Calculus Ia MATH 1125
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Date Created: 09/17/15
Section 25 1 Section 25 Continuity De nition A function f is continuous at a number a if hm we M 404 Interpretation De nition A function f is discontinuous at a number a or has a discontinuity at a if f is de ned in an open interval containing 17 but f is not continuous at 1 Example 1 For the function whose graph is given below7 nd the points of discontinuity of f Section 25 2 Three Types of Discontinuities 1 removable discontinuity 3 7 2 2 7 8 Example f W x 2 2 jump discontinuity Example f the greatest integer function 3 in nite discontinuity 1 Exam le z 7 p f 27 4 De nition A function f is continuous from the right at a number a if nm n m maa De nition A function f is continuous from the left at a number a if gg n m De nition A function f is continuous on an interval if it is continuous at every number in the interval At the end points of an interval7 f should be continuous from the left or right Example 2 Determine the intervals where the function in Example 1 is continuous Section 25 3 Theorems About Continuous Functions Theorem 1 If f and g are continuous at a and c is a constant then the following functions are continuous at a f i 9 cf fg i if 9a 7 0 9 Theorem 2 The following types of functions are continuous on their domains polynomials rational functions root functions trigonometric functions inverse trigonometric functions exponential functions and logarithmic functions 5 if S 72 Example 3 Consider the function f 7s 3 if 7 2 lt x S 1 2x 7 1 if 1 lt z a At what values of z is the function discontinuous b Give the intervals where the function is continuous Be careful about whether to include the endpoints Example 4 For what value of c is the following function continuous on 700 00 x2 7 9 f 96 z 7 3 x2cxc if3 ifzlt3 Theorem 3 If f is continuous at b and limgz b then lim fgz fb 3 Example 5 Evaluate lim cos L 1773 x2 5x 6 Section 25 4 Theorem 4 If g is continuous at a and f is continuous at 9a7 then the composition function f o 9 given by f o fgx is continuous at 1 Example 6 Determine the intervals where the function f lnz2 7 4 is continuous The Intermediate Value Theorem Suppose that f is continuous on the closed interval 17 and let N be any number between fa and fb7 where fa 31 fb Then there exists a number 0 in a7b such that fc N Example 7 Show the equation cos x x has a root in the interval 01 Section 12 1 Section 12 Mathematical Models A Catalog of Essential Functions Mathematical Modeling Find a mathematical description often using a function or equation of a real world phenomena such as average daily temperature in Hartford cost of gas the height of an object in free fall or the demand for cars Two Main kinds of Models 1 Physical Model based on a physical law or laws Example 2 Empirical model based on recorded data Example Empirical models are determined by a statistical technique called regression Although we will not pursue this method in this class the idea is that you can graph the data and look for a function which best ts77 the data Lines A linear function where m is the slope of the line and b is the y intercept can be written in the slopeintercept form yfx mb The slope m of a line through the points zhyl and x2y2 is M yryz 2791 77177 liig 271 i Ax An alternate formula for a line is the pointslope form of a line with slope m that goes through the point zhyl y iyl mx7 1 Section 12 2 Example 1 Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature and the relationship appears to be very nearly linear A cricket produces 113 chirps per minute at 70 F and 173 chirps per minute at 80 F a Assuming a linear model nd the function that gives the number of chirps per minute N as a function of the temperature T b What is the slope with units of the line What does the slope represent c Sketch a graph of the line labeling the axis intercepts d What is a reasonable domain for this model Polynomials A polynomial function is of the form Pz mm aw nil 1ng alz 10 where n 2 0 is an integer and an owl a2a1a0 are real constants called the coe icients If an 31 0 then 71 is the degree of the polynomial Note that a line is a polynomial of degree one Examples Section 12 3 A quadratic function is a polynomial of degree 2 which can be written f azz bx c 0 Its graph is a parabola which opens upwards if a gt 0 and opens downwards if a lt 0 o The y intercept is c and the z intercepts7 if they exist7 are given by the quadratic equation 7 ibi xbz 7 4ac 7 2a 39 0 Another form of a quadratic function is f az 7 h2 k where a is the leading coef cient as before and h7 k is the vertex of the parabola Example 2 Free Fall Galileo discovered that the distance fallen by any freely falling body is proportional to the square of the time it has been falling ignoring air resistance by a constant