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# AFAMST MUS EXAM 1 guide

OSU

GPA 2.71

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This 7 page Study Guide was uploaded by CMDChiimeh on Wednesday January 21, 2015. The Study Guide belongs to a course at Ohio State University taught by a professor in Fall. Since its upload, it has received 53 views.

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Date Created: 01/21/15

SPRING 2015 10 MARCH MIDTERM STUDY GUIDE WORKING DERIVATIVES Definitions of Derivatives MATH 1151 CALCULUS k39Tke cler39m dve 0 I 03 O E ua lioni CICQ3QKM r R90 R DERIVATIVE SHORTCUTS ddx M moon1 ddx X1 1x11I ddx CX c ddx 2x I f 39x dydx WM 00 1 Sum Difference Rule ddX fX gX f X g X 11 Product Rule 3 ddX fX gX f 002200 fXg X 111 Quotient Rule 3 ddX fX gX f XgX fXg X g 002 IV Chain Rule 3 ddX fgX f 2300 g 00 V Exponential Rule a ddx ex 6quot VI Trigonometry Rule a ddx sin x b ddXcos x c ddx tan x Extra Shortcuts REMEMBER gt ddx ey ey dydx gt ddx ex ex ddx I gt ddxlogcy logcltegty1dydx gt ddxlogcx 10gce x clde gt ddX Cy Cy1nc dydx 410v C09 d ddxcscx e ddXsec x f ddXcot x ddx C 0 ddX 3 0 SPRING 2015 MIDTERM STUDY GUIDE MATH 1151 10 MARCH CALCULUS gt ddX ex cquot1nc de1 gt ddx lnX 1X dydx gt ddx lnX 1x ddx LOGARITHMIC DIFFERENTIATION Differentiation Steps 1 Take In of both sides 2 Differentiate to get ddX lnfx 3 Find f X using the rule ddX lnfx f X fX LX git q52 S x 3911 1m a 3 4quot 4 00 3 Lv 1 r 5 inbc LI 2mm 3 312 c 3 AXLRhLICQ E39 T 357 may will u cvgt 9nccg A A erfJXHW 3 5 2 s I Wei3 T W E39W E39Ezs 1quot INVERSE TRIGONOMETRIC FUNCTIONS DERIVATIVES a ddx sin1 1 1 X6 1 ddxcsc1 I 1IXI1 X2 b ddx cos1 1 1 X2 e ddx sec1 1 IXI1 X2 c ddXtan39111IX2 f ddXcot391I11IX2 DERIVATIVES AS RATES OF CHANGE I l ndistance travelled d t sty velocity over I04 Iv I quot 7 w v quotI LIIIi 4 first derivative as ddX st or s t acceleration over My me second derivative as ddX ddXst V s t eX1 Suppose an object is moving horizontally with position function given by st t36t29t a Find the velocity function vt a ddx st 3t339162t239191t1391 vt 3t212t9 b Find the acceleration function at a ddX s t 32t2391121t13910 at 6t12 c When is the object as rest a The object is a rest when its velocity is zero so vt 3t212t9 o 3t24t3 o SPRING 2015 MIDTERM STUDY GUIDE MATH 1151 10 MARCH CALCULUS 3t3t1 O Att 1 3 d When is the object moving in the positive direction a When vt gt O 3t3t1 gt 0 At 00 1U3 00 e Sketch the motion of the object 56 36a 9 3 we 6 M3 w 11 l l n l l L quot quot l I39 I I I I 4 1 u a I z 3 eX2 A company is producing tablet computers The total cost of producing X tablets is given by Cx5x215x200 Marginal Cost The First Derivative IMPLICIT DIFFERENTIATION Finding dydx Step 1 Take ddX of both sides Step 2 Solve for dydX or y ex Find dydX of x3 y3 4xy through implicit differentiation Step 1 dydx x3 V3 dydx 4xy Step 2 3x2 3y2l 4y 4x 3x2 4y 4x 3y2y Related Rates Shape Measurements gtkThe measurements of shapes tend to be very necessary for dealing with most related rates problems SHAPES PERIMETER AREA VOLUME SURFACE AREA circumference Circle 27tr 7td Sphere of insolence Triangle sidel sidez side3 triangle 12bh SPRING 2015 MIDTERM STUDY GUIDE 10 MARCH Rectangle Sldel Sldez Slde3 side4 Rectangular Prism A Box Trapezoid sidel sidez side3 1 Cy nder MATH 1151 CALCULUS 2lw