Calc for BusLife Sciences
Calc for BusLife Sciences MATH 1314
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This 54 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1314 at University of Houston taught by Marjorie Marks in Fall. Since its upload, it has received 56 views. For similar materials see /class/208393/math-1314-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
Math 1314 Online Week 1 Notes Instructor Marjorie Marks Email mmarksc mathuhedu Course Website onlinemathuheducourses Click on the link to Math l3 l4 CASA Website casauhedu My personal website mathuheduNmmarksc For Course Orientation see the Orientation video posted under Week 1 at onlinemathuheducourses on the 1314 page Open the Popper titled popl at casauhedu Section 104 Limits What is calculus The body of mathematics that we call calculus resulted from the investigation of two basic questions by mathematicians in the 18 century 1 How can we nd the line tangent to a curve at a given point on the curve 2 How can we nd the area ofa region bounded by an arbitrary curve The investigation of each of these questions relies on the process of nding a limit so we ll start by informally de ning a limit and follow that by learning techniques for nding limits Limits Informal definition Finding a limit amounts to answering the following question What is happening to the yvalue of a function as the x value approaches a speci c target number If the y value is approaching a speci c number then we can state the limit of the function as x gets close to the target nurnber Look at these graphs and nd the limit as X gets really close to l in both cases y Example 1 Find lin21fx Find ling fx It does not matter whether or not the x value every reaches the target number It might or it might not Example 2 Find lim fx xgt1 When can a limit fail to exist We will look at two cases where a limit fails to exist note there are more but some are beyond the scope of this course Case 1 The y value approaches one number from numbers smaller than the target number and it approaches a second number from numbers larger than the target number Case 2 At the target number for the xvalue the graph of the function has a vertical asymptote 79724757574444 34557291 4N For either of these two cases we would say that the limit as x approaches the target number does not exist More Formal De nition We say that a function f has limit L as x approaches the target number a written lim f x L Ha if the value x can be made as close to the number L as we please by taking x sufficiently close to but not equal to a Note that L is a single real number Evaluating Limits There are several methods for evaluating limits We will discuss these three 1 substituting 2 factoring and reducing 3 finding limits at infinity To use the first two of these methods we will need to apply several properties of limits Properties of limits Suppose 1imfx L and limgx M Then 1iinfx39 lini fx39 L for any real number r cfx clilnafOc CL for any real number c 1331ku ggx gigmi lxigalgW L M Iguana 1333 1 00 133 gx 1M lim fx im fx L L providedM 2 0 Ha gx 11mgx M bUJN UI We ll use these properties to evaluate limits Substitution Example 3 Evaluate 1imx2 6x 5 XHS x25 3 2x Example 4 Evaluate lim xgt72 k What do you do when subst1tut1on g1ves you a value 1n the form 6 where k 15 any non zero real number Example 5 Evaluate lim 2 KHZ Indeterminate Forms 0 What do you do when subst1tutlon glves you the value 6 This is called an indeterminate form It means that you are not done with the problem You must try another method for evaluating the limit See if you can factor the function If you can you may be able to reduce the fraction and then substitute 2 4 3 Example 6 Evaluate llm Al x 3 x x 2 Example 7 Evaluate lim XHO x x Now we ll look at limits as X gets big without bound approaches in nity 2 Consider the function f x 22x 1 What happens to x as we let the value of x get x larger and larger x 10 50 100 1000 10000 100000 1000000 10000000 x We say that a function fx has the limit L as x increases without bound or as x approaches in nity written lim f x L if x can be made arbitrarily close to L by taking x large enough We say that a function fx has the limit L as x decreases without bound or as x approaches negative in nity written lim f x L if x can be made arbitrarily close to L by taking x to be negative and suf ciently large in absolute value We can also nd a limit at in nity by looking at the graph of a function 2x77 x Example 8 Evaluate lim We can also nd limits at in nity algebraically or by recognizing the end behavior of a polynomial function Example 9 Evaluate lim 74x3 7 7x 5 Limits at in nity problems often involve rational expressions fractions Here s a technique we can use to evaluate limits at in nity 0 divide every term in the numerator and the denominator of the rational expression by xquot where n is the highest power of x present in the denominator of the expression 0 apply this theorem Theorem Suppose n gt 0 Then limin 0 and lim in 0 provided i is de ned I M x X iw x x After applying this limit we can determine what the answer should be YOUMUST KNOW THIS PROCEDURE 2 2 5 1 Example 10 Evaluate llmxz x Hm 3x 2x 7 Often students prefer to just lea1n some rules for finding limits at infinity The highest power of the variable in a polynomial is called the degree of the polynomial