope o the tangent line, ou approimate the slope o the secant line b choosing points closer an closer to the point o tangenc. () c

2 .1 Notes: Derivatives an Tangent Lines 9/16/1 The slope o the tangent line is calle the erivative. The limit use to eine the slope o the tangent line is also use to eine one o the two unamental operations o calculus ierentiation. The erivative o at is given b: A unction is ierentiable at i its erivative eists at. Notations or the erivative o ( ) : a.) Fin the erivative o ( ) b.) Fin the slope o ( ) at 1. c.) Fin the equation o the line tangent to ( ) at 1.

3 a.) Fin the erivative o ( ) b.) Fin the slope o ( ) at 1,, an 0. c.) What is (1)? Write the equation o the tangent line at 1. Dierentiabilit an Continuit A unction is ierentiable on an open interval ( ab, ) i it is ierentiable at ever point in (, ) **I a unction is not continuous at c, then it is not ierentiable at c.** However, just because a unction is continuous at c oes not mean the unction is ierentiable. ab. The unction coul have a cusp: or a vertical tangent line RECAP: A unction is NOT ierentiable i it is not continuous, has a cusp, or has a vertical tangent. Theorem: I is ierentiable at c, then is continuous at c. Dierentiable Continuous Continuous oes not impl Dierentiable

4 Given these graphs a) etermine the interval or which () is ierentiable b) sketch the graph o the erivative

5 . Notes Basic Dierentiation 9/18/1 The einition o the erivative, '( ) ( + ) ( ) lim 0, eplains wh the erivative represents slope. Fortunatel there are some quick an eas rules that make taking erivatives much less time consuming! Fin the pattern: ( ) ( ) 7 an ' Constant Rule: [ c ] Power Rule: n [ ] Constant Multiple Rule: c ( ) g t t 5 1. Fin the erivative o ( ) an the slope o the graph at t.. Fin the erivative o t t. Fin the erivative o ( ) 1 5

6 . Fin [ sin ] MEMORIZE!!!! [ sin ] [ cos ] 5. Fin '( ) or ( ) sin cos +. You still nee to know the limit einition o a erivative or problems like 6. Evaluate ( h) ( ) 8+ 8 lim h 0 h A) 8 B) 19 C) 6 D) E) The limit oes not eist Review: Match the graph o with its erivative.

7 .b Notes: Applications o Derivatives 9//1 Slope Rate o Change Distance Rate Time Rate (or Velocit) Average Velocit 16t A billiar ball is roppe rom a height o 100 eet. It s height in eet at time t secons is ( ) Fin the average velocit over the time interval [ 1, ]. s t The Instantaneous Velocit is oun b calculating the average velocit over a ver small interval. Instantaneous Velocit Derivative o the Position Function Spee Velocit 1 s ( t) gt + vot + so where g- eet/sec or g-9.8 meters/sec v initial velocit o s initial height o Eample. At time t0 a iver jumps o the iving boar that is eet above the water. Her position is given b ( ) s t 16t + 16t +. a.) When oes she hit the water? b.) What is her velocit on impact? c.) When is her velocit equal to zero?.) What is her velocit at 1 secon? At 1.5 secons?

8 Higher Orer Derivatives First Derivative: Secon Derivative: Thir Derivative: Fourth Derivative: ' '( ) '' '' ( ) ''' '''( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) nth Derivative: ( n) ( n ) ( ) n n n n ( ) 6 Eample: Fin '''( ) i ( ) 5 sin Linear Approimations Given ( ) 11+ 1, in the value o (1.1) without a calculator just kiing! (We can approimate the value using erivatives though.) a) in () b) write the equation or the line tangent to () at 1 c) evaluate that line or 1.1 ) evaluate (1.1) using the calculator an compare to the approimation

9 . Notes: Prouct an Quotient Rules 9//1 Prouct Rule: ( g ) ( ) Fin the erivative: k sin 1. ( ) ( h + + )( 5). ( ) + 5. ( ) ( ) Quotient Rule: ( ) g( ) ( ) tan

10 [ sin ] [ tan ] [ sec ] [ cos ] [ cot ] [ csc ] MEMORIZE these trig erivatives!!!!! 7. sec 8. ( ) ( )( + 1)( + + 1) h( ) ( + 5 7) 10. csc g ( ) 1 cot