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# Math 103, midterm 2 study guide MATH 103 001

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This 0 page Study Guide was uploaded by Cambria Revsine on Saturday March 12, 2016. The Study Guide belongs to MATH 103 001 at University of Pennsylvania taught by William Simmons in Spring 2016. Since its upload, it has received 56 views. For similar materials see Intermediate Algebra Part III in Mathematics (M) at University of Pennsylvania.

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Date Created: 03/12/16

32 Math lO3 Midterm 2 32 3 ll Derivatives The derivative of a function fX at any point X f39x1im fxh fltxgt f39x1im fZ fltxgt z X h or the second point Derivative notations flx 2y jfxDfxDxfxgt One can graph a derivative by estimating the slope at several points on the function plotting the points as x f x on a second graph and connecting the dots with a smooth curve Differentiable when a function has a derivative at every point or at a specific point I a function is differentiable on an open interval if it has a derivative at every point on the interval I a function is differentiable on a closed interval ab if it has a derivative on the interior ab and a right hand derivative at a and left hand derivative at b Where is a function not differentiable at a point gtkWhenever the right and left hand derivatives are not equal 1 2 3 4 5 A corner like f x x at XO xV f x Z A vertical tangent where the slope is vertical undefined A discontinuity where there is a jump or point missing A cusp like at XO 1 Rapid oscillation of the slope like at f x s1n at XO gt If f is differentiable at a point it is continuous at the point 33 I the converse is not necessarily true Differentiation Rules Derivative of a Constant Function If fxc then CO Power Rule 1 xquot mcquot1 dx Derivative Constant Multiple Rule du d If u is a differentiable function of x and c is a constant then aka CE Sum Rule If u and v are differentiable functions of x then their sum u v is differentiable and iuv d u dx dx dx amp Find the points where the curve y x4 2 x2 2 has horizontal tangents lHorizontal tangents occur where the slope of an equation equals 0 i x4 2x2224x3 4x dx 4 x3 4 x 0 4 x x2 1 0 x O l 1 Plug the X values back in to original equation 0211lt 11gt Derivative of e 1 dx X 6 X 6 Product Rule If u and v are differentiable at x then their product uv is differentiable and dv du W u v dx dx dx EI First times derivative of the second plus second times derivative of the first Quotient Rule If u and v are differentiable at x and if v x i O then the quotient 1s v differentiable and v u dx dx d dx Bottom times derivative of the top minus top times derivative of the bottom all over bottom squared M V Second derivative derivative of the first derivative Second derivative notations x i Q d 39 quot 2 2 dx2 dx dx iy D fxDxfltxgt dx gt You can keep taking the derivative any subsequent number of times to the nth derivative 34 Rates of Change Instantaneous rate of change off with respect to x at x0 equals the derivative of x0 fxohfxO f x0 lim hgt0 h gtkInstantaneous rates are limits of average rates rate of Change usually means instantaneous rate of Change Position 5 f I Velocity derivative of position with respect to time t tAt t V hm fl fl dt A gt0 A t gt Positive velocity means it is moving forward negative velocity means it is moving backward Speed absolute value of velocity MIN dtvc Acceleration derivate of velocity with respect to time If a object s position at time t is sft then its acceleration at time tis dv d 2 S a I 2 dt dz gtkrate of Change of an object s velocity how quickly an object increases or decreases speed Jerk derivative of acceleration with respect to time 39 z L3 J dt d 3 gtksudden change in acceleration Derivatives in Economics Marginal cost of production derivative of cost of production with respect to x the number of units produced I AKA average cost of each additional unit produced dc dx Sensitive a term for when a small change in X produces a large change in fX 0 When X is small its change produces a larger 1 change in y 05 I 39 0 In the derivative graph a higher value means V it is more sensitive at that point a 39 05 35 Trig Derivatives Derivative of sin x cos x Derivative of cos x sin x Derivative of tan x sec2x Derivative of csc x csc x cot x Derivative of sec x sec x tan x Derivative of cot x csc2 x 36 The Chain Rule When differentiating a composite function find the derivative of the outside function first keeping the original inside functions and then multiply by the derivative of the inside functions I Composite functions f g fgX y f u u 8 x Derivative f gxg39x In other terms dy dy du E E39E E Find the derivative of y 3x212 yflwlbt2 ugx3x2l f 39ugt g 39xgt 2u6x z23x2i6x a36x312x amp Find the derivative of sin fxz ex with respect to X 6x2excos fx2ex2xex isin 2 dx 37 Implicit Differentiation Implicit Differentiation is used when differentiating a function with X and y where it is difficult or impossible to manipulate it into a y 2x form Examples yz x 0 x2y225 1 Differentiate both s1des of the equation With respect to X The derivative of y is y or dx y I dy d 2 Collect the terms With E on one s1de of the equation and solve for E Cl 2 2 E F d f X in dx 0 x 32 25 x 222 dy 2dy dx 2x2y x20 x dy 2 2 y dx x Ly 1 dx y d kFind of y22x2sin xy 2yy392x008wxy39y 2yyl cosxyxyl2xcosxyy xylx 2y cos22xcosxyy y39Z 2xycosxy 2y xcosxy gt To find the second derivative follow normal derivative rules to differentiate the first derivative y 39 39 Z and then plug in the first derivative into the 3239 values of the second derivative Normal line Line perpendicular to the tangent line of a point take negative reciprocal of the derivative to find the slope of the normal line E Show that the point 24 lies on the curve x3 y3 9 xy 0 Then find the tangent and normal lines to the curve To show the point lies on the curve plug in the point into X and y 2343 9240 v Find derivative 3x23y2yl 9xyly0 3y2yl 9xyIZ 3x29y y 3y2 9x 3x29y 3 x29 y y 3 y2 9x v3y x2 322 3 x Plug in point into the derivative to find the tangent slope 42 32gt 39 y 5 4 12 4 2bb 5 5 4 12 T tl39 angen me 2 5 x 5 Find normal line slope mi 4 5 13 4 2bb 4 2 5 13 Normal llne 2 4 x 2 38 Derivatives of Inverse Functions and Logarithms f The Derivative Rule for Inverses 22 1 39 x 11 f 39f M c 1 Ex fxx3 2 xgtO Find at x6f2 X Derivative 0f the Natural Log Function d 1 du lnu dx x dx Derivative of a u u du a a not dx dx Derivative 0f logau d 1 du 10gau dx ulna dx xn enlnx In xquot nlnx E Differentiate f x xx xx exlnx 1 dx xlnx i dx xlnx rm 2 e xlnx xlnx 1 lnxx x 26 xxlnx1 e as a Limit x 18 4 6 e1im Z xgt0 39 Inverse Trig Functions Graphs of Inverse Trig Functions 39 vl39t l l 39I 3 EILEEN 39ll llmmm e L 11mm e HIE l f 2 i Huang 391 7339 y lhmnmir l39 I r I Ir J W l I ll Ham Et mil L figquot2 Mrmam I T L l quotr 39I Hung 39 Mu IIJ gtkInverse and regular trig functions have opposite inputsoutputs 39 jrmtsum Li V 39 I 3 I l y Y Uunmm l i I 1 ll Range 1 fl fffli jI39 mean 17 I 1 2 l I 1 Ihmmin II 9 2 39I Rmngr It Tr gt Graphs of inverse trig functions are one cycle of trig function graphs rotated clockwise 90 Derivative Rules If x falls within each inverse trig function s domain l u2ZZ d l Sln uT dx 6 l 3 u2 d 1 1 cos u dx 6 tan1u 1 d x 1u2dx 1csclu 1 dx lul Muz l dx 1 a x lu2 dx 310 Related Rates Related rates problems Finding a rate of change from other known rates of change by differentiating multiple variables that are functions of time Steps 1 Draw a picture and name the variables and constants t time all variables are differentiable functions of t Write down additional numerical information Write down what you are asked to find I usually a rate derivative Write an equation that relates the variables Might have to combine equations to eliminate variables Differentiate with respect to t Plug in the known rates and then evaluate for unknown rate I I l39 39 I39m amp A hot air balloon rising straight up from a level field is tracked by a range finder 150 m from H the liftoff point At the moment the range finder s elevation angle is Z the angle is increasing at the rate of 014 radmin How fast is the balloon rising at that moment l Balloon d9 6 dt 14 radmin d yZ dt Range finder 150 m Variables 9 angle in radians the range finder makes with the ground and y height in meters of the balloon off the ground 2 More information 62 2 014 radmin when 92 3 We want to find d y when 92E dt 4 4 Equation tan9 I y2150tan9 dy 2 d9 150 9 5 Differentiate dt sec dt d 6 Plug in known rates and evaluate 1 3 150seczg 014 22201442 21502 dz I At the moment in question the balloon is rising at the rate of 42 mmin amp A police cruiser approaching a right angled intersection from the north is chasing a speeding car that has turned the corner and is now moving straight east When the cruiser is 06 mi north of the intersection and the car is 08 mi to the east the distance between them is increasing at 20 mph If the cruiser is moving at 60 mph at the instant of measurement what is the speed of the car Police cruiser dS E 20 Situation when XO8 y06 dy 60 dt d d f Car Variables X position of car at time t y position of cruiser at time t and s distance between car and cruiser at time t d d d Wewanttofind 61 when x08mi y06mi d ZZ 6Omph jZZOmph d d i is negative because 2 1s decreas1ng Equation 52 x2 y2 25 2xd x2yd y dt dt dt dSldxd lt ygt 5 XE ydt ds 1 g Q dtx2y2ltx dty dzgt 1 dx 20 O8 O6 6O 082O62lt dt lt D 2008d x 36 dt dx 56208 dt dx 7O dt I At the moment in question the car s speed is 70 mph 311 Linearization Approximating complicated functions with simpler functions linearizations at a point I Results in same equation as tangent line but found in different way Linearization off at a Lxfaf39ltagtltx agt l kFind the linearization of Zx at x20 fxv73 1 lx 2 1 x 2 6 f gt 2 l 102 2 1 2 Linearization Lxfaf ax a l x L 1 0 21 x 2 x 2 I Plugging in numbers close to 0 show that the linearization is more accurate the closer to O the input is k N 1X kx x near 0 any number k Differentials With the differentiable function y f x the differential dx is an independent variable The differential dy is dy f xdx Sometimes also called df f xdx d I different set up for f 39x Ema dy if yx537x 5x 437dx dyz amp d tan2x S6622 xd2x 2 seCZZxdx I Same as derivative of the function times dx Estimating with Differentials When dszx fadxfaAy The differential estimation gives f adxz f ady I a is a usually whole number on the x aXis near point a dx that is used to estimate f a dx because f a is easier to compute a plus the change in x equals the new x l k Use differentials to estimate 7973 1 65de We make a 8 because it is the closest whole number to 797 fadxzfady Tomake adx797 dx is 003 f797zf8f39xdx B because dyf xdx gal3L 0o3 1 62E O03 19975 1 gtkThe true value of 7 97 is 1997497 Error in Differential Approximation True Changein f AffaAx fa Estimated Change in f df f 39aAx A f and df are not the same Approximation error A f df tfaAx fa f39aAx Mama ma Ax f aAx eAx i K faAx fa fla isealled e Ax Changein yfx near xa2 Ayf aAxeAx true Change estimated Change error gt0 as Ax gtO

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