MATH CONCEPTS CALCULUS
MATH CONCEPTS CALCULUS MATH 131
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This 45 page Class Notes was uploaded by Evert Christiansen on Wednesday October 21, 2015. The Class Notes belongs to MATH 131 at Texas A&M University taught by Yvette Hester in Fall. Since its upload, it has received 35 views. For similar materials see /class/226024/math-131-texas-a-m-university in Mathematics (M) at Texas A&M University.
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Date Created: 10/21/15
53 Evaluating the De nite Integral Reminder de nite Integral is a family of functions DEFINITE Integral is a number Evaluation Theorem If f is continuous on ab then b j fxdx Fb Fa where F f Recall Represents Net Change Examples HW 53 1 27 odd 33 43 SECOND DERIVATIVE AND CONCAVITY Critical numbers from the second derivative are possible in ection points change of concavity fquotx 0 or fquotx is unde ned Test each side of the partition number using a sign chart fquot 3 f is concaveup fquot 3 f is concave down THE SECOND DERIVATIVE TEST COMPUTE f quot0 for EACH critical point 0 found from f RESULTS negative confirms local maximum at 0 positive confirms local minimum at 0 zero test fails and is inconclusive Information from the Derivative Terminology If f39x gt 0 on I then f is increasing on I If f39x lt 0 on I then f is decreasing on I A change from 39 39 to J 39 or J 39 t0 39 39 indicates a local extremum If f x gt 0 on I then f is concave upward on I If f quot16 lt 0 on I then f is concave downward on I A change from concave up to concave down or concave down to concave up indicates in ection An antiderivative of f is a function F such F 39 f HW 29 1311151617 46 OPTIMIZATION APPLICATIONS OF ABSOLUTE EXTREMA I Preliminary word problems MUST be done using CALCULUS Trial and error methods will earn no credit Example HW 2 Find two numbers whose difference is 100 and whose product is a minimum 11 Word Problems a Read through the problem sketch a picture and label if possible b De ne variables create a function and nd the domain c Find critical values i If only one use second derivative test ii If not the right one test endpoints d Answer the question posed in the problem Don t forget units Example A fence is built to enclose a rectangular area of 800 square feet The fence along three sides is made of material that costs 6 per foot The material for the fourth side costs 18 per foot Find the dimensions of the rectangle that will allow the most economical fence to be built You must use calculus HW 46 19 21 INTRODUCTION TO THE TANGENT LINE I A GEOMETRIC INTERPRETATION Recall from geometry that a tangent line to a circle is a line that passes through one and only one point on the circle But for functions in general this is not a satisfactory de nition Estimating the slope of the tangent line Finding velocity 25 Infinite Limits and Limits at Inf39mitV 1 Vertical Asymptotes Infinite Limits A The vertical line x a is a veltical asymptote if lim fx i 00 as x approaches a from the left or the right x T or L without bound X9117 EX Q B For fx n d if 610 0 but nc 0 then x c is avertical asymptote EX II Horizontal Asymptotes Limits at In nity xp For 7 if xq Apgtq B qgtp Example 25 1 9 odd 15 33 odd 32 DERIVATIVES PRODUCTS AND QUOTIENTS 1 Product Rule y fx goo hoe y39 foe gm hoe W goo OR y39 foe goo m hoe gm since multiplication is commutative Examples II Quotient Rule hoe Hx toe b39rx 3939 y x Mx y fx may don t forget that subtraction is not commutative so you cannot switch the order in the numerator Examples HW 32 3 29 odd 13 gcon t The Algebra of Functions Let f be a function Whose domain is A and g be a function Whose domain is B Then fgx fx gx Sum domain AnB f39gx fx gx Di erence domain AnB f gx f gx Product domain Am B JOC fx Quotient domain Am B gx 2 0 g gx Examples Graphical Addition COIVIPOSITION OF FUNCTIONS f o gx f gm g 0 f x gfx In general f o gx fgx at g o fx gfx Examples HW 13 123odd262955odd6163 28 The Derivative Function 1 The Derivative Function Recall we de ne the derivative of f at x denoted by f 39x to be fxhfx h f 39x rate of change of f at x hlino if the limit exists The derivative of a function f 39 is a new function whose domain is a subset of the domain of f Alternate notations f 39x m f m y m dy The notation a reminds us that the derivative is a rate of change The derivative of a