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# samplequestions - QUESTION 1 Find the derivatives of the...

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QUESTION 1: Find the derivatives of the following functions. DO NOT TRY TO SIMPLIFY. (a) f ( x )=tan(1 /x )( b ) f ( x )=s in( e x c ) f ( x )=(1+cos( x )) 1 / 3 (d) f ( x )= ln x x +1 QUESTION 2: Using only the deﬁnition of the derivative ﬁnd f 0 ( x ) for the function f ( x x 2 +1. QUESTION 3: Shown below is the graph of a function y = f ( x ). Sketch the graph of the function f 0 ( x ). QUESTION 4: Suppose that a bird foraging for a time t derives, from the food gathered, an amount of energy equal to G ( t 3 t 1+ t . At the same time, because of its foraging eﬀort, it loses an amount of energy equal to L ( t )=2 t . (a) What is the net gain in energy, E ( t ), after foraging for a time t ? (b) For which values of t> 0 (that is, for which amounts of time spent foraging) is the net gain in energy zero? (c) Determine that time t for which the net gain in energy is greatest. QUESTION 5: Let f ( x x 2 - sin x, - π 2 <x< π 2 . (a) Find all critical ponts of f ( x ) and determine if they are local maxima, local minima or neither. 1

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(b) Find all intervals where f ( x ) is increasing (resp. decreasing). (c) Find all intervals where f ( x ) is concave up (resp. concave down). QUESTION 6: A certain function f ( x )sat isﬁes f (1) = 2 ,f 0 (1) = - 1. (a) What is the equation of the tangent line to the graph of y = f ( x )when x =1? (b) Find an approximation for f (1 . 01). Now assume in addition that f 00 ( x ) > 0 for all x . (c) Sketch the graph of y = f ( x ) in the neighbourhood of x =1 . (d) Will your approximation in (b) be larger or smaller than the actual value of f (1 . 01)? QUESTION 7: (a) Derive a formula for x 1 in terms of x 0 . See the diagram below. x 0 x 1 y = f ( x ) ta ngent line (b) Use Newton’s method to ﬁnd the critical point(s) of the function g ( x )= x 2 + e - x . QUESTION 8: An airplane travelling at 400 km/hr and at a constant altitude of 3 km passes directly over a radar station on the ground which is tracking the plane. (a) How far does the plane ﬂy in t hours? (b) Determine the angle of elevation α ( t ), see the diagram. (c) Determine the rate of change of the angle of elevation 20 sec after the plane passes overhead. 2
plane alpha 3 km x(t) (t) QUESTION 9: Cultures of two diﬀerent cell types, A and B, are studied in a biotech lab. Let y A ( t )and y B ( t ) be the concentrations of the two at time t . Initially, the two concentrations are identical, i.e. y A (0) = y B (0) = y 0 . After 1 day, both y A and y B doubled, i.e. y (1) A = y (1) B =2 y 0 .Du r ing the 2nd day, y A doubled again, i.e. y A (2)=4 y 0 , but y B (2) = y B (1) + y 0 =3 y 0 . This trend continued, i.e. y A doubled every day while y B increased by y 0 each day. (a) Find the two concentrations at any time t (in days), that is y A ( t y B ( t ). (b) How long does it take for each population to reach 100 times the initial concentration? QUESTION 10: A room containing 1200 ft 3 of air is orginally free of carbon monoxide. At time t =0 cigarette smoke that contains 4% of carbon monoxide is produced at the rate of 0.1 ft 3 /min by a couple of smokers. The well-circulated mixture is allowed to leave the room at the same rate.

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## This note was uploaded on 01/12/2010 for the course STAT 100 taught by Professor Denissjerve during the Winter '06 term at San Jose State.

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samplequestions - QUESTION 1 Find the derivatives of the...

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