We can compare the degree of the numerator with the degree of the denominator and come up with some generalizations If the degree of the numerator is smaller than the degree of the denominator then lim fx 0 H gx If the degree of the numerator is the same as the degree of the denominator then you can find lim ff by making a fraction from the leading coefficients of the numerator and denominator and then reducing to lowest terms If the degree of the numerator is larger than the degree of the denominator then it s best to work the problem by dividing each term by the highest power of x in the denominator and simplifying You can then decide if the function approaches 00 or oo depending on the relative powers and the coefficients The notation lim f x 00 indicates that as the value of x increases the value of the Hm function increases without bound This limit does not exist but the co notation is more descriptive so we will use it Example 11 Evaluate lim 5x2 3x4 em4x2 2x8 2x4 5x4 Example 12 Evaluate lim 2 HA x x1 Example 13 Evaluate 1im24x 5 Hm x 9x 9 Section 1057 Oneisided Limim and Continuity Sometimes We are only interested in the behavior of a mction When We look from one side and not from the other Consider the function fx Find thfx 6 x Now suppose We are only interested in nding a limit from one side of the target number For example We might look only at the values of x that are bigger than 0 In this case We are looking at a oneisided limit If We are looking at values of x that are bigger than 0 then We are considering a right hand limit Here s the notation for a onesided limit Where We are interesting only in values of x that are bigger than We could also be interested in looking at the values of x that are smaller than zero If We are interested in looking only at the values of x that are smaller than 0 then We would be nding the le hand limit Here s the notation for a lefthand limit Our de nition of a limit from the last lesson is consistent with this information We say that lim f x L if and only if the function approaches the same value L from both the xgta left side and the right side of the target number This idea is formalized in this theorem Theorem Let f be a function that is defined for all values of x close to the target number a except perhaps at a itself Then lim fx L if and only if lim fx lim fx L Example 1 Consider this graph Find lim fx lim fx and ling fx if it eXists xaz Hr x We can also nd onesided limits from piecewise de ned functions x l x gt 3 Example 2 Suppose f x F1nd x2 5 x S 3 lim fx lim fx and lin21fx if it exists xgt239 xgt2 xgt 2x 3 x lt 1 x 2 x 2 139 Find lim fx lim fx and lim fx if it exists xarl xarf r1 Example 3 Suppose fx Continuity We will be interested in nding where a function is continuous and where it is discontinuous We ll look at continuity over the entire domain of the function over a given interval and at a speci c point Continuity at a Point Here s the general idea of continuity at a point a function is a continuous at a point if its graph has no gaps holes breaks or jumps at that point Stated a bit more formally A function f is said to be continuous at the point x a if the following three conditions are met 1 at is defined 2 lim fx exists lim fxfa LA You ll need to check each of these three conditions to determine if a function is continuous at a specific point 2x 3 x 2 l Example 4 Determme 1f f x 1s contmuous at x l 2 x 4 x lt l If a function is not continuous at a point then we say it is discontinuous at that point We nd points of discontinuity by examining the function that we are given A function can have a removable discontinuity a jump discontinuity or an infinite discontinuity Example 5 Find any points of discontinuity State why the function is discontinuous at each point of discontinuity 7973275754444 12155 291 Continuity over an Interval A function is continuous over the interval a b if it is continuous at every point in the interval We ll state answers using interval notation 2 4 4 Example 6 Flnd the 1ntervals on wh1chf1s c0nt1nu0us f x x x Example 7 State where f is continuous using interval notation Example 8 State where f is continuous using interval notation Sometimes we consider continuity over the entire domain of the function For many functions this is the entire set of real numbers Example 9 State where fx 3x4 5x2 2x 7is continuous Math 1314 Lesson 20 Evaluating De nite Integrals We will sometimes need these properties when computing de nite integrals Properties of De nite Integrals Suppose fand g are integrable functions Then 1 fxdx 0 2 j fxdx fxdx 3 b cfxdx c fxdx 4 I fx i gxdx I fxdx i I gmdx 5 b fxdx fxdx b fxdx where a lt c lt b We will need to use substitution to evaluate some problems Example 1 Evaluate 1034xx2 35 dx Example 2 Evaluate xzexzdx 2 x2 Example 3 Evaluate II 73 3 6dx x Applications Example 4 A company purchases a new machine for which the rate of depreciation is given by 10000t 6 How much value is lost over the rst three years that the machine is in use Example 5 Suppose you are driving a car and that your velocity can be approximated by vt 2N 25 t2 where tis measured in seconds and v is measured in feet per second How far will you travel in the 5 seconds from t 0 to t 5 Example 6 The marginal daily pro t function associated with production and sales of a Video game is estimated to be