Calc for BusLife Sciences
Calc for BusLife Sciences MATH 1314
Popular in Course
Popular in Mathmatics
This 56 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 Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/208412/math-1314-university-of-houston in Mathmatics at University of Houston.
Reviews for Calc for BusLife Sciences
Report this Material
What is Karma?
Karma is the currency of StudySoup.
You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!
Date Created: 09/19/15
Math 1314 Lesson 14 Optimization Now you ll work some problems where the objective is to optimize a function That means you want to make it as large as possible or as small as possible depending on the problem The first task is to write a function that describes the situation in the problem Here are some suggestions to help make this easier 1 Read the problem carefully to determine what function you are trying to nd 2 If possible draw a picture of the situation Choose variables for the values discussed and put them on your picture 3 Determine if there are any formulas you need to use such as area or volume formulas If you have a right triangle in your picture decide if the Pythagorean Theorem will help In many problems you ll state the domain before you work the problem Once you have the function and its domain you ll nd the critical points and see if the critical points fall within the domain of the function You can also use the second derivative test to verify that you have an absolute max or an absolute min in many problems Example 1 A man would like to have a rectangular shaped garden in his back yard He has 100 feet of fencing to use to fence in the garden Find the dimensions for the largest possible garden he can make if he uses all of the fencing Example 2 If you cut away equal squares from all four comers of a piece of cardboard and fold up the sides you will make a box with no top Suppose you start with a piece of cardboard the measures 4 feet by 5 feet Find the dimensions of the box that will give a maximum volume Example 3 A homeowner wants to fence in a rectangular vegetable garden using the back of her garage which measures 20 feet across as part of one side of the garden She has 110 feet of fencing material and wants to use that to build the fence What should be the dimensions of the garden if she fences in the maximum possible area Example 4 Postal regulations state that the girth plus length of a package must be no more than 104 inches if it is to be mailed through the US Postal Service You are assigned to design a package with a square base that will contain the maximum volume that can be shipped under these requirements What should be the dimensions of the package Note girth of a package is the perimeter of its base Example 5 A power station and a factory are on opposite sides of a river 60 m wide A cable must be run from the power station to the factory The factory is 1000 m downstream from a point directly across the river from the power station It costs 25 per meter to run the cable under water and 15 to run the cable on land Determine how far downstream from the power station the construction crew should begin laying the cable under land in order to minimize construction costs Example 6 You are assigned to design some shipping materials at minimum cost The package will be a closed rectangular box with a square base and must have a volume of 50 cubic inches The material used for the top costs 35 cents per square inch the material used for the bottom of the box costs 45 cents per square inch and the material used for the sides costs 20 cents per square inch Find the dimensions of the box that will minimize the cost Example 7 When an apartment leasing company sets its rent for a onebedroom apartment at 600 per month all of the 50 onebedroom apartments are leased Research shows that for every 25 increase in monthly rent they will lose one tenant What should be the rent if the leasing company wants to maximize its revenue Example 8 Suppose you wish to fence in a pasture that lies along the straight edge of a river You will divide the pasture into two parts by means of a fence that runs perpendicular to the river and parallel two of the sides of the pasture You have 1500 meters of fencing to use and you wish to fence in the maximum possible area Determine the dimensions of the pasture that will provide the maximum area What is that area From this section you should be able to Write a function from a verbal description Optimize a function Math 1314 Lesson 21 Area Between Two Curves Two advertising agencies are competing for a major client The rate of change of the client s revenues using Agency A s ad campaign is approximated by fx below The rate of change of the client s revenues using Agency B s ad campaign is approximated by gx below In both cases x represents the amount spent on advertising in thousands of dollars In each case total revenue is the area under the curve given in thousands of dollars Agency A Agency B L y 1000 2000 30 1000 2000 30 This graph shows the relationship between the two revenue functions We see that one function is above the other The area between the two functions represents the projected additional revenue that would be realized by using Agency B s ad campaign Y 1000 2000 so This is an example of the kinds of problems you will be able to solve with the techniques you learn in this lesson We can compute the area between the two curves The general