function at a point tells you the rate of change at which the value of the function is changing at that point Example If f 39 exists for each x in the open interval a b then f is differentiable over the interval a b But when does the derivative NOT exist 11 NoneXistence of the Derivative If the limit dne at x a then f is nondifferentiable at x a or f a dne So ifthe graph off 1 has a sharp corner at x a thenf a dne 2 has a vertical tangent line at xa and has no tangent line at xa thenf39a dne since slope is unde ned 3 Ifthe graph off is broken at x a not continuous at x a then f a dne NOTE If f is differentiable then f is continuous But f continuous DOES NOT IMPLY f differentiable III Higher Derivatives HW 28 l23l3l925 odd3l3337 42 Information from the First Derivative c is a critical number forf if f39c 0 0r f39c is undefined FIRST DERIVATIVE TEST Critical numbers and sign charts f f local maximum at c c 2 local minimum at c c 3 no local extremum c 4 no local extremum c EXAMPLES De nition Absolute MAX if f C 2 f x for all x in the domain of f Absolute MIN if S for all x in the domain of f Extreme Value Theorem A function f continuous on a closed interval a b assumes both an absolute maximum and an absolute minimum on that interval I All absolute extrema if they exist must always occur at critical values or at endpoints f continuous over a b Find critical values in a b Find fafbfc Absolute maximum is the largest of step 3 Absolute minimum is smallest of step 3 EXAMPLES HW 42 1 17odd 54 Fundamental Theorem of Calculus Reminder de nite Integral is a family of functions DEFINITE Integral is a number Fundamental Theorem of Calculus If f is continuous on ab and F is any antiderivative of f then If gx ftdt then g x fx and fxdx Fb Fa represents the change in F from x a to x b Examples HW 54 19 odd 1321odd 24 Onesided Limits and Continuity x l x Z 0 Example f x x lxlt0 Onesided limits help de ne continuity A function f is continuous at a number x a if 1 f a is de ned 2 lim x exists xgta 3 JETx 170 If not the function f is discontinuous at x a Further f is continuous on an interval I if it is continuous at every number in I Jaxgt0 xxS0 Example f x x 2 x at 2 Example hx 1 2 Graph h x Is h x cont1nuous at x 2 a x Properties of Continuous Functions A constant function f x c is continuous everywhere 2 A polynomial function f x P x is continuous everywhere 3 Arational function R x is continuous everywhere 9 x 7 0 q x 4 If f x and g x are both continuous at x a then a f x quot is continuous at x a b f x i g x is continuous at x a c f x g x is continuous at x a x is continuous at x a g 90 Examples 24 HW 341316 25 31 odd N Linear Regression Go to 2nd zero Catalog and select DiagnosticsOn Select ENTER until the word Done appears To enter data into the lists STAT EDIT ENTER To obtain a Scatterplot and Regression equation 2nd Y STATPLOT ENTER Select ON ENTER and the rest of your scatterplot choices using the arrow keys Before graphing the scatterplot you must clear out the Y list ZOOM 9 for scatterplot STAT CALC LinRegaXb Note defaults to data in LlL2 Y VARS Statistics EQ RegEQ GRAPH 23 Properties of Limits Let xli nfoc L and xli ngoc M 1 ailW gem L 1imcfx 7 c 1 i 1 x CL 2 xgta 7 xgta 1 i Magi gx 1imfx i limgx 3 xaa xaa xgta 4 1 i n fxgx 1imfx limgltxgt xaa xaa xgta lim x HIE MCI m 7 A 539 xgta SOC lim 05 7 M xgta 1 i m 0 H S to 639 xgta 23 125792231333540 L M 15 EXPONENTIAL FUNCTIONS fx bx b gt 0 b 1 b is called the base and x is the exponent Graph Characteristics 1 Domain 2 Range All Laws of Exponents still hold 1 Miaka 4 abab 2 Lara 5 3 L by b 3 3 b Examples 1 x 2x f1 853 239 8713 3 Solve 22x71 16 Base 6 APPLICATIONS Growth and Decay yce kgt03growthandklt03decay Exponential Growth Examples Populations grow exponentially The national debt grows exponentially Speci cally the population of Mexico can be modeled exponentially by P 67381026 Initially the year of beginning interest was 1980 t 0 the population was 6738 million and grows by 26 each year Most often however the exponential growth model with base 6 used Example Exponential Decay 15HW17 26 RATES OF CHANGE RECALL GEOMETRIC INTERPRETATION To de ne atan ent line for ata oint P A point P is given on f N Pickapoint Q on f E Draw a line through PQ this is the secant line 4 Let the distance from Q to P gt 0 3 h gt 0 fx1hfx1 h mm slope of the tangent line to f at P THUS lim