which is half the gravitation constant 98 msZ Suppose a coin is dropped from the upper observation deck of the CN Tower in Toronto which is 450 meters above the ground a Find a function that gives the height7 h7 of the coin from the ground as a function oftime t in seconds b How long does it take the coin to hit the ground Power Functions A power function is of the form f x where a is any real constant Special Cases 0 a n Where n is a positive integer polynomials with one term Section 12 4 1 o a 7 Where 71 1s a pos1tlve Integer the root functlon n o a 71 the reciprocal function Rational Functions where Pz and Qz are polyHOmjaIS39 The domain is all real numbers except those that are zeroes of A rational function is of the form f Examples Algebraic and Transcendental Functions An algebraic function is any function which can be constructed from polynomials using algebraic operations such as addition subtraction multiplication division and taking roots Most of the functions discussed so far are algebraic the exception is the power function x when a is an irrational number Examples A transcendental function is any function which is not algebraic The functions discussed in the remainder of this section are all transcendental Note Exponential and Logarithmic functions are important transcendental functions which will be discussed in detail in Sections 15 and 16 53m 12 Triganametnc Momms Buaf Renew at 39l tlganametry SEE App nmx D 7nedlsns 80 we cmg at Camman Values By the dmnmnn ehme we have um dism2i mm mm c y m 1 en saute sage 111 Mg mengg um an an my 7 my 7 Mi ed my 3 mi an an Section 12 The Trigonometric Functions 0 f sin o f Cosx sinx o f tanx 1 cos x 1 1 o The reciprocals See Appendix D cscx secx COt smx cosx tarw sum Section 23 Section 23 Calculating Limits using the Limit Laws Limits Laws Suppose c is a real constant 71 is a positive integer and lim f L and lim g M both exist m7nz m7nz 1 2 4 U 03 5 3331mm m 7 33 me 33 gltzgt 7 L M limf 7 lim f 7 limgz L 7 M lim cfz 0 lim fx CL 33 fltzgtgltzgt 7 33 we gm 7 L M provided that 77 0 Igmnr 7 3331mm 7 L 6 7 a lim x 1 4H1 lim W 5 if n is even then we must have a gt 0 m7nz lim 7 f n lim fx VL if n is even then we must have lim fx gt 0 Example 1 Use limit laws to evaluate the following limit x3 7 z 1 llrn 173 2 1 Direct substitution property If f is a polynomial or a rational function and a is in the domain of f then lim fx fa This follows by applying Limit Laws 179 above Section 23 2 5 if x S 72 Example 2 If f 7c 3 if 7 2 lt x S 17 nd the following limits if they exist 2x 7 1 if 1 lt z enggm mug m How to Compute lim f When f Is Not De ned at a Key Property If f g when x 31 17 then lim fx limgx provided both limits exist Example 3 Rational Functions Factor the numerator and denominator and cancel common factors if possible Use limit laws to evaluate the following limit z 7 12 7 9 H W Example 4 Use limit laws to evaluate the following limit 2 7 7 1712 2 7 Example 5 Functions with Square Roots Rationalize the numerator or denominator where the radical is located Use limit laws to evaluate the following limit 1 7 xt 7 3 1511 4t7t2 Example 6 Functions with Addition of Fractions Find a common denominator and add the frac tions Use limit laws to evaluate the following limit 1 1 lim 4 x 774 4 x Section 23 3 Example 7 Functions With Absolute Values The strategies vary but it usually comes down to determining when the expression inside the absolute values is negative or positive Use limit laws to evaluate the following limit 2x 12 396 w 6 Important Limit Theorems Comparison Theorem lf f S g when x is near 1 except possibly at a and the limits lim f and lim g both exist7 then lim f 3 lim Squeeze Theorem lf f S g S h when x is near 1 except possibly at a and the limits lim f lim h L7 then lim g L Fact lim M 1 eao 6 5 Example 8 Algebraically nd the value of the limit lin S1 95 ma Section 11 1 Section 11 Four Ways to Represent a Function Example 1 Tori Amos will be playing at Radio City Music Hall which seats 2500 people For simplicity7 lets say that all tickets cost 50 each How much revenue will the concert earn What are the different ways we can express this information De nition A function f from a set D to a set E is a rule that assigns to each element x in D exactly one element y in E 0 independent variable 0 dependent variable 0 domain 0 range 0 functional notation Other Examples Example 1b Determine the independent and dependent variables of the function in Example 17 and nd the domain and range De nition A graph of a function f with domain D is the set of all ordered pairs z for all z in Section 11 2 De nition The z intercepts of a function are the numbers if they exist where the graph intersects the x axis or equivalently the values of z for which y 0 if y 0 is in the range De nition The y intercept of a function is the