P39x 00003x2 004x 17 where x is the number of units produced and sold daily and P39x is measured in dollars per unit Find the additional daily pro t realizable if production and sales is increased from 200 units per day to 300 units per day The Average Value ofa Function We can use the de nite integral to nd the average value of a function Suppose f is an integrable function on the interval a b Then the average value of f over 1 b a the interval is r f xdx This is what average value represents Example 7 Find the average value of f x J over the interval 1 16 Example 8 Find the average value of fx x2 3x 5 0n 2 5 Example 9 The sales of ABC Company in the rst I years of its operation is approximated by the function S t txl02t2 4 where S t is measured in millions of dollars What were the company s average annual sales over its rst ve years of operation From this lesson you should be able to Use the FTOC to compute de nite integrals including problems that require substitution De ne average value Use the FTOC to nd average value Solve word problems using FTOC Math 1314 Lesson 4 Basic Rules of Differentiation We can use the limit de nition of the derivative to nd the derivative of every function but it isn t always convenient Fortunately there are some rules for nding derivatives which will make this easier First a bit of notation d 5 fx 1s a notatlon that means the der1vat1ve of f w1th respect to x evaluated at x Rule 1 The Derivative of a Constant 1H 0 where c is a constant dx Example 1 If fx l7 nd f39x Rule 2 The Power Rule xquot me for any real number 71 Example 2 If fx x5 nd f39x Example 3 If fx 6 nd f39x Example 4 If fxi nd f39x 3 a x Rule 3 Derivative of a Constant Multiple of a Function d d 7 cfx 0 fx where 0 IS any real number dx dx Example 5 If fx 3xquot nd f39x Example 6 If fx 5 nd f39x Rule 4 The SumDifference Rule immigmlfxigx Example 7 Find the derivative fx 4x3 2x2 4 x Rule 5 The Derivative of the Exponential Function 61 x x 7 e e Example 8 Find the derivative fx J 4x3 2 66 Rule 6 The Derivative of an Exponential Function base is not 2 d 7 a In a r a I i l Example 9 Find the derivative f x 4x Rule 7 The Derivative of the Logaritlm c Function id 111 i x i 1 provided x 7 0 dx x Example 10 Find the derivative fx 5x 2 6 lnx From this lesson you should be able to State the basic rules for nding derivatives Select the appropriate rule to use for a given problem Find the derivative of a function using the basic rules Math 1314 Lesson 13 Absolute Extrema In earlier sections you learned how to nd relative local extrema These points were the high points and low points relative to the other points around them In this section you Will learn how to nd absolute extrema that is the highest high andor the lowest low on the domain of the function or on a speci c closed interval Absolute Extrema 0n the Domain of f De nition If f x S f c for all x in the domain off then fc is called the absolute maximum value off If fx 2 fc for all x in the domain off thenfc is called the absolute minimum value of Sometimes you Will be asked to nd the absolute extrema over the interval ltgtltgt 00 Example 1 State the absolute maximum andor absolute minimum values 1234557291 Example 2 State the absolute maximum andor absolute minimum values y 79 72 77 76 is 74 7 72 71 1234557291 As you can see from these two examples the absolute extrema may or may not exist To nd absolute extrema on 0 00 or on the entire domain of the given function algebraically you must graph the function using the guide to curve sketching Absolute Extrema on a Closed Interval More often you will be asked to find the absolute extrema over a closed interval a b and for a continuous function those will always exist and they are much easier to find Theorem If a function f is continuous on a closed interval a b then f has both an absolute maximum value and an absolute minimum value on a b Finding the Absolute Extrema of f on a Closed Interval 1 Find the critical points of f that lie in a b 2 Compute the value of the function at every critical point found in step 1 and also compute at and fb 3 The absolute maximum value will be the largest value found in step 2 and the absolute minimum value will be the smallest value found in step 2 Note The absolute maximum value or the absolute minimum value refers to the y value of the point on the graph or x Read the question carefully Example 3 Find the absolute extrema of fx x2 on the interval 1 3 Example 4 Find the absolute extrema of fx x3 3x2 1 on the interval 2 2 Example 5 Find the absolute maximum value and the absolute minimum value of the function fx x2 2x over the interval 3 l Example 6 Find the absolute maximum value and the absolute minimum value of the function f x 3x4 4x3 on the interval 1 2 Example 7 Find the x coordinate of the absolute extrema of the function 2 fx 3x3 2x on the interval 1 l Example 8 An apartment complex has 100 onebedroom apartment units Research shows that the monthly pro t in dollars realized from renting out x apartments is given by Px 12x2 2256x 48000 How many units should be rented out to maximize the monthly pro t What is that monthly pro t Example 9 A company that produces digital cameras wants to minimize its production costs They estimate that their total monthly cost for producing the camera is given by Px 00025x2 80x 10000 Find the average cost function