[Ezra Halleck]

3.5: The graphs of and are shown below. De­ter­mine which graph cor­re­sponds to and . Be sure to pro­vide sup­port (ev­i­dence) for your choices.

[Abdoulaye]

How to solve the prob­lem:

There are two in­di­ca­tors that one should look for when iden­ti­fy­ing the graphs of higher de­riv­a­tives — both in­di­ca­tors are the re­sult of recognz­ing cer­tain facts about de­riv­a­tives. The first in­di­ca­tor is to rec­og­nize that the points at which the the tan­gent line is hor­i­zon­tal. These hor­i­zon­tal tan­gent lines occur at min­ima and max­ima of graphs (the peaks and troughs of a graph). It is ex­tremely im­protant to re­mem­ber that when the tan­gent line is hor­i­zon­tal the de­riv­a­tive is equal to zero. The sec­ond in­di­ca­tor is whe­hter a par­tic­u­lar in­ter­val on the graph is in­creas­ing or de­creas­ing. If an in­ter­val along the graph is in­creas­ing the de­riv­a­tive has a pos­i­tive value, while if an in­ter­val is de­creas­ing the de­riv­a­tive has a neg­a­tive value. The graph of the de­riv­a­tive along the in­ter­val of the func­tion that is de­creas­ing should be below the x-axis, i.e, neg­a­tive. In con­trast the graph of the de­rivat­ice along the in­ter­val of the func­tion that is in­creas­ing should above the x-axis, i.e, pos­i­tive . Putting these to­gether, one could de­ter­mine what graphs are de­riv­a­tives of oth­ers.

In the prob­lem given no­tice that the graph of func­tion A has a max­i­mum that is right on the y-axis i.e, the tan­gent line across this max­i­mum is hor­i­zon­tal. In other words, at this point the de­riv­a­tive is equal to zero. There­fore the graph of the de­riv­a­tive of the func­tion of A should be cross­ing the x-axis at pri­cisely the same x-value that the max­i­mum is lo­cated. And in­deed, one no­tices that the graph of the func­tion B obeys this be­hav­ior: It crosses the x-axis at pri­cisely the same x-x-axis value that the max­i­mum for the graph of A is lo­cated. Eval­u­at­ing the ex­trema for the graph of B should drive this point home and dis­play the ease at each the graphs could be de­ter­mined. No­tice that the graph of B has max­i­mum and and min­i­mum. At both these points, the tan­gent line hor­i­zon­tal, again, in­di­cat­ing that the de­riv­a­tive is zero. There­fore, the graph of the de­riv­a­tive of B should have two points at which it crosses the x-axis at pri­cisely the same x-value that ex­trema (max­i­mum or min­i­mum) is lo­cated. If one looks care­fully it is clearly ev­i­dent that the graph of C obeys these fac­tors. These ob­ser­va­tion alone should be able to con­vince any­one that graph B is the de­rivat­ice of A and graph C is the de­riv­a­tive of the graph of C: A=f, B=f’, and C= f”. How­ever, for more com­plex graphs an­a­lyz­ing in­creas­ing and de­creas­ing in­ter­vals could be nec­es­sary.

At any rate, this my analy­sis (it may be a lit­tle con­fus­ing and not very clear). Any feed­back and ques­tions are wel­comed.

-Ab­doulaye

[V. Panov]

This is all I see….. “3.5: The graphs of and are shown below. De­ter­mine which graph cor­re­sponds to and . Be sure to pro­vide sup­port (ev­i­dence) for your choices.” I don’t see any graphs.

[Abdoulaye]

@1475­panov

Fol­low this link to get a pdf of the prac­tice exam: http://​openlab.​citytech.​cuny.​edu/​mat​1475​sp20​13/​exam_​review/

[V. Panov]

@Ab­doulaye Thanks so much man, I ap­pre­ci­ate it.

[Dom]

Dude, you could’ve just said that since C has a neg­a­tive value where B has a neg­a­tive slope and since at both changes in slope di­rec­tion for B re­sults in C cross­ing the x-axis: C is the de­riv­a­tive of B. But since B has a pos­i­tive value when A’s slope is pos­i­tive and a neg­a­tive value when A’s slope is neg­a­tive: B is the de­riv­a­tive of A. So you’re right.