formula is 17 J top function bottom functiondx a Example 1 Find the area between the two curves fx xz 2 and gx x lfromx 0 tox 2 v msyxmz v v 7x7 i Example 2 Find the area between the function fx x3 x and the x axis from x l to x l Example 3 Find the area between the functions f x x2 4x and gx 4x Example 4 Find the area between the functions f x 3x2 2 and gx x 3 and the veItical lines x 1 and x 3 Example 5 Without any effort to curb population growth a government estimates that its population will grow at the rate of 608 02 thousand people per year However they believe that an education program will alter the growth rate to t2 60 thousand people per year over the next 5 years How many fewer people would there be in the country if the education program is implemented and is successful Example 6 The management of a hotel chain expects its profits to grow at the rate of 2 1 13 million dollars per year Iyears from now If the renovate some of their existing hotels and acquire some new ones their profits would grow at the rate of 1 24 4 million dollars per year Find the additional profits the company could expect over the next ten years if they proceed with their renovation and acquisition plans I ll use a graphing utility to generate the graphs of these two functions From this lesson you should be able to Find the area between two curves Sketch the graphs Set up the necessary integrals Find points of intersection if necessary Integrate and evaluate Solve word problems involving the area between two curves Math 1314 Lesson 3 The Derivative The Limit De nition of the Derivative We now address the rst of the two questions of calculus the tangent line question We are interested in nding the slope of the tangent line at a speci c point W I I We need a way to nd the slope of the tangent line analytically for every problem that will be exact every time We can draw a secant line across the curve then take the coordinates of the two points on the curve P and Q and use the slope formula to approximate the slope of the tangent line Consider this function Now suppose we move point Q closer to point P When we do this we ll get a better approximation of the slope of the tangent line Suppose we move point Q even closer to point P We get an even better approximation We are letting the distance betweenP and Q get smaller and smaller What does this sound like Now let s give these two points names We ll express them as ordered pairs Now we ll apply the slope formula to these two points This expression is called a difference quotient The last thing that we want to do is to let the distance betweenP and Q get arbitrarily small so we ll take a limit This gives us the de nition of the slope of the tangent line Definition The slope of the tangent line to the graph of fat the point Px f x is given by hm fx hgt7 fx hgt0 h provided the limit exists The difference quotient gives us the average rate of change We nd the instantaneous rate of change when we take the limit of the difference quotient The derivative of f with respect to x is the function f 39 read fprime defined by f39x The domain of f39 x is the set of all x for which the limit eXists We can use the derivative of a function to solve many types of problems But first we need a method for nding the derivative The Four Step Process for Finding the Derivative Now that we know what the derivative is we need to be able to find the derivative of a function We ll use an algebraic process to do so We ll use a FourStep Process to find the derivative The steps are as follows 1 Find fxh 2 Find fxh fx 3 Form the difference quotient fx h fx h fxhfx h 4 Find the limit of the difference quotient as h gets close to 0 lim hgt0 fxh fx SOfx ll1 01 h Example 1 Find the rule for the slope of the tangent line for the function f x 3x 2 Step 1 Step 2 Step 3 Step 4 Example 2 Use the fourstep process to nd the derivative of f x x2 Step 1 Step 2 Step 3 Step 4 1 Example 3 Find the derivative of f x 7 x Example 4 Find the derivative if f x 4x2 2x 3 Example 5 Suppose f x x2 2x 5 Find the average rate of change of f over the interval 2 5 From this lesson you should be able to Explain what a derivative is State the limit de nition of the derivative Use the fourstep process to nd the derivative of a polynomial function Find the average rate of change 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 f 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 fxx2 3x2 Example 2 Suppose Hx x3 10 and Kx x3 27 If fx 3x2 show that each ofH andK 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 I f x dx F x C This indicates that the inde nite integral of f 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 jkdxkxc Example 3 Jde Rule 2 The Power Rule n 1 x Ixquotdx Cn l n1 Example 4 Ix dx Example 5 dx Example 6 dx x2 Rule 3 The Inde nite Integral of a Constant Multiple of a Function Icfxdx 0J fxdx Example 7 I4x3dx Example 8 dx x Rule 4 The Sum Difference Rule 1 fx i gxdx j fxdxi goodx Example 9 J2x2 5x 1dx Rule 5 The Indefinite Integral of the Exponential Function Ie dx 6x C Example 10 156 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 10973 76 Babc x Differential Equations A differential equation is an equation that involves the derivative or differential of some function So if we write f x 3x 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 f x 5x2 5x 3 1s a solutlon of the d1fferent1al equatlon s1nce the der1vat1ve off is 3x 5 The general solution of a differential