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# Study Guide for Midterm 1 MATH 1951

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This 11 page Study Guide was uploaded by Ashley Aquino on Saturday October 10, 2015. The Study Guide belongs to MATH 1951 at University of Denver taught by Dr. Riquelmi Cardona in Fall 2015. Since its upload, it has received 42 views. For similar materials see Calculus 1(Section 1) in Applied Mathematics at University of Denver.

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Date Created: 10/10/15

Calculus 1 Midterm 1 Outline Section 2 to Section 33 1 Section 21 The Tangent and Velocity Problems a h i Review Pre Calc a Vertical Line Test b Inverse functions fa a a fa cDomain set of input X values of a function 1 Range set of output y values of a function eGraphs of Sine Cosine and Tangent sin x f tan x E Every non vertical line in the plane has a slope 3 2 y1 962 361 Given 2 points on the line use this equation to find slope m PointSlope Form yy1 mxx1 SlopeIntercept Form y mx b Graphs Given fxa the graph of f is the set of points a fa for all in the domain off yfx Tangent Given a curve at a point p on that curve the line through p that touches the curve at a single point is called the tangent line at p Change E position Average Velocity Time elapsed Instant Velocity Velocity at a particular moment 2 Section 22 a Defn Suppose fx is defined when x is near the number a Nearnmeans thaw is de ned on Dquot aninterval that contains a except possibly for a itself Then We Write KI afx L and say the limit of fx as x approaches a is equal to L If we can make the values of fX as close to L as we want by taking x to be close enough to a but not equal to a Defn We write limit as x approaches a from the left a39 and from the right 61 and say the limit of fX as x approaches a from the left is equal to L if we can make the values of fx as close as we want to L by taking x to be close to a and xlta a EX Lim Km fX L xlta Lim XDafx L altx c Rule lim XD a x L if and only if Lim ma fX L AND Lim Kg fx L 1 Infinite Limits when a function keeps going toward positive or negative infinity which are not numbers it has no limit thus the limit DNE Does Not Exist e Defn Let f be a function defined on both sides of a except at a itself Then lim XD afx 00 means that we can make the values of fx as large as we want by taking values of x very close but not equal to a 39quot Similarly lim XD afx 00 means the values of fx can be made very large negative numbers ONE sided limits are defined in the same way g Defn The line xa is called a vertical asymptote of the curve yfx is at least one of the following is true lim Xma x 00 lim Xgafx 00 Elim XIII a x 3900 hm XIII afx 3900 3 Section 23 Calculating Limits Using Limit Laws 3 Suppose that lim XD afx and lim XD a gx exist and a is a constant e Kl a Limxna fx gx limxna fx limxna gx Limxna fx gx11imxna fX 1im xna gX cLim Xga cfx c lim XDafX Lim xnafx gX 1imxna fX limxna gX lim Lxgag x f x gX if the limxga gx 0 z b If we use the project rule iv repeatedly with fx gx we get v1 vii viii iX X Xi 4 Section 24 a Direct Substitution Property If f is a polynomial or a rational function Qlt Limxna fx Limxna x n Xna C C Lim Xna xa where n is a positive integer C is a constant let fxx combine vi amp viii we get iX lexna xquot lexna xn a we also get X amp Xi 7 7 L1m Xna x a n pos 1nteger 1f n 1s even agt0 V lim am f x L1m Kl a f x gt n pos integer if n is even lim xna fx gt0 Px x where Px and Qx are polynomials and a is in the domain of f then lim Xna f x a a in domain when it makes denominator equal to 0 b If a is NOT in the domain of we will look for a function that is the same as f in all points except for a c Greatest Integer Function Floor Function quotxi x J the largest integer that is less or equal to x 5 Section 25 Continuity a Defn a function f is continuous at a number a if lim XDa x fa b NOTE For this to be true a f a is defined a must be in domain of f b limfx exists lim Xlafx A lim X311730 exists cIf i and ii are true then lim XDa x has to be f a c In a continuous function small changes in X cause small changes in fx d If lim XD1 ctfa then we say that fx is discontinuous at xa e NOTE It s easier to