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# Calc for BusLife Sciences MATH 1314

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This 31 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 Rebecca Heeth in Fall. Since its upload, it has received 17 views. For similar materials see /class/208370/math-1314-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

Math 1314 Lesson 16 Antiderivatives So far in this course we have been interested in nding derivatives and in the applications of derivatives In this chapter we will look at the reverse process Here we will be given the answer and we ll have to nd the problem This process is generally called integration We can use integration to solve a variety of problems Antiderivatives De nition A function F is an antiderivative of f on interval I if F x x for all x in I The process of nding an antiderivative is called antidifferentiation or nding an indefinite integral Example 1 Determine if F is an antiderivative of f if F x g gxz 2x 5 and xx23x2 Example 2 Suppose Hx x3 10 and Kx x3 7 27 If x 3x2 show that each ofH and K is an antiderivative off and draw a conclusion Notation We will use the integral sign I to indicate integration antidifferentiation Problems will be written in the form l x dx F x c This indicates that the inde nite integral of x with respect to the variable x is F x c where F x is an antiderivative off Basic Rules Rule 1 The Inde nite Integral of a Constant j k dx kx C Example 3 l 5 dx Rule 2 The Power Rule n1 x 1Cn l Ixquotdx n Example 4 lx4 dx Example 5 dx Example 6 dx x Rule 3 The Indefmite Integral of a Constant Multiple of a Function Icfxdx cjfxdx Example 7 I 4x3 dx Example 8 dx x Rule 4 The Sum Difference Rule 1 fx i gxdx 1 fx dx i 1 gm dx Example 9 J 2x2 5x 1 dx Rule 5 The Inde nite Integral of the Exponential Function Ie dx 6x C Example 10 Sex 7 4x3 dx Rule 6 The Indefinite Integral of the Function f x l x jldx1nxC x 0 x Example 11 Ex 2 2 dx x T x Applying the Rules 3x 4x2 5x3 dx Example 12 J x Example 13 1x2 l 3jabc x x Example 14 JT3 76 2 dx x Differential Equations A differential equation is an equation that involves the derivative or differential of some function So if we write f x 4x 5 we have a differential equation We will be interested in solving these A solution of a differential equation is any function that satisfies the differential equation So for the example above x 2x2 5x 3 is a solution of the differential equation since the derivative of f is 3x5 The general solution of a differential equation is one which gives all of the solutions so the general solution for the example above will befx 2x 5x c If we know a point that lies on the function we can find a particular solution that is we solve for c If we know that Z 1 we can substitute this information into our general solution and solve for c The point 2 1 orf2 l is called an initial condition Initial Value Problems An initial value problem is a differential equation together with one or more initial conditions If we are given this information we can find the function f by first finding the general solution and then finding the value of c that satisfies the initial condition Example 15 Solve the initial value problem f39 x 2x 5 f 2 3 Example 16 Solve the initial value problem f x 3e 2x f 0 7 Example 17 A cable television provider estimates that the number of its subscribers will grow at the rate of 100 2101 34 new subscribers per month tmonths from the start date of the service Suppose 5000 subscribers signed up for the service before the start date How many subscribers will there be 16 months after the start date From this section you should be able to Explain what we mean by an antiderivative indefinite integral a differential equation and an initial value problem Determine if one function is an antiderivative of another function Use the basic rules to find antiderivatives Simplify if necessary before applying the basic rules Solve initial value problems M 1314 lesson 1 1 Math 1314 Lesson 1 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 centuIy 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 arbitraIy curve M 1314 lesson 1 2 W L we39ll onee we39 we39ll Core Probl em 1 Amajor natura1 foods retauerbegan an aggressrve aeqursrtron earnpatgn m 2006 acqumng tva eornpetrng natura1 foods retauers dunng that year The graph grven ear below represents the acqumng company Revenues m brlhons of dollars 5 gross revenues dunng that y 1 2 3 4 Months State the acqumng eornpany39s gross revenues at the begmmng of 2006 State the acqumng eornpany39s gross revenues at the begmmng oprnl 2006 State the acqumng eornpany39s gross revenues at the