Find the level of production that results in the smallest average production cost Use the second derivative test to verify that you have found a minimum cost From this section you should be able to Identify absolute extrema on 00 00 from a graph off Find absolute extrema on 61 b algebraically Solve word problems having to do with absolute extrema Math 1314 Lesson 12 Curve Sketching One of our objectives in this part of the course is to be able to graph functions In this lesson we ll add to some tools we already have to be able to sketch an accurate graph of each function From prerequisite material we can nd the domain y intercept and end behavior of the graph of a function and from the last two sections we can learn much about a function by analyzing the first and second derivatives We also know how to find the zeros of some functions We ll expand that group of function before we continue to curve sketching The Rational Zeros of a Polynomial Function The rational zeros of a function are the zeros of the function that can be written as a fraction such l as 2 or Somet1mes we can find the ratlonal zeros of a functlon by factor1ng Example 1 Find the rational zeros fx x3 16x Example 2 Find the rational zeros fx x4 1 1x2 18 Note that zeros that are square roots are NOT rational roots Imaginary solutions to the equation fx 0 such as i 3139 are NOT rational roots Sometimes we won t be able to factor the function Then we ll need another method We ll use a theorem called the Rational Zeros Theorem First we ll nd all of the possible rational zeros of a given function using the Rational Zeros Theorem Then we can use a calculator or synthetic diVision to determine which 7 if any 7 of the possible rational zeros are actually zeros of the function Here s the theorem Rational Zeros Theorem Suppose fx anxquot a x 1 a0 where an 72 0 and a0 72 0 and all ofthe coefficients of the polynomial are integers If x E is a rational zero of the function where p and q have no common factors then p is a q factor of the constant term a0 and q is a factor of the leading coefficient an Example 3 Find the possible rational zeros of fx 2x3 3x2 8x 4 Example 4 Find all rational zeros of f x x3 6x2 3x 10 or state that there are none Example 5 Find all rational zeros of f x x3 7x2 l6x 12 or state that there are none Example 6 Find all rational zeros of f x x4 4x3 3x2 4x 4or state that there are none Example 7 Find all rational zeros of f x x3 3x2 l or state that there are none Example 8 Find all rational zeros of f x x4 5x3 4x2 or state that there are none Example 9 Find all rational zeros of f x x3 4x2 4x 16 or state that there are none Curve Sketching Now we ll turn our attention to graphing functions You will need to be able to use the following guide to sketch the graphs of functions A Guide to Curve Sketching 1 Determine the domain off 2 Find the rational x intercepts and y intercept of the function If there are no rational x intercepts say so 3 Determine the end behavior of the function 4 For an exponential function determine any horizontal asymptotes 5 Determine where the function is increasing and where it is decreasing 6 Find the x and y coordinates of any relative extrema 7 Determine where the function is concave upward and where it is concave downward 8 Find the x and y coordinates of any points of in ection 9 If necessary plot a few additional points to determine the shape of the graph 10 Sketch the function Recall the generalizations about end behavior of a polynomial function from College Algebra PEHH NELL POLH NOHL Example 10 Use the guide to curve sketching to sketch f x x4 4x3 Sometimes a function has some zeros that are not rational We may occasionally give you the approximate zeros of the function and ask you to complete the rest of the guide to curve sketching Example 11 Use the guide to curve sketching to sketch fx x3 6x2 15x 3 Note the approximate zeros of the function are 022 172 and 7 94 Example 12 Use the guide to curve sketching to sketch fx x3 7x2 16x 12 Note we found the rational zeros in example 5 Example 13 Use the guide to curve sketching to sketch fx x3 8x2 19x 12 Example 14 Use the guide to curve sketching to sketch f x xe For the next two problems you are given all of the information listed in the guide to curve sketching You just need to use it to graph the function Example 15 Sketch the function if you are given the following information X concave concave Example 16 Sketch the function if you are given the following information X concave concave Example 17 Here is the graph of a polynomial function Which of the statements below isare true The function has three zeros The graph of the function is increasing on one interval and decreasing on two intervals The graph of the function has one relative maximum and one relative minimum The graph of the function has two in ection points The function could be a quartic function 4 11 degree with a positive leading coefficient V eP Nf From this section you should be able to Find any rational zeros of a 3ml or 4th degree polynomial Use the guide to curve sketching to sketch the graph of a polynomial or exponential Sketch a graph of a function given all of the information from the guide to curve sketching Answer questions about the graph of a function given the graph of the function
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