determine points of discontinuity aRem0vable Discontinuity can change the value of the fxn at 1 point to make it work limit exists b In nite Discontinuity can t be made to fit fxn no matter how hard you try cJump Discontinuity one sided limits exist but they are NOT the same f Defn A function is continuous from the right at a if lim Xg1fxfa and continuous from the left at a if lim XE fx fa g Defn A function is continuous on an interval if it is continuous at every number in the interval h Thm If f and g are continuous at 39a and 39c is a constant then these functions are also continuous at afg b fg g cf d fg efg ifga 0 i Sine and Cosine are continuous exponential and logarithmic lnx functions are continuous in their domains j Thm The following types of functions fxn are continuous at every number in their domain aPolynomials b Rational Fxn gRoot Fxn d Trigonometric Fxn gLogarithmic Fxn k Thm If f is continuous at b and lim Xna gxb then Iim xna fgxfb In other words Iim xna f gx f Iim xna gx 1 NOTE Continuous functions allow you to push lim into the expression m Thm If g is continuous at a and f is continuous at ga then the composition off 9 is continuous at a af ga fga b Note Composition of continuous functions is continuous II Intermediate Value Theorem IVT Suppose that f is continuous on the cross interval from a b and let N be any number between f a and f b where fa fb then there exists 6 in a b I such that f cN altcltb 6 Section 26 Limits at In nity Horizontal Asymptotes a b Defn Let f be a function defined on some interval a 00 Then lim XDw x L means that values of fx can be as close to L as we want if we pick x sufficiently large Defn Line yL is called a horizontal asymptote HA of yfx if 1im xnoo x L OR1im xnoo x 7 Section 27 Derivatives and Rates of Change a g Slope of a Tangent Line For a curve yfX the slope of the tangent line at the fah fa fx fa P0111t a 170 I 11m hEIO h 1m xna h fr2 fn t2 t1 t1 a h t2 t1 Average Velocity fah fa Instantaneous Velocity va lim hgo h Defn Derivative of a function f at a number a denoted by f 39 a is fah fa f a hm hm h if this limit eXists NOTE velocity is the derivative of the position function Rates of Change a If y is a quantity that depends on another quantity X we can write yfX b If X changes from X1 to X2 then the change in X is called AX X2 X1 corresponding change in y is called AyfX2fX1 Average Rate of Change Ay AX h Instantaneous Rate of Change lim AXDO Ay AX 8 Section 28 The Derivative as a Function a b Defn A function is a rule f that assigns to each element X in a set D domain exactly one element that we call fX in a set E codomain ie range Domain of fX Can only have elements in the domain of fX It may be smaller than the domain of f Notations a Second Derivative f quotX b Third Derivative f quot39x cFourth Derivative f 4x d Nth derivative f quotx Other notations dv difixii d1 a f x y dx d dx Dfx Dmx DLfn A function f is differentiable meanmgade va vecan becalculatedfromit at a if f a exists It is differentiable on an open interval a b if it is differentiable at every number in the interval m If f is differentiable at a then f is continuous at 39a If f is discontinuous at 39a then f is not differentiable at 39a NOTE Continuity does NOT result in differentiability 1 Section 31 Differentiation Formulas a b c l39quot If we have a constant function fXc then f 39x0 Vertical tangents result in NO derivatives General Power Rule If fXXIl and n is an integer then f 39x nxn391 d icin Constant Multiple Rule If c is a constant and f is differentiable then W m M dx If f and g are differentiable difixigixii difixii digixii 1 Sum Rule dx dx dx difixigixii difixii digixii 11 D1fference Rule dx dx dx eh 1 Definition of the number 6 e is the number such that lim W h 1 d ex eh 1 Thm dx exlimhm T gtCX 2 Section 32 The Product and Quotient Rules a b Product Rule If f and g are differentiable then f g 39 f 39g f g 39 gffg39 2 Quotient Rule If f and g are differentiable then E 39 g d Uni 10dUnJadUo a W lo2 i In other words 3 Section 33 Derivatives of Trigonometric Functions a Assumptions made when calculating trigonometric derivatives ii iii iv cosG l e 0 DO sin 9 e 1 DO NOTE This sign stands for the Greek letter theta EX Limit of sin