begmmng ofAugust 2006 Where 15 the graph ofthe eornpany39s revenues ducontmuou Explam the 15 V101 ated at eaeh pornt of drseontrnurtw Use hrnrt notataon to desenbe the hrnrts from the 1e t and nght at eaeh pornt of dummian M 1314 lesson 1 3 Limits Finding a limit amounts to answering the following question What is happening to the y Value of a function as the xValue 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 number Example 1 9 s 7 455 4 3 2 1 2 3 4 5 5 7 a 9 1 10 M 1314 lesson 1 4 It does not matter Whether or not the x value every reaches the target number It might or it might not Example 2 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 M 1314 lesson 1 5 Case 2 At the target number for the x value the graph of the function has a vertical asymptote 79724757574472 3455729 For either of these two cases we would say that the limit as x approaches the target number does not exist Definition We say that a function f has limitL as x approaches the target number a written lim f x L ifthe value x can be made as close to the numberL as we please by taking x suf ciently 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 substitutin 2 factoring and reducing 3 nding limits at in nity To use the rst two of these methods we will need to apply several properties of limits M 1314 lesson 1 Properties of limits Suppose lim fx L and limgx M Then liiralfx39 fx39 L for any real number r cfx fx CL for any real number c gym i gx gigmi giggoc L M g3fxgx 1333 1 00 133 gx 1M lim fx lim fx L L providedM 2 0 Ha gx 11mgx M bUJN UI We ll use these properties to evaluate limits Substitution Example 3 Evaluate linzl3x2 4x 5 2 Example 4 Evaluate 11m x H0 x l 3 Example 5 Evaluate lim3x2 xgt4 M 1314 lesson 1 7 k What do you do when subst1tutlon glves you a value 1n the form 6 where k 1s any nonzero real number 2 5 Example 6 Evaluate 11m x xgt3 95 3 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 x2 4x 5 Example 7 Evaluate lim 2 xgt1 x 1 2 5 6 Example 8 Evaluate llm Xgt2 x M 1314 lessonl 8 So far we have looked at problems Where the target number is a speci c real number Sometimes we are interested in nding out What happens to our function as x increases or decreases Without bound Limits at In nity 2 Example 9 Consider the function f x 22x 1 What happens to x as we let the value x of x get larger and larger 1000 10000 100000 1000000 10000000 We say that a function x has the limitL as x increases Without bound or as x approaches in nity Written lim f x L if x can be made arbitrauly close to L by taking x large enough We say that a function x has the limitL as x decreases Without bound or as x approaches negative in nity Written lim f x Liffx can be made arbitrauly close to L by taking x to be negative and su iciently large in absolute value We can also nd a limit at in nity by looking at the graph of a function 2x77 Example 10 Evaluate lim M 1314 lesson 1 9 We can also nd limits at in nity algebraically Example 11 Evaluate lim4x3 7x 5 Limits at in nity problems often involve rational expressions fractions The technique we can use to evaluate limits at in nity is to 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 Then we can 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 12 Evaluate llmxz x Hm 3x 2x 7 Often students prefer to just lea1n some rules for finding limits at in nity 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 M 1314 lesson 1 10 If the degree of the numerator is smaller than the degree of the denominator then If the degree of the numerator is the same as the degree of the denominator then you can nd 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 2x4 5x4 Example 13 Evaluate lim 2 Hm x xl 5 2 3 4 Example 14 Evaluate llme x Hm 4x 2x 8 M 1314 lesson 1 11 Example 15 Evaluate lim24x 5 Hm x 9x 9 From this lesson you should be able to State the two basic problems of the calculus De ne limit indeterminate form Find a limit as x approaches a target number from a graph off State when a limit fails to exist k Evaluate 11m1ts where subst1tution glves 6 k 72 0 Evaluate limits by substitution or by factoring Evaluate limits at in nity Math 1314 Lesson 8 Some Applications of the Derivative Equations of Tangent Lilies The rst applications of the derivative involve nding the slope of the tangent line and writing equations of tangent lines Example 1 Find the slope ofthe line tangent to fx x2 5x 15 at the point 2 3 Example 2 Find the equation of the line tangent to x 4x 7 2x2 at the point 3 6 Example 3 Find the equation of the line tangent to x 5 7 2x 7 3x2 when x 2 Example 4 Find the equation of the line tangent to x 1n2x 7 x2 when x 1 Horizontal Tangent Lines etc Some other basic