hgt0 also called the slope of the graph of f at P the instantaneous rate of change and velocity Knowing the slope of the tangent line and the coordinates of P enables us to use the pointslope form of a line to write the equation of the tangent line Example Find the equation of the tangent line to 8X 2X2 at X l HW712l7 55 Method of Substitution Reversing the Chain Rule n1 JLfxquotf39xdx c quotgt 1 n Select u such that du is afactor in the integrand N Rewrite the integrand entirely in terms of u and du E Evaluate the new integral 4 Rewrite the antiderivative in terms of the original variable Examples HW 55 153 odd 4STEP METHOD FOR ANALYZING THE BEHAVIOR OF A FUNCTION STEP 1 Use fx A Domain B Intercepts C Asymptotes STEP 2 Use f x A Partition Numbers 1 Equal to Zero 2 Unde ned B Sign Chart C Intervals Where f is increasing and decreasing D First Derivative Test for Critical ValuesLocal Extrema STEP 3 Use f x A Partition Numbers 1 Equal to Zero 2 Unde ned B Sign Chart C Concavity of f D In ection E Second Derivative Test for Critical ValuesLocal Extrema STEP 4 Back tofx A Start graph With Step 1 information B Sketch in Step 2 information C Sketch in Step 3 information D Use graphing utility to check accuracy 33 A quot quot ofthe Derivative to Biolog The derivative is a rate of change A average rate of change 3 7y instantaneous rate of change 3 11m 7 Ax Axao Ax From physics velocity is f and acceleration is f For Biology some application of the derivative include the rate of growth for animal or plant populations or the rate of ow of blood through a blood vessel Example If we consider the ow of blood through a blood vessel we can take to shape of the blood vessel to be cylindrical with radius R and length l The velocity of the blood is greatest blood ows the fastest along the center of the vessel central axis and decrease closer to the walls of the vessel because of friction At the walls the velocity becomes 0 Let r the distance from the center aXis to the wall of the vessel The relationship between v and r is given by the law of laminar ow discovered by the French physician Poiseuille in 1840 as P v 7 R2 r2 where 77 is the viscosity ofthe blood and P is the pressure difference between the ends of the tube So Av dv the 1nstantaneous rate of change 3 11m 7 7 r 7gt0 Ar d1 of velocity with respect to r So forasmall human artery we let 77 0027 I 2 cm P 4000 Lainey 2 r 0002 cm cm instantaneous velocity HW 33 3 13 25 11 Functions 1 A function is a rule that assigns to each element x in a set A domain or input set represents the independent variable exactly one element called f x in a set B range or output set represents the dependent variable A We usually consider functions for which the sets A and B are sets of real numbers Functions arise whenever one quantity depends on another 11 We visualize a function by its graph If f is a function with domain A then its graph is the set of ordered pairs x fx l x e A A Vertical Line Test 7 An equation de nes a function if and only if no vertical line in the xy plane intersects the graph more than once III Finding the domain and range and function notation IV Evaluating a function 11 HW part 5 7 13 odd 23 7 39 odd 45 47 35 The all important General Power Rule or Chain Rule a y fx ux y39f39x nux 1u39x Notice that we now have a function raised to a power and notjust x b yfx W y39 f 39x em u x Again notice that we now have functions raised to a function and notjust x c y fx bx y39f39x bx lnb Examples HW 35 1 39 odd 22 Introduction to Limits Notation um x L This is read quotas x approaches a x approaches Lquot xx20 x xlt0 Recall fx x x3 xlt 2 Example fx xgt 2 1 x20 Example hx 1 0 x lt 1 1 continued I Piecewise De ned Functions De ned in pieces graphed in pieces x x20 fltxgtx 7x xlt0 x xlt0 fx x20 11 Symmetry A f x f x 3 f is an even function symmetric about the yaXis B 7 f x f x 3 f is an odd function symmetric about the origin III Increasing and Decreasing Functions A f is increasing if fx1 lt fx2 when x1 lt x2 in an interval B f is decreasing if fx1 gt fx2 when x1 lt x2 in an interval HW 11 41 44 61769 odd 16 artZ LOGARITHMS Graph of f 31 9 x gb gt 0 b 3 5 1 Properties Continuous on 090 Passes through 1 0 Domain 000 Range 5R b gt 1 increasing 0 lt b lt 1 decreasing PROPERTIES OF LOGARITHMS 1 Oaxgc 0 lac x M 1 logbglogbxlogby 2 1 Obkggj1 01ng Oby 3 10bgpp10bg 410gbquotx loblgOb e