( ) + ( ) ( x) ax. = f x g x. = ax. Then find the derivative. 5 x. y = 6. values at x = 1: Recitation Worksheet 5A

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1 Recitation Worksheet 5A I First rewrite the function in the form y n = a Then find the derivative II 5 1 y = y 1 5 = y = y = 6 10 Rewrite if necessary until you have the sum of a few terms, each of the form Then find the derivative (Do not use the product or quotient rule for these) y = = y n a III Find the derivative You will want the product or quotient rule Do not simplify your answer 7 y ( )( 5 7 1) = y = IV Suppose the functions f ( ) and g( ) and their derivatives have the following values at = 1: 9 Find h ( 1) if h = f g f ( ) g( ) f g Find h ( 1) if h = + f g + 1

2 Recitation Worksheet 5B I Find the derivative of each of the following Do not simplify your answers 5 1 y = (Rewrite first!) 7 5 y = y = II Suppose f and g and their first derivatives have the following values at = and = : a Find h if h = f + g b Find h if h = f g f ( ) g( ) f g III Suppose f and g and their first derivatives have the following values at = 1 and = : Find h if h f g = + Then find the equation of the tangent line to the graph of y h f ( ) g( ) f g = at = IV Find the third derivative of y = +

3 Recitation Worksheet 6A I Find the derivative of each of the following Do not simplify your answers 1 e π y e e = ln + ln 7 = ( + ln ) y e ln y = 7 y = e y = ln ( + 5 ) 6 y = ln ( 8+ 0) 7 y = ln ln ( + e ) II Find and simplify the second derivative of y e ( 5 ) = + III Suppose g = 7 and g = 6 Find h if h ln g = +

4 Recitation Worksheet 6B 1 Suppose $10,000 is invested at an annual interest rate of 5% compounded continuously a How long will it take for the investment to double in value? b How long will it take for the investment to triple in value? A recent college graduate decides he would like to have $0,000 in five years to make a down payment on a home a How much money will he need to invest today in order to have $0,000 in five years, given that he can invest at an annual interest rate of % compounded continuously? b Suppose instead the best interest rate he can find is only 5% (instead of %) Now how much will he need to invest? c Suppose the interest rate is % again, but now he would like to have the $0,000 in only four years How much does he need to invest? The half-life of caffeine is 5 hours This means the amount of caffeine in your bloodstream is reduced by 50% every five hours A grande French Roast has 0 mg of caffeine Let Q( t ) denote the amount of caffeine in your system t hours after drinking your grande French Roast (For simplicity, assume the entire drink is consumed instantly) a How many milligrams of caffeine will be in your system after 5 hours? after 10 hours? b Let Q( t) = Qe kt 0 Find Q 0 and k c How many milligrams of coffee will be in your system after hours? A bacteria culture triples in size every 7 hours Three hours from now, the culture will have 8000 bacteria If Q( t ) denotes the number of bacteria at time t, then Q( t) 0 Find Q 0 and k Qe kt = 5 The graph of y = f passes through the point ( 0, ) The slope of f at any point P is three times the y coordinate of P Find an epression for f ( ), and find f

5 Recitation Worksheet 7 1 Carbon monoide (CO) binds about 00 times more effectively than oygen to hemoglobin, forming the comple called carboyhemoglobin This tight binding affinity makes death by carbon monoide poisoning a problem in industrial settings as well as from running cars in a garage Let p by the partial pressure of CO measured in torrs, then a dissociation curve for CO and hemoglobin is given by y( p) p = p hemoglobin bound by CO a Differentiate y( p ) What are the units for this epression? b Find the second derivative, y ( p) c Find the value(s) of p satisfying y ( p) 0, where y is the fraction of, and identify the appropriate units = The price-demand and cost function for the production of shirts is given by 5 0 p = C = and a Find the revenue and profit functions, R( ) and P( ) b Find the marginal cost function and marginal profit function c Find the eact cost of producing the 81 st shirt d Use marginal cost to approimate the cost of producing the 81 st shirt e Find the average cost function, C( ), and the marginal average cost function f Find the average cost and marginal average cost at a production level of 80 shirts, and interpret your answer

6 Recitation Worksheet 8A f = Let a Find the critical numbers (the values where f = 0 or f DNE), if any b Use your answer to part (a) find the maimum and minimum values of f ( ) on the interval [,5 ] Let g ln ( 8 0) = + Find the critical numbers, if any, and use them to find maimum and minimum values of f ( ) on the interval [ 0,10 ] Let h + + = > 1 a Is h( ) continuous at = 1? b Is h( ) differentiable at = 1? c Find the critical numbers of h( ) (Hint: there are three) d Find the maimum and minimum values of h( ) on the interval [ 0, ] e Find the maimum and minimum values of h( ) on the interval [,5]

7 Recitation Worksheet 8B 1 On the same graph, plot both f = 5 and its derivative on the interval [, ] What do you notice? In particular, what appears to be true about f ( ) when its derivative is zero? What appears to be true about f ( ) when its derivative is positive? is negative? Let g + = + 9 a Find the critical numbers of g( ), if any b Find the maimum and minimum value of g( ) on the interval [ 1, 6 ] = Let h e ( 5) a Find the critical numbers of h( ), if any b Find the maimum and minimum value of h( ) on the interval [ 0,6 ] Let g = Find a value c in the interval [,9 ] such that g ( c) average rate of change of g( ) on the interval [,9 ] equals the