applications involve nding where the slope of the tangent line is equal to a given number Example 5 Find all values of x for which the line tangent to x x3 7 4x2 4x 9 is horizontal Example 6 Find all values of x for which the slope of the line tangent to x 3x 7 21nx is equal to 0 Example 7 Find all values of x for which the slope of the line tangent to x x3 7 4x2 7 5x 7 is 3 Rates of Change Sometimes we re interested in nding the rate of change of a function at a specific point Example 8 A Chamber of Commerce commissions a study on population growth indicating that the city s population is growing according to the function Pt 40000 50f 150t where P represents the population tyears from now How fast is the population increasing in 4 years Example 9 Developers of a masterplanned community estimate that the population in thousands of the community I years from now will follow the function Pt 25t2 125t 200 t2 5t 40 will be the population after 10 years At what rate will the population be changing when t 10 Find the rate at with the population is changing with respect to time What Velocity and Acceleration A common use of rate of change is to describe the motion of an object The function of the position of the object is with respect to time so it is usually a function of t instead of x Ifthe object changes position over time we can compute its rate of change which we refer to as velocity We can find either the average rate of change or the instantaneous rate of change depending on the question posed The average velocity on the interval x x h will be the difference quotient The instantaneous velocity at the point x c will be the derivative of the position function evaluated at c Velocity can be positive negative or zero If you throw a rock up in the air its velocity will be positive while it is moving upward and will be negative while it is moving downward We refer to the absolute value of velocity as speed When you accelerate while driving you are increasing your speed This means that you are changing your rate of change Acceleration then is the derivative of velocity 7 the rate of change of your rate of change It follows that the second derivative of a position functions gives an acceleration function So if position is given by ft and instantaneous velocity is given by vt f t then acceleration is given by at f 39t Example 10 The distance 3 in feet covered by a car I seconds after starting from rest is given by the function st t3 12t2 36L a Find the average velocity over the interval 1 2 b Find the instantaneous velocity when t 2 seconds c Find the acceleration when t 2 seconds From this lesson you should be able to Find the slope of a tangent line at a point Write an equation of a tangent line at a point Determine values of x for which the rst derivative is zero ie the tangent line is horizontal Determine values of x for which the rst derivative is some constant k Solve problems involving velocity and acceleration Solve problems involving other rates of change Math 1314 Lesson 9 Marginal Functions in Economics Marginal Cost Suppose a business owner is operating a plant that manufactures a certain product at a known level Sometimes the business owner will want to know how much it costs to produce one more unit of this product The cost of producing this additional item is called the marginal cost Example 1 Suppose the total cost in dollars per week by ABC Corporation for producing its bestselling product is given by Cx 10000 3000x 7 04x2 Find the actual cost of producing the 101st item The cost of producing the 1015 item can be found by computing the average rate of change that C101 C100 is by computing 101 100 wherex 100 and h 1 Note that w W 01 100 xh x The right hand side of this equation is the formula for average rate of change of the cost function This will give us the actual cost of producing the next item This however is often inconveni ent to use For this reason marginal cost is usually approximated by the instantaneous rate of change of the total cost function evaluated at the specific point of interest That is to say we ll nd the derivative and substitute in our point of interest Example 2 Suppose the total cost in dollars per week by ABC Corporation for producing its bestselling product is given by Cx 10000 3000x 7 04x2 Find C 100 and interpret the results Note that the answers for examples 1 and 2 are very close This shows you why we can work with the derivative of the cost function rather than the average rate of change The derivative will be much easier for us to work with So we de ne the marginal cost function as the derivative of the total cost function You will nd that by a marginal