To do this which theorem did you use? b) Determine which points are inflections and mark the concavity on a number line for f.

Transcription

1 Math 13, Lab 11 1 a) Let f() = + 4 Determine which critical points are local maima, minima, and which are not etreme and mark this on a number line for b) Determine which points are inflections and mark the concavity on a number line for Plot and label the critical and inflection points Indicate coordinates f() = + 4 is not defined at = What happens as +? Evaluate lim What happens as? Evaluate lim + 4 Use an appropriate scale and aes so that the action 1

2 2 a) Let f() = ( 3 8) 1/3 Determine which critical points are local maima, minima, and which are not etreme and mark this on a number line for b) Determine which points are inflections and mark the concavity on a number line for Hint: To save time use: () = 16 ( 3 8) 5/3 Plot and label the critical and inflection points Indicate coordinates Plot and label y (and ) intercepts Use appropriate scales and aes so that the action near critical numbers can be seen The scales may be different on each ais if necessary 2

3 3 a) f() = 6 1/3 2 Determine which critical points are local maima, minima, and which are not etreme and mark this on a number line for b) Determine which points are inflections and mark the concavity on a number line for Plot and label the critical and inflection points Indicate coordinates Plot and label y (and ) intercepts Use appropriate scales and aes so that the action near critical numbers can be seen The scales may be different on each ais if necessary 3

4 4 Below I give you information about the derivatives of two continuous functions First determine where each function is increasing and decreasing Net classify each critical point as a relative ma, relative min, or neither Then determine where the function is concave up and down and the location of any inflections Then for each, sketch the graph function that would have derivatives like those given Indicate on your graph which points are local etrema DNE indicates that the derivative does not eist at the point (though the original function does) You will need to make up function values for the critical points that are consistent with the given information The only point you know in each case is that the function passes through (, ), which is marked on each graph a) b) DNE ++ g 1 2 g ++ DNE

5 To the left is the graph of (), the derivative of f() This is the graph of (), NOT the graph of f() You can read off the values of for your number line directly from it Eg, (2) = 18 (NOT ) a) Convert the graph of () into number line information for () Use the number line below the graph Mark where is () positive, negative, and b) Use the number line to classify each critical point of f (loc etrema or not) Use the number line to find where f is increasing and decreasing d) Now create a number line for Remember is the derivative (slope) of So where is the slope of positive? Negative?? Use this number line to determine concavity and inflections e Sketch the graph of the original function y = f() Use the same aes You will have to make up values for the function consistent with () Start your graph at ( 5, 2) () () 6 This is a combination graphing and optimization problem A patient takes a 2 mg dose of a drug (in pill form) The drug is absorbed and metabolized by the patient so that after t hours there are f(t) = 2te t mg in the patients bloodstream, where for the time interval [, ) a) Find and classify the critical numbers of this function What theorem did you use? b) Find the absolute maimum value or amount of medication in the bloodstream At what time does this occur? Justify your answer with a theorem that we discussed yesterday Sketch a graph of the function on the interval [, ) Include critical points and endpoint Make sure to justify your graph by showing your work and carefully using the first and second derivatives 5

6 Math 13, Lab 11: Answers 1 a) f() = + 4 Notice = is not in the domain of f (so it cannot be a critical number) Critical points: () = 1 4, DNE at = 2 () = = 1 = = 4 So = 2, 2 and watch out for = on the number line since is not continuous b) R Ma not in domain R Min Incr 2 Decr Decr 2 Incr First Derivative Test () = d (1 4 ) d 2 = d ( ) = 8 3 = 8 d 3 = Never! () DNE at = DNE Con Dn Con Up Concavity Test VA: = At the critical numbers: f( 2) = = 4 and f(2) = = 4 There are no intercepts Finally, and lim + + lim + 4 = + = VA }{{} + 4 = = VA }{{} 9 2 a) Critical numbers: () = 1 3 (3 8) 2/3 3 2 = 2 (d) below) ( 3 8) 2/3 = at = and DNE at = 2 (See number line in part b) Potential inflections: () = 16 ( 3 8) 5/3 = at = and DNE at = 2 (See number line in part (d) below) Evaluate f at critical and inflection points: f() = 2, f(2) = These are also the intercepts 3 Neither Neither d) INFL INFL CP & ect: (, 2) CP & ect: (2, ) 6

7 3 a) f() = 6 1/3 2 Critical numbers: () = 2 2/3 2 = 2 2/3 2 = 1 = 2 2/3 2/3 = 1 = 1, 1 and () DNE at = R Min CP R Ma ++ DNE ++ Decr 1 Incr Incr 1 Decr First Derivative Test b) () = 4 5/3 = 4 5/3 = Never! () DNE at = ect DNE Con Up Con Dn Concavity Test (1, 4) Rel Ma (, ) CP ( 1, 4) Rel Min At the critical numbers: f() = and f( 1) = 4 f(1) = 4 intercept: 6 1/3 2 = = and 6 1/3 = 2 3 = 2/3 = ±3 3/2 = ± a) There s a local min at = 2 Note: There is a horizontal tangent at = 2 but no ma or min CP, ect a) R Min CP, Neither Decr 2 Incr 2 Decr ect Con Up ++ Con Dn 2 Con Up R Min b) There s a local min (and a corner) at = 1 and a local ma at = 2 7

8 R Ma b) g Decr R Min R Ma DNE ++ 1 Incr 2 Decr g ++ Con Up DNE ++ 1 Con Dn 3 Con Up R Min, R Ma 5 Neither R Ma R Min Incr 4 Incr Decr 4 Incr Con Dn 4 Con Up 2 Con Dn 2 Con Up f R Min 6 a) Use the First Derivative Test (t) = 2[e t te t ] = 2e t (1 t) = at t = 1 R Ma ++ [ Incr 1 Decr b) We cannot apply CIT because the interval is the wrong type Nonetheless, the function is differentiable on [, ) On the interval [, ) the function is differentiable and has a single critical point at t = 1, where there is a relative ma By the SCPT (Single Critical Point Theorem), if a differentiable function on an interval has a single critical point and it is a relative etreme point, then it is an absolute etreme point Here, we have a single critical point at t = 1, its a relative ma, so it must be an absolute ma And the ma value is f(1) = 2e The first derivative info is above (t) = 2[e t (1 t) + 2e t ( 1) = 2e 1 (2 t) = at t = 2 Evaluate f at critical points and inflections: f(1) = 2e and f(2) = 4e RMa (, 7357) POI Con Dn 2 Conc Up (, 7357)

9 Post-Lab 11: Take-Home Quiz Names: There s a two point bonus if you do this with a partner Hand in only one copy 1 a) f() = 3 2/3 Determine which critical points are local maima, minima, and which are not etreme and mark this on a number line for b) Determine which points are inflections and mark the concavity on a number line for Plot and label the critical and inflection points Indicate coordinates Plot and label y (and ) intercepts Use appropriate scales and aes so that the action near critical numbers can be seen The scales may be different on each ais if necessary 9