wh 1h 2739E1392 2hnr Method Steps to Solution Step 1 Step 2 Step 3 Step 4 Step 5 Read amp Draw amp Label Write an equation Take ddX Like you would in implicit di erentiation Plug in values given then solve Check your work m i gfwc mam a sauna mm as a at L 0 Rank A4 wlwl39 1 is Le radius 39IMch inj when ne radius is Sun f K F 5 V F VI a i5 51 v1m3 v Spkue 3 2 vquot53n3r r V39L11r1r39 r39quot Mtg 39139L15r 25 2 WM quot55quot MAXIMA AND MINIMA Basic Definitions I Extrema singular Extremum maximums andor minimums I Critical Point A number c in the interior of domain f is a critical point of a graph Where it changes direction or Where its slopes or derivatives are 0 I Extreme Value Theorem E VT If f is continuous on a closed interval a b then f has a max value amp min values I Absolute Maximum Global Max f has a global max at x c if for all x in domain f fx I fc I Absolute Minimum Global Min f has a global min at x c if for all x in domain f fx I fc I Relative Maximum Local Max f has a local max at x c if fc I f x for all x nearby x c SPRING 2015 MIDTERM STUDY GUIDE MATH 1151 10 MARCH CALCULUS I Relative Minimum Local Min f has a local min at x c if fc I f x for all x nearby x c global Inimimauuri A K Inncal nmainnrm local Jii mmuu lowest global I39niJiimnu Finding Critical Points A number c in the interior of domain f is a critical point of a graph where it changes direction or where its slopes or derivatives are 0 Step 1 Take the first derivative of given function Step 2 Set equal to 0 Step 3 Solve for all possible X values that fit in interval if one is given Step 4 Plug in numbers that are to the left and right of these possible critical points into original function SPRING 2015 10 MARCH MIDTERM STUDY GUIDE MATH 1151 CALCULUS GRAPHING FUNCTIONS r7 39 9 I What do we need to graph y fx 1 II III IV VI 39omain Steps Step 1 Find X coordinates where the function is undefined Step 2 State our domain in interval notation ex a b Intercepts X amp y Rules I xintercept where fX or y O I yintercept where X O lymmetry even amp odd Test I If f X fX gt symmetric over yaxis I If f X fX gt symmetric over xaxis I If neither gt NO SYMMETRY Isymptotes vertical and horizontal Rules I Vertical Asymptote where the fX NotOJo I Horizontal Asymptote llmxgtoroo fX rational function trick local Extrema minimums amp maximum Steps Step 1 Take the first derivative of the function and set it 0 Step 2 Determine critical points by solving for all possible X points Step 3 Test around critical points by plugging other values into y Use line graph Step 4 Determine local maX and local min with x amp y values in notation IncreasingDecreasing Step After testing around critical points by plugging other values into y SPRING 2015 MIDTERM STUDY GUIDE MATH 1151 10 MARCH CALCULUS Step 1 Write increasing and decreasing in interval notation VII Ioints of ln ection S eps Step 1 Take the second derivative of the function then set equal to 0 Step 2 Solve for all possible X values Step 3 Test around In ection Points by plugging other values into yquot VIII Ioncavity Test A er resting momma in ection poim s by plugging ozher Wines Mm y 9 1 Iffquotx gt 0 gt fis Concave Up 21ffquotx lt 0 gt fis Concave Down 3 If f quotX changes signs at c gt x c is the In ection Point

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