function we mean the derivative of the function So the marginal cost function is the derivative of the cost function the marginal revenue function is the derivative of the revenue function etc Example 3 A company produces noisecanceling headphones Management of the company has determined that the total daily cost of producing x headsets can be modeled by the function Cx 00001x3 7 003x3 135x 15000 Find the marginal cost function Use the marginal cost function to approximate the actual cost of producing the 21st and 181st headsets Average Cost and Marginal Average Cost Suppose Cx is the total cost function for producing x units of a certain product Ifwe divide this function by the number of units produced x we get the average cost function We denote C x The derivative x this function by 5x Then we write the average cost function as 5x of the average cost function is called the marginal average cost Example 4 A company produces office furniture Its management estimates that the total annual cost for producing x of its top selling executive desks is given by the function Cx 400x 500000 Find the average cost function What is the average cost of producing 3000 desks What happens to 5x when x is very large Marginal Revenue We are often interested in revenue functions as well The basic formula for a revenue function is given by Rx px where x is the number of units sold and p is the price per unit Often p is given in terms of a demand function in terms of x which we can then substitute into Rx The derivative of Rx is called the marginal revenue function Example 5 A company estimates that the demand for its product can be expressed as p 004x 800 where p denotes the unit price and x denotes the quantity demanded Find the revenue function Then nd the marginal revenue function Use the marginal revenue function to approximate the actual revenue realized on the sale of the 4001st item Marginal Pro t The nal function of interest is the pro t function This function can be expressed as Px Rx 7 Cx where Rx is the revenue function and Cx is the cost function As before we will nd the marginal function by taking the derivative of the function so the marginal pro t function is the derivative of Px This will give us a good approximation of the pro t realized on the sale of the x lst unit of the product Example 6 A company estimates that the cost to produce x of its products is given by the function Cx 0000003x3 7 008x2 500x 250000 and the demand function is given byp 600 7 08x Find the pro t function then nd the marginal pro t function Use the marginal pro t function to compute the actual pro t realized on the sale of the 51st unit Example 7 The weekly demand for a certain brand of DVD player is given by p 02x 300 0 S x S 15000 where p gives the wholesale unit price in dollars and x denotes the quantity demanded The weekly cost function associated with producing the DVD players is given by Cx 0000003x3 7 004x2 200x 70000 Compute C 3000 R 3000 and P 3000 Interpret your results From this lesson you should be able to Explain what a marginal cost revenue pro t function is and how it is used Find a marginal cost function and use it to approximate the cost of producing the x lSt item Find an average cost function Find lim C x and explain what it means Hm Find a revenue function Find a marginal revenue function and use it to approximate the revenue realized on the sale ofthe x lSt item Find a profit function Find a mgrginal profit function and use it to approximate the profit realized on the sale of the x 1 item Math 1314 Lesson 22 Functions of Several Variables So far we have looked at functions of a single variable In this section we will consider functions of more than one variable You are already familiar with some examples of these Px y 2x 2y dim xa 13 C 2 at These formulas are functions of several variables We have just never called them that before We will for the most part limit our discussion to functions of two variables Functions of Two Variables Definition A real valued function of two variables f consists of a setA of ordered pairs of real numbers x y called the domain of the function and a rule that associates with each ordered pair in the domain of f one and only one real number denoted by z x y You will need to learn several skills using functions of several variables 1 Evaluating a function of several variables at a given point Example 1 Suppose x y 3x2y 7 4xy 6 Computef0 0f2 l andfl 3 Example 2 The volume of a cylindrical tank with radius r and height h is given by the formula V r h nrzh Find the volume of a tank with radius 6 and height 20 feet

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