theta over theta as theta goes to zero equals one sin 9 e is an EVEN function limits 1 on both sides Trigonometric Derivatives cot cotangent csc cosecant d 1 a sin x cos x d 2 3 cos x sin x d 3 a tan xsec2x d 4 a cot x csc2 x d 5 a sec x sec x tan x d 6 a csc x csc x cotx 1 Sectio a rushing h i 2 Sectio a Calculus 1 Midterm 1 Outline Section 2 to Section 33 n 21 The Tangent and Velocity Problems Review Pre Calc aVertica1 Line Test b Inverse functions fa a a fa cDomain set of input X values of a function 1 Range set of output y values of a function eGraphs of Sine Cosine and Tangent sin x f tan x OS x Every non vertical line in the plane has a slope Given 2 points on the line use this equation to find slope m PointSlope Form y y1 mx x1 SlopeIntercept Form y mx b Graphs Given fxa the graph of f is the set of points a fa for all in the domain off yfx Tangent Given a curve at a point p on that curve the line through p that touches the curve at a single point is called the tangent line at p Change in position Average Velocity Time elapsed Instant Velocity Velocity at a particular moment 11 22 Defn Suppose fx is defined when x is near the number a Near means that f is defined on an interval that contains a except possibly for a itself Then We write x9 a f X L and say the limit of fx as x approaches a is equal to L If we can make the values of fx as close to L as we want by taking x to be close enough to a but not equal to a Defn We write limit as x approaches a from the left a39 and from the right 61 and say the limit of fx as x approaches a from the left is equal to L if we can make the values of fx as close as we want to L by taking x to be close to a and xlta a Lim Xea fx L xlta Lim X9afx L altx Rule lim x9 afx L if and only if Lim xea fX L A Lim X9afx L Infinite Limits when a function keeps going toward positive or negative infinity which are not numbers it has no limit thus the limit DNE Does Not Exist Defn Let f be a function defined on both sides of a except at a itself Then lim X9 afx 00 means that we can make the values of fx as large as we want by taking values of x very close but not equal to a Similarly lim x9 afx 00 means the values of fx can be made very large negative numbers ONE sided limits are defined in the same way g Defn The line xa is called a vertical asymptote of the curve yfx is at least one of the following is true alim x9 1 00 b lim X afx 00 x afX D 1 lim X9 afx 00 3 Section 23 Calculating Limits Using Limit Laws 3 Suppose that lim X9 afx and lim X a gx exist and a is a constant aLim X9 a fx gx lim X9 a fx lim X9 a gx b Lim x9 1 fx gx lim x9 a fx lim x a gx cLim X9 a cfx c lim x9 afx d Lim X9 afx gx lim X9 a fx lim X9 a gx fX Limxaa fX gcx Limxaagcx b If we use the project rule iv repeatedly with fx gx we get vi Lim X9 a fXn Lim X9 afxn where n is a positive integer e x9 a thC x9 a Vii Lim x9 a C C C is a constant viii Lim x a xa let fxx combine vi amp viii we get ix ix Lim X9 a xquot Lim X9 a xn a we also get X amp xi X Lim X9 ark W n pos integer if n is even agt0 xi Lim X9 and f x n Limxga f x gtquotn pos integer if n is even lim Xea fx gt0 4 Section 24 Px a Direct Substitution Property If f is a polynomial or a rational function where Px and Qx are polynomials and a is in the domain of f then lim X9 a fx a a75 in domain when it makes denominator equal to 0 b If a is NOT in the domain of we will look for a function that is the same as f in all points except for a c Greatest Integer Function Floor Function xxj the largest integer that is less or equal to x 5 Section 25 Continuity a Defn a function f is continuous at a number a if lim xea fx f a b NOTE For this to be true a fa is de ned a must be in domain of f blimfx exists lim X9a x w lim Xea x exists cIf i and ii are true then lim x9afX has to be f a c In a continuous function small changes in x cause small changes in fx d If lim xea fx7 f a then we say that fx is discontinuous at xa e NOTE It s easier to determine points of discontinuity a Removable Discontinuity can change the value of the fxn at 1 point to make it work limit exists b1nfinite Discontinuity can t be made to t fxn no matter how hard you try cJump Discontinuity one sided limits exist but they are NOT the same o Defn A