( ) ( ) ( ) PAL Session Stewart 3.1 & 3.2 Spring 2010

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1 PAL Session Stewart 3. & 3. Spring Key Terms/Concepts: Derivative of a Constant Function Power Rule Constant Multiple Rule n Sum/Difference Rule ( ) Eercise #0 p. 8 Differentiate the function. f() t t t 3. Formulas What oes each mean? () c 0 n n ( ) f( ) ± g( ) f '( ) ± g'( ) cf ( ) c f ( ) cf '( ) ( ) ( ) Eercise #4 p. 8 (moifie) Differentiate the function f( ) + 4π Eercise #54 p. 8 Fin an equation of the tangent line to the curve y that is parallel to the line y + 3.

2 PAL Session Stewart 3. & 3. Spring Key Terms/Concepts: The Prouct Rule One ee two plus two ee one The Quotient Rule Low ee high minus high ee low over low square 3. Formulas What oes each mean? ( f( g ) ( )) f( g ) '( ) + g ( ) f'( ) f( ) g( ) f '( ) f( ) g'( ) g( ) g ( ) ( ) Differentiate the following functions: Eercise #4 p 87 + f( ) 3 + Eercise # p 87 Rt () ( t+ e t )(3 t ) Eercise #6 p 88 (moifie) as + b f() s cs + Eercise 9 & p 88 (moifie) Fv () 3 ( v v v)( v v) + v + v

3 PAL Session Stewart 3.3 & 3.4 Spring Formulas ( sin ) cos ( cos ) sin ( tan ) sec ( ) sec sec tan ( ) csc csc cot ( cot ) csc Eercise #0 p sin Differentiate y + cos Eercise #6 p. 95 Differentiate y sin tan Eercise #4 p. 96 cosθ Compute the limit lim θ 0 s in θ Eercise Differentiate cosαcscα f ( α) secα + cotα

4 PAL Session Stewart 3.3 & 3.4 Spring Key Terms/Concepts: Chain Rule & Applications Derivative of the outer times the erivative of the inner Eercise #6 p. 03 Differentiate y 3cot ( nθ ) 3.4 Formulas what oes each mean? F ( ) f g ( ) F'( ) f'( g ( )) g'( ) ( ) ( c ) c ln c Eercise #0 p. 03 Differentiate y 3 ( + ) + Eercise #6 p ( y ) Differentiate G( y) ( y + y) 5 Eercise #5 p. 04 Fin an equation of the tangent line to the curve at the given point: y sin + sin ; (0,0)

5 PAL Session Stewart 3.5 & 3.6 Spring Key Terms/Concepts: Implicit Differentiation Derivatives of Inverse Trig Functions 3.5 Formulas ( sin ) ( cos ) ( tan ) ( sec ) ( csc ) ( cot ) Eercise #48 p. 4 Fin the erivative of the function. Simplify as much as possible. y sec Eercise #4 p. 3 Fin y/ by implicit ifferentiation y s in( ) sin( y ) Eercise #0 p. 3 Fin y/ by implicit ifferentiation 5 3 y + y + ye Eercise #6 p. 3 Fin y/ by implicit ifferentiation + y + y

6 PAL Session Stewart 3.5 & 3.6 Spring Key Terms/Concepts: Derivative of Log base a Derivative of Natural Log ***LOG LAWS ***Remember the CHAIN RULE Eercise #8 p. 0 Differentiate the function a z H( z) ln a z Formulas what oes each mean? ( log a ) ln a ( ln ) Eercise # p. 0 Differentiate the function y log e cosπ ( ) Eercise #38 p. 0 Use logarithmic ifferentiation to fin the erivative of the following function 0 y e + ( ) Eercise #50 p. 0 y Fin y if y

7 PAL Session Stewart 3.7 & 3.8 Spring Key Terms/Concepts: Average Rate of Change Instantaneous Rate of Change Velocity Acceleration Optional: Cost Function, laminar flow, current, compressibility 3.7 Formulas y f( ) f( ) y y lim 0 at () v'() t s''( t) Section 3.7 #5 p. 3 A spherical balloon is being inflate. Fin the rate of increase of the surface area ( S 4π r ) with respect to the raius r when r is (a) ft., (b) ft., an (c) 3 ft. What conclusion can you make? Section 3.7 #8 p. 3 If a tank hols 5,000 gallons of water, which rains from the bottom of the tank in 40 minutes, then Torricelli s Law gives the volume V of water remaining in the tank after t minutes as t V t Fin the rate at which water is raining from the tank after (a) 5 min (b) 0 min (c) 0 min () 40 min Section 3.7 #30 p. 33 The cost function for prouction of a commoity is C ( ) (a) Fin an interpret C '(00). (b) Compare C '(00) with the cost of proucing the 0 st item. 3

8 PAL Session Stewart 3.7 & 3.8 Spring Key Terms/Concepts: Law of Natural Growth/Decay Half-Life Optional: Continuously Compoune, Newton s Law of Cooling 3.8 Formulas what oes each mean? y ky t yt ( ) y(0) e kt Section 3.8 # p 39 A population of protozoa evelops with a constant relative growth rate of per member per ay. On ay zero the population consists of two members. Fin the population size after si ays. Section 3.8 # 4 p A bacteria culture grows with a constant relative growth rate. After hours there are 600 bacteria an after 8 hours the count is 75,000. (a) Fin the initial population (b) Fin an epression for the population after t hours. (c) Fin the number of cells after 5 hours. () Fin the rate of growth after 5 hours. (e) When will the population reach 00,000? A sample of tritium-3 ecaye to 94.5% of its original amount after a year. Section 3.8 #0 p. 40 (a) How long is the half-life of tritium-3? (b) How long woul it take the sample to ecay to 0% of its original amount?

9 PAL Session Stewart 3.9, 3.0, & 4.8 Spring Key Terms/Concepts: Relate Rates Process:. Rea the problem carefully. Draw a picture 3. Convert everything into math lingo erivatives, etc.; note which quantities are time epenent 4. Write an equation that relates all the quantities given in the problem. Use geometry of problem to eliminate unknown quantities. 5. Use chain rule to ifferentiate each sie with respect to (w.r.t.) time 6. Substitute given info into this ifferentiate equation. Eercise Two cars are traveling on long straight roas that meet at right angles. Car A leaves the intersection traveling east at 48 mph an car B leaves the intersection 3 hours later an travels north at 50 mph. At what rate is the istance between the two cars increasing hours after car B leaves the intersection? Eercise A stone roppe into a still pon sens out a circular ripple whose raius increases at a constant rate of 4 ft/sec. How rapily is the area enclose by the ripple increasing at the en of 8 secons? Eercise 3 3 San is being umpe from a conveyor belt at the rate of 8π ft / mn i. The coarseness of the san is such that it forms a pile in the shape of a cone with the raius of the base always /3 the height. How fast is the height increasing when the pile is 5 feet high?

10 PAL Session Stewart 3.9, 3.0, & 4.8 Spring , 4.8 Key Terms/Concepts: Linear approimation Linearization Differentials Relative Error Newton s Metho Eercise #4 p. 5 Fin the ifferential of each function. tan( πt) (a) y e (b) y (+ ln z) 3.0, 4.8 Formulas what oes each mean? f( ) La ( ) f( a) + f ( a) ( a) y f( + ) f( ) y RE.. y f( n ) n+ n f ( ) n Eercise #8 p. 5 Use a linear approimation (or ifferentials) to estimate the given number Eercise #4 p. 338 Use Newton s metho to approimate the inicate root of the equation to si ecimal places: The root 5 3 of in the interval [, ].

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