Revision Topic 7: Introduction to differential calculus

Transcription

1 Revisio Topic 7: Itroductio to differetial calculus Chapter 0: Itroductio to differetial calculus The derivative The derivative is the rate at which a quatity chages. If the quatity is modelled by a fuctio, the derivative ca be foud by workig out the gradiet of the graph. A lie has a costat gradiet, m. A horizotal lie has a gradiet of 0. A vertical lie has udefied gradiet. O a curve, the gradiet chages from poit to poit ad may be egative, positive or zero. To fid the gradiet at ay poit o the curve, you eed to draw a taget to the curve at that poit. A taget is a straight lie that touches the curve at a sigle poit ad has the same gradiet as the curve at that poit. Differetiatio Rather tha draw tagets wheever you eed to fid the gradiet or rate of chage, you ca use some rules to differetiate the fuctio ad fid its derivative. You oly eed to kow the rules for differetiatig a polyomial fuctio. To differetiate f ( x) = ax the method is: Multiply by the power ax = ax The reduce the power by oe 1 ax Copyright Cambridge Uiversity Press 014. All rights reserved. Page 1 of 7

2 Note that: If = 1, the power becomes 0 ad, as 0 x = 1, the derivative will be just a. If < 0, the derivative will have a opposite sig to the origial fuctio. If you have a ratioal fuctio, first rewrite it as a egative power. You ca write the derivative usig two differet types of otatio. If the fuctio was give as f ( x) = ax the you would write f ( x) = ax 1 If the fuctio was give as y = ax the you would write dy = ax dx 1 Depedig o the cotext of the questio, you may have to write such expressios usig differet letters for the variables or a differet letter for the fuctio. If the polyomial is made up of multiple expressios of the form separately. ax, you differetiate each of them For example, if separately: f( x) = 4x x+ 1, the you differetiate the three parts 4x, x ad +1 4x becomes 4x = 1x x becomes 1 = = ( ) x x +1 becomes 0 (a horizotal fuctio has a gradiet of 0) Puttig these back together gives f ( x) = 1x. Rates of chage You eed to work with cotexts ad models that have variables differet from x ad y. The method is the same, but the letters chage. The most commoly see situatios are: Situatio Height i relatio to time Volume i relatio to radius Ay two variables, say A i relatio to b Derivative otatio dh dt dv dr da db Copyright Cambridge Uiversity Press 014. All rights reserved. Page of 7

3 Gradiet of a curve at a give poit If you are give a x value at which to fid the gradiet, you should: Differetiate the fuctio. Substitute i the value for x ad calculate the aswer. Use the otatio give i the questio to write dow the gradiet. For example, give f( x) = 4x x+ 1, fid the gradiet whe x =. The derivative is f ( x) = 1x, so you eed to work out f () = 1 () = 106. So the gradiet is 106 at this poit. Usig your GDC to fid the gradiet After plottig the fuctio o your GDC, you ca fid the value of the derivative at a particular value of x: Texas TI-84 Access the d y dx fuctio. The type i the x value for which you wish to fid the gradiet ad the press [ENTER]/[EXE]. Casio fx-9750gii First make sure your GDC is set up with derivatives o. Fidig x- ad y-coordiates from the gradiet You ca work out the coordiates of a poit o a curve if you kow the gradiet at that poit. To do this: Differetiate the fuctio to get the derivative. Set the derivative equal to the give gradiet value. Solve this equatio to fid the x-coordiate. Substitute this value for x ito the origial fuctio to fid the y-coordiate. Copyright Cambridge Uiversity Press 014. All rights reserved. Page of 7

4 Equatio of the taget at a give poit You should use your GDC to do this. First, eter the fuctio ad draw its graph. The do the followig to fid the equatio of the taget at the give poit: Texas TI-84 Access the taget fuctio. The put i the x value of the give poit ad press [ENTER]/[EXE]. The equatio of the taget is displayed o the scree. Casio fx-9750gii Equatio of the ormal at a give poit The ormal is the lie which is perpedicular to the taget at the give poit. If you have a Casio GDC, there is a ormal fuctio just below the gradiet fuctio. If your GDC is ot able to give you the equatio of the ormal directly, you should first fid the equatio of the taget ad the use it to work out the equatio of the perpedicular lie, as doe i Chapter 14. Copyright Cambridge Uiversity Press 014. All rights reserved. Page 4 of 7

5 Chapter 1: Statioary poits ad optimisatio Icreasig ad decreasig fuctios These are defied as follows: A fuctio is icreasig where the gradiet is positive, i.e. f ( x) > 0 A fuctio is decreasig where the gradiet is egative, i.e. f ( x) < 0 A poit at which the gradiet is zero, i.e. f ( x) = 0, is called a statioary poit. A statioary poit where the curve chages directio from icreasig to decreasig or vice versa is also called a turig poit. Statioary poits, maxima ad miima I the above graph there are four statioary poits. All of these are local maximum or miimum poits. They do t give the overall largest or smallest values for f( x ), but each gives the largest or smallest value o that portio of the graph. You may be asked to fid local maximum ad miimum poits of a give fuctio or to determie whether a particular statioary poit is a local maximum or miimum. To work this out, first draw the graph of the fuctio o your GDC; use the maximum ad miimum fuctios to fid the turig poits. The steps are summarised i the followig table. Copyright Cambridge Uiversity Press 014. All rights reserved. Page 5 of 7

6 Texas TI-84 To get the maximum poit: Casio fx-9750gii To get the miimum poit: The Texas GDC eeds you to tell it which part of the graph you wat to aalyse. The Casio GDC will assume you mea a particular poit. Use the arrow keys to move the cursor to a x value o the left of the turig poit; the press [ENTER]/[EXE]. Now move the cursor to the right of the turig poit ad press [ENTER]/[EXE] agai. The press [ENTER]/[EXE] agai, ad you should get the poit you wat. Copyright Cambridge Uiversity Press 014. All rights reserved. Page 6 of 7

7 Optimisatio You may be asked to work out the optimal solutio for a situatio. This meas the best solutio ad will ormally be oe that gives a maximum or miimum value. To solve such problems: Put the fuctio ito your GDC. Draw the graph o your GDC ad sketch it o paper to provide a explaatio of your aswer. Fid the maximum or miimum poits. Use the differetiatio fuctio o your GDC to fid rates of chage if eeded. Use the table o your GDC to fid particular values if eeded. Do ot forget to aswer the questio i the cotext i which it is set. Copyright Cambridge Uiversity Press 014. All rights reserved. Page 7 of 7