cboaclz 5 19 39 x lobbglb e 0116sz Examples Use properties of logs to simplify Recall that the population of Mexico can be modeled exponentially by P 67381026t where initially in 1980 t 0 the population was 6738 million and grows by 26 each year Question When will the population be 100 million 1 k 6 7 1 3 3 t We could use trial and error t 10 2 6 71 3 X5 5amp3 6 not enough t 20 2 6 71 35 g 2 6 too much back up and try t 15 2 6 7 1 3 338 N29 close go one more t l6 2 6 71 3X g B 6 close So what we know that the population will be 100 million somewhere between 15 and 16 years THIS COULD GET TIME CONSUMINGH Logarithm functions allow us to solve for the exact t 10 is a special base called the common log and is written 1 0x 6 0 P 50 x This says that the logarithm to base 10 of x is the power of 10 you need to get x Examples of base 10 Richter scale pH The most frequently used base is base 6 or the natural logarithm c 1 9x gl XII C 0 er x lnx is the power of e neededto get x Solve 1 2 523t Example Find the halflife of P Po0 8 1 rx Change of Base Formula 1 0 gr E and e bx Function Hierarchy e grows the fastest x p In x grows slowest HW 21 39 odd 47 48 49 57 SIMPLE DERIVATIVE RULES function I Constant y f x constant II Power Rule yfx xquot 7161 III Constant Times a Function Rule y fx kgx IV Sum and Difference Rule y fx gx i Wx V The Exponential Function y HW 31 3 31 odd 37 38 derivative formula yfw0 Recall derivative is slope and slope of a horizontal line y const is 0 y39af39cw nx yJWwk1w ffwg wilfw yfww 34 TRIG FUNCTIONS AND THEIR DERIVATIVES Identities sin x 1 tan x sec x cosx cosx cosx 1 1 cot x f 050 x 5111 x cotx 5111 x II Derivatives d d iSIle cosx icosx smx dx dx d 2 d 2 7 tan x sec x 7 cot x 050 x d d isecx secxtanx icscx cscxcotx dx dx Examples HW 34 119odd 25 27 THE DERIVIATIVE For y f we de ne the derivative of f at x denoted by f 39x to be fxh fx h f 39x rate of change of f at x hlino if the limit exists De nition The tangent line to y f x at a f a is the line through a f a whose slope is equal to f 39x the derivative of f at a Examples HW 1 7odd 13 18 11W I INTRODUCTION TO REGRESSION METHOD OF LEAST sg ZAURES Calculator exercise Interpolation when the estimated value lies between within the observed values Extrapolation when the estimated value lies outside the observed values II POLYNOMIAL FUNCTIONS 1 Elma a0 quotEIa1X 7 Elan1xlan30 0 I polynomialofdegree n n normegative 2 degree 1 11 near degree 2 2 quadratic graph is aparabola 2 degree 3 cubic III ALGEBRAIC FUNCTIONS Recall from section 13 that these are combinations of functions using sum difference product and quotient rules with restricted domains IV RATIONAL AND POWER FUNCTIONS D I f Ck I Rational R g D I quotient of two polynomials automatically algebraic 1 Power PD EX FUN EX V Trigonometic Functions VI F quot 39 and I ogarithmic Functions Transcendental HW 12 1 15 odd 19 20 21 23 24 61 AREA BETWEEN TWO CURVES Let f and g be continuous functions such that f x Z gx on a b Then the area bounded by x and gx on a b is b Woe gx dx Example Find the area bounded by the xaXis and y x2 4x 8 on 1 4 Sketch a graph and shade the appropriate area Example Find the area bounded by y x and y 2 x2 on 1 3 Sketch a graph and shade the appropriate area RIGID AND NONRIGID TRANSLATIONS RIGID TRANSLATIONS f x k shifts the graph of x to the right k units f x k shifts the graph of x to the left k units f x k shifts the graph of x down k units f x k shifts the graph of x up k units f x re ects the graph of x across the x 7 aXis f x re ects the graph of x across the y 7 aXis NONRIGID TRANSLATIONS k fx two cases 1 if0ltklt1 atter graph smaller y Values farther from the yaXis 2 ifkgt1 steeper graph larger yValues closer to the yaXis 16 Dart 1 INVERSE FUNCTIONS I A function f is called one to one if it never takes on the same yValue twice Horizontal Line Test A function is onetoone if and only if no horizontal line intersects its graph more than once Onetoone functions have inverses 11 Let f be a onetoone function with domain A and range B Then it s inverse function denoted f 391 has domainB and rangeA and is defined by f 391yx when fxy HW1 15odd 37 DERIVATIVE RULES FOR LOG FUNCTIONS m derivative formula a y1nx xgt0 y frx l x 39 1 1 b y fx 10gb x y f x 17 1113 x r 1 I c y 1111100 y f x 7u x ux Examples 37 HW 1 25 odd
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