equation is one which gives all of the solutions so the general solution for the example above will be fx gxz 5x C If we are given a point that lies on the function we can find a particular solution that is we can find C Ifwe know that f 2 1 we can substitute this information into our general solution and solve for C f 2 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 f x 2x 5 f23 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 3 rate of 100 210t2new 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 Math 13 14 Maxima and Minima of Functions of Several Variables Relative Extrema of Functions of Two Variables We learned to nd the maxima and minima of a function of a single variable earlier in the course Although we did not use it much we had a second derivative test to determine whether a critical point of a function was a maximum or a minimum or possibly that the test was not conclusive We will use a similar technique to find relative extrema of a function of several variables Since the graphs of these functions are more complicated determining relative extrema is also more complicated At a specific critical number we can have a max a min or something else That something else is called a saddle point p a t g 9 v 3 r a a wig g5 55 1 It 0 WWII awareI ll WWI The method for finding relative extrema is very similar to what you did earlier in the course First nd the first order partial derivatives and set them equal to zero You will have a system of equations in two variables which you will need to solve to find the critical points Second you will apply the second derivative test To do this you must find the second order partial derivatives Let Dx y fmfyy ny 2 You will compute Da b for each critical point a b Then you can apply the second derivative test for functions of two variables IfDa b gt 0 and f a b lt 0 thenfhas a relative maximum at a b IfDa b gt 0 and f a b gt 0 thenfhas a relative minimum at a b If Da b lt 0 then fhas neither a relative maximum nor a relative minimum at a b ie it has a saddle point which is neither a max nor a min IfDa b 0 then this test is inconclusive Example 1 Find the relative extrema of the function f x y x2 yz Example 2 Find the relative extrema ofthe function fx y x3 y2 6x2 8y 15x 7 Example 3 Find the relative extrema ofthe function fx y x3 2y2 2xy 3x 2y 5 Example 4 Find the relative extrema of the function f x y 3x3 4xy 2 y2 7 Example 5 The total daily revenue in dollars that a publishing company realizes in publishing and selling its English language dictionary is given by Rx y 005x2 003y2 002xy 20x 17y where x denotes the number of deluxe copies and y denotes the number of standard copies published and sold The total daily cost of publishing these dictionaries is given by C x y 6x 3y 200 dollars Determine the number of standard copies and the number of deluxe copies that the publishing company should publish per day to maximize its pro ts What is the maximum pro t realizable From this section you should be able to Find relative extrema of lnctions of two variables Solve word problems involving extrema of functions of two variables Math 1314 Lesson 15 Exponential Functions as Mathematical Models In this lesson we will look at a few applications involving exponential functions We ll first consider some word problems having to do with money Next we ll consider exponential growth and decay problems Interest Problems From previous course work you may have encountered the compound interest formula mt A P 1LJ m P principal amount invested A accumulated amount r interest rate m number of times interest is compounded per year t time in years Now suppose we let the number of compounding periods increase that is we ll take the limit of this function as m goes to in nity mt lim P 1 L mam m This is a fairly complicated limit to evaluate so we will omit the details r mt lim P l i Pequot mam m You may also have seen this formula before This is the interest formula to use when interest is compounded continuously We ll be interested in two kinds of problems those that ask for an accumulated amount and those that ask for present value We ll use two formulas Accumulated amount A Pequot Present value P Ae quot All values are as defined above Example 1 a Find the accumulated amount when 3000 is invested for 5 years in an account that pays 3 annual interest compounded continuously b Suppose you want to have 3000 in your savings account in 5 years The bank will pay 3 annual interest compounded continuously How much money should you deposit today so that you will have 3000 in 5 years Exponential Functions Recall the graph of an exponential function such as f x 3 l 79724757574444 1234557291 This is an exponential growth function The function increases rapidly This kind of growth will occur for any exponential function where b gt 1 including f x 8 1 If f x we ll have the re ection of this graph about the y axis y 79724757574444 1234557291 This is an exponential decay function This kind of decay will occur for any exponential function where 0 lt b lt l We ll look at a function Qt Qoek for exponential growth problems and a different function Qt Qoe k for exponential decay problems In these formulas Q0 is the original amount of the substance or population under study Qt is the amount of the substance or population at time t and k is the growth constant or 7k is the decay constant depending on whether your problem is a growth problem or a decay problem We can find the rate of growth or rate of decay by finding the derivative of the growth or decay functions Thus the growth rate can be found using Q t kQOek and the decay rate can be found using Q t kQOe k Exponential Growth Example 2 A biologist wants to study the growth of a certain strain of bacteria She starts with a culture containing 25000 bacteria After three hours the number of bacteria has grown to 63000 How many bacterial will be present in the culture 6 hours after she started her study What will be the rate of growth 6 hours after she started her study Assume the population grows exponentially Example 3 A think tank began a study of population growth in a small country 5 years ago At the beginning of the study the population was 4500000 Three years later it was 6200000 What will the population be in 2 years What will the growth rate be in 2 years Assume the population grows exponentially Exponential Decay Example 4 At the beginning of a study there are 50 grams of a substance present After 17 days there are 387 grams remaining How much of the substance will be present after 40 days What will be the rate of decay on day 40 of the study Assume the substance decays exponentially Example 5 A certain drug has a halflife of4 hours Suppose you take a dose of 1000 milligrams of the drug How much of it is left in your bloodstream 28 hours later Example 6 The halflife of Carbon 14 is 5770 years Bones found from an archeological dig were found to have 22 of the amount of Carbon 14 that living bones have Find the approximate age of the bones From this lesson you should be able to Solve problems involving continuously compounded interest including problems that ask for accumulated amount and present value Solve problems involving exponential growth Solve problems involving exponential decay Find a rate of growth or decay Math 1314 The Fundamental Theorem of Calculus In the last lesson we approximated the area under a curve by drawing rectangles computing the area of each rectangle and then adding up their areas We saw that the actual area was found as we let the number of rectangles get arbitrarily large Computing area in this manner is very tedious so we need another way to nd the area The fundamental theorem of calculus allows us to do just this It establishes a relationship between the antiderivative of a function and its definite integral The Fundamental Theorem 0f Calculus Letfbe a continuous function on a b Then F fxdx Fb Fa where Fx is any antiderivative off If you are interested in seeing why this works see the link for the proof of the fundamental theorem of calculus on the class notes page Example 1 Suppose f x 3x 2 Find the area under the graph of f from x 0 to x 4 2 Example 2 Evaluate 023x 56 dx Example 3 Evaluate 1123x2 8x ldx Example 4 Evaluate Ii Ljabc x J3 7x24x 3dx x Example 5 Evaluate 64 Example 6 Evaluate II J 3 dx V Example 7 Evaluate 0306 4x ldx Example 8 Evaluate 102x2 4Xx 3dx Example 9 Find the area of the region under the graph of f x 7x x2 over the interval 0 5 Example 10 Find the area of the region under the graph of f x e 2x over the interval 1 2 From this lesson you should be able to State the fundamental theorem of calculus Use the FTOC to compute de nite integrals Use the FTOC to nd area under a curve Math 1314 Lesson 18 Area and the De nite Integral We are now ready to tackle the second basic question of calculus 7 the area question We can easily compute the area under the graph of a function so long as the shape of the region conforms to something for which we have a formula for geometry Example 1 Suppose f x 5 Find the area under the graph of f x from x 0 to x 4 Approximating Area Under a Curve Now suppose the area under the curve is not something whose area can be easily computed We ll need to develop a method for nding such an area Example 2 Here we ll draw some rectangles to approximate the area under the curve We can nd the area of each rectangle then add up the areas to approximate the area under the curve by l Example 3 Next We ll increase the number of rectangles Example 4 And We ll increase th number of rectangles again What you should see is that as the number of rectangles increases the area we compute using this method becomes more accurate The Area Under the Graph of a Function Let f be a nonnegative continuous function on a b Then the area of the region under the graph of f is given by A giggfx1 fx2 foe 1Ax where x1 x2 xquot are arbitrary points in the interval a b of equal width Ax b1 n The sums of areas of rectangles are called Riemann sums and are named after a German mathematician Example 4 Use left endpoints and 4 subdivisions of the interval to approximate the area under fx 2x2 l on the interval 0 2 Example 5 Use right endpoints and 4 subdivisions of the interval to approximate the area under f x 2x2 l on the interval 0 2 Example 6 Use midpoints and 4 subdivisions of the interval to approximate the area under fx 2x2 l on the interval 0 2 Example 7 Suppose f x 1 3x Approximate the area under the graph of f on the interval 0 12 using 6 subdivisions and left endpoints The De nite Integral Letfbe de ned on a b If limfx1 fx2 fxn Axexists for all choices of b a representative points in the n subintervals of a b of equal width Ax then this limit is called the de nite integral of f from a to b The de nite integral is noted by 17 fxdxlimfx1 fx2 fxn Ax The number a is called the lower limit of integration and the number b is