number if it exists where the graph intersects the y axis or equivalently the value of y when x 0 if x 0 is in the domain Vertical Line Test A curve in the xy plane will be the graph of a function of x if and only if every vertical line intersects the graph at most once Four Ways to Represent a Function Example 2 Algebraic to Numeric to Visual Consider the algebraic real valued equation f 2 2 where z is a real number a Find the values f71 and f0 b Find fx 3 7 f and simplify c Sketch a graph of the function f What are the z and y intercepts Section 11 3 d Find the domain and range of f using the graph above Example 3 Visual to Verbal Consider the graph below Where t is the number of hours since 4 an and C is the number of cars that have crossed the George Washington Bridge into Manhattan on a Weekday Ct cars 2000 1000 t hours a Why do We know C is a function of t b About hoW many cars have crossed the bridge by 9 am Represent this With functional notation c Describe in Words What the graph tells you about the ow of cars across the GW bridge into Mani hattan Section 11 Implied Domain of Algebraic Function Unless stated otherwise7 the domain of a function given by a formula is the set of all real numbers which can legally be put in for the independent variable What7s illegal Example 4 Find the domain of each of the following functions Give your answer using interval notation ewm 2 c x2x72 wwm fi Piecewise De ned Functions Functions which are given by two or more different formulas on different parts of their domain Example 5 The absolute value of a real number x denoted m is the distance from x to 0 a Write this as a piece wise de ned function b Sketch a graph of the function f Section 11 A Few More Concepts De nition A function f is called an even function if for every x in its domain Graphical Interpretation De nition A function f is called an odd function if for every x in its domain Graphical Interpretation Example 6 Determine if the following functions are even7 odd or neither a f96 96396 Section 11 6 De nition A function f is increasing on an interval I if for every 1 and 2 in I if 1 lt x2 then fx1 lt zz Graphical Interpretation De nition A function f is decreasing on an interval I if for every 1 and 2 in I if 1 lt x2 then fx1 gt zz Graphical Interpretation De nition A function f is constant on an interval I if for every 1 and 2 in I 951 952 Graphical Interpretation Take Home Problem Look at all the examples in this section exclude Examples 4 and 7 and determine the intervals where each function is increasing7 decreasing7 or constant Section 22 1 Section 22 The Limit of a Function 2 7 5 7 6 Example 1 Consider the function f Notice that it is unde ned at z 71 However7 let us look at values of f when x is close to 71 Are the function values getting close to some number De nition Let f be a function de ned on both sides of 17 except possibly at a We say that the limit of a function fx as x approaches 1 equals a number L7 written lim f L7 if we can make values of fx arbitrarily close to L by taking values of z suf ciently close to a but not equal to 1 Limits Graphically 4 Example 2 Use the graph to guess the value of lin w ma Section 22 When does a limit not exist Example 3 Use a table of values to guess liH sin Be careful ma X X Example 4 Use a table of values to guess liH ma x Left Hand Limit We say that the limit of a function fx as x approaches a from the left7 equals a number L7 written limi f L7 if we can make values of f arbitrarily close to L by taking values of z lt a suf ciently close to a Right Hand Limit We say that the limit of a function fx as x approaches a from the right7 equals a number L7 written lim f L7 if we can make values of f arbitrarily close to L by taking values of z gt a suf ciently close to 1 Theorem lim f L if and only if limi fx limJr f L Example 5 Given the graph below7 nd the following Section 22 3 In nite Limits 1 Example 6 Use a table of values to guess mlirnL X De nition Let f be a function de ned on both sides of 17 except possibly at a We say that the limit of a function fx as x approaches 1 tends to 007 written lim f 007 if we can make values of f arbitrarily large by taking values of z suf ciently close to 17 but not equal to 1 De nition Let f be a function de ned on both sides of 17 except possibly at a We say that the limit of a function fx as x approaches 1 tends to 700 written lim f 700 if we can make values of f arbitrarily large negative by taking values of z suf ciently close to 17 but not equal to 1 De nition A function f has a vertical asymptote of s a if any of the following limits hold limi f ioo mac Example 7 Given the graph below7 nd the following a Ere b hm M 1472 c hm we za72 d ma72fm Example 8 Determine the in nite limits a lim 1473 2 7 9 b lim 27 maifd z 7 9
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