Skills Needed for Success in Calculus 1

Transcription

1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell ou that the eason that students ae not successful in Calculus is not because of the Calculus, it's because thei algeba and tigonomet skills ae weak. You see, Calculus is eall just one additional step beond algeba and tig. Calculus is algeba and tigonomet with limits and limits aen't eall that had once ou figue them out. Thee is often onl one step in the poblem that actuall involves calculus, the est is simplifing using algeba and tigonomet. That's wh it is cucial that ou have a good backgound in those subjects to be successful in calculus. Moe good news about calculus is that we live in the eal wold, we don't deal with imagina numbes (ecept fo section 9.4, which isn't in Calculus ). Also, in Calculus, we don't deal with logaithmic o eponential functions, which seem to give some people geat difficult. The pupose of this document is to help identif some of those aeas whee ou will need good algeba and tigonomet skills so that ou calculus epeience can be successful, pleasant, and ewading. Algeba Skills Needed Factoing You need to be able to facto epessions and equations like it was second natue to ou. Man of the poblems in calculus will involve finding the oots of a function and fo the most pat that means factoing. Don't just concentate on polnomial factoing, eithe; ou need to be able to facto epessions with ational eponents. Hee is an eample of factoing out the geatest common facto, which is involves taking the smallest eponent on all of the common tems. ( ) ( + ) + ( ) ( + ) ( / / ) ( ) ( ) 8( ) ( / / ) ( ) / / / / / / ( ) ( ) ( 0 9 6) Know how to ecognize and facto the special pattens of the diffeence of two squaes,

2 the diffeence of two cubes, and the sum of two cubes. Know that the sum of two squaes usuall doesn't facto in the eal wold. Diffeence of two squaes: ( )( + ) Sum / diffeence of two cubes: ± ( ± )( + ) Completing the Squae Anothe task that ou will be called on to pefom occasionall is completing the squae. You need to be able to do this with both an equation and an epession. Eamples of both ae shown below. Completing the squae b adding to both sides of the equation ( ) ( ) ( ) ( ) ( ) ( ) Completing the squae b adding and subtacting on the same side ( ) f f ( ) ( ) f ( ) f ( ) + 4 Basic Functions and Tansfomations Algeba (and calculus) can be simplified if ou undestand that thee ae basic functions and that man of the othe functions ae tansfomations of those basic building blocks. You should be able to sketch the gaph and know the domain and ange of the basic functions upon sight. You should also be able to ecognize and appl tansfomations to the basic functions. Conside ( ) quadatic function. You should be able to ecognize that the basic shape is the. To that basic shape, ou have eflected it about the -ais, shifted it up two units, and shifted it ight thee units ( ).

3 Constant k Linea Quadatic Range: { k} Range: (, + ) Cubic Squae Root Absolute Value Range: (, + ) Domain: [ 0, ) Simplifing Epessions Much of ou time in this couse will be spent simplifing the esults of an epession that ou obtained. Know how to combine simila o like tems and know the popeties of eponents like adding eponents when multipling factos that have the same base o multipling eponents when aising to a powe. Fomula Manipulation You need to be able to wok with fomulas as well as have a good ecall of basic geomet fomulas fo aea and volume fo common figues. Thee ae geometic fomulas on the inside font cove of ou tet as a esouce. Fomula manipulation is much moe than just memoizing fomulas and plugging the values into them, howeve. You will need to solve fo diffeent vaiables and ou will need to combine fomulas togethe to come up with new fomulas. Eample: A ight cicula clinde has a volume of 0 cm and its height is twice its cicumfeence. Find the adius and height of the clinde. The fomula fo the volume of

4 an clinde with paallel bases is V Bh B π, whee B is the aea of the base. Since the base is a cicle, the aea of the base is. The cicumfeence of a cicle is and in this clinde, the height is twice the cicumfeence, so the height is C π h C π 4π ( ) ( )( ) V π 4π 4π. The volume becomes. Since V 0 0 4π we know that the volume is, we get. Solving that fo gives π π o cm. The height is cm, but that h 4π 4π simplifies to be π h 64π 0π cm. π Using You Calculato This ma sound like a given condition b the time ou get to calculus, but ou need to be able to gaph functions and get a pope viewing window. You should be able to use the Calc menu on ou calculato to find oots, minimums, maimums, and intesections. You should also know how to use the table mode on ou calculato. You should also know how to change the mode on ou calculato and leave it in Radian mode fo most of this couse. Tigonomet Skills Needed Appendi A in the tetbook contains a eview of Tigonomet. You eall need to know evething in it with the eception of the poduct to sum and sum to poduct fomulas (fomulas 47-). Most of this will need to be memoized so that it is available fo instant ecall. Othe things can be deived b undestanding the elationships between the tigonometic functions and the diffeent quadants. You should be able to sketch the basic tigonometic functions and be able to appl tansfomations to them. Fo eample, conside the function. You should be able to pick out that the basic gaph is the sine function sin ( ) sin + sin, that π π sin ( ) the amplitude is because of, the peiod is fom, sin ( ) sin ( ) + the phase shift is unit to the ight, and the entie gaph has been shift up five units.

5 Sine sin Cosine cos Tangent tan Range: [,] Peiod: π Range: [,] Peiod: π Domain: π ± kπ, + Range: ( ) Peiod: π Memoize the values of the thee tigonometic functions fo the special angles! In this class, ou will be epected to give eact answes in most cases. That means + 7 π witing instead of.44 o and not The good news is that ou will not usuall have to ationalize ou denominatos and witing is oka.