function is continuous from the right at a if lim X921fxfa and continuous from the left at a if lim X921 xfcz Defn A function is continuous on an interval if it is continuous at every number in the interval M If f amt g are cowtthuous at a amt c is a COWSJCalLJC them these uwct tows are aLso cowt muous at a fg bfg ccf dfg efg ifga7 0 Sine and Cosine are continuous exponential and logarithmic lnx functions are continuous in their domains M The oLLow utg tapes of uwcttows ow are cowt muous at everg lLltllLb l LlL their dammitL a1gtoL5 momrims Mentioth Fxh 0120 PM dTr Lg0w0hLetlLc FXlL e Logartth mic FXlL M If f is cowt muous at b amt L thL x901 g x b them L LhL x901 g 0 b Em other words L LhL x901 g 0 LLhL mm a 0 NOTE Continuous functions allow you to push lim into the expression M If g is cowt muous at a amt is cowt muous at g a them the compostttow 0151 g is cowt muous at a a a a Wan b Note Composition of continuous functions is continuous Intermediate Value Theorem IVT Suppose that f is continuous on the cross interval from a b and let N be any number between f a and f b where fa7 fb then there exists 6 in a b 9 such that f cN altcltb 6 Section 26 Limits at Infinity Horizontal Asymptotes a b Defn Let f be a function de ned on some interval a 00 Then lim x900fX L means that values of fx can be as close to L as we want if we pick x sufficiently large Defn Line yL is called a horizontal asymptote HA of yfx if lim Xeoo x L OR1im x oofX 7 Section 27 Derivatives and Rates of Change a b Slope of a Tangent Line For a curve yfx the Slope of the tangent line at the point a fa a 11m 1190 W OR hm m fofa h ft2 ft1 Average Velocity t t 61 t1 h t2 t1 2 1 0 f ahf a Instantaneous Velocity va lim h h d Defn Derivative of a function f at a number a denoted by f a is g h f a lim hso W if this limit exists NOTE velocity is the derivative of the position function Rates of Change aIf y is a quantity that depends on another quantity X we can write yfx b If X changes from X1 to X2 then the change in X is called AX X2 X1 corresponding change in y is called Ay fX2fX1 Average Rate of Change Ay AX Instantaneous Rate of Change lim Axeo Ay AX 8 Section 28 The Derivative as a Function a b Defn A function is a rule f that assigns to each element X in a set D domain exactly one element that we call fX in a set E codomain ie range Domain of fX Can only have elements in the domain of fX It may be smaller than the domain of f Notations aSecond Derivative f quotX b Third Derivative f quot x c Fourth Derivative f 4x d Nth derivative f quotx Other notations Emma dx dx dx Dfx Dxf DLfn A funetion iS differentiable meaning a derivative can be calculated from it at a fa eXists It is differentiable on an open interval a b if it is differentiable at every number in the interval M If f is dt erettt tabte at a them f is cowt tttttous at a If f is discowt tttuous at a them f is wot dt erettt tabte at a NOTE Continuity does NOT result in differentiability af X y 1 Section 31 Differentiation Formulas a If we have a constant function fXc then f 39X0 b Vertical tangents result in NO derivatives c General Power Rule If fXXn and n is an integer then f 39X me 1 Constant Multiple Rule If c is a constant and f is differentiable then M C e If f and g are differentiable i Sum Rule dfxgx dfx dL9x dx dx dx ii Difference Rule dfxgx dwxn dLgOm dx dx dx h f Definition of the number 6 e is the number such that lim hao e 1 1 h g TWAL M eh l 6quot lim heo gt ex dx 2 Section 32 The Product and Quotient Rules a Product Rule If f and g are differentiable then f g f g f g b Quotient Rule If f and g are differentiable then 939 i Z 9 f f 939 92 d M lodhi hidlo In other words a lo 102 3 Section 33 Derivatives of Trigonometric Functions a Assumptions made when calculating trigonometric derivatives i ii iii iv cos 0 1 0 0 size 1 NOTE This sign 6 stands for the Greek letter theta EX Limit of sin theta over theta as theta goes to zero equals one sin 6 6 is an EVEN function limits 1 on both sides Trigonometric Derivatives cot cotangen csc cosecan Lim 990 Lim 990 d 1 5s1n x cos x 1 cos x sin x dx d 2 dx tan x sec x 2 3 4 icot csc2 dx x x d gt1lt 5 dx sec x secx tanx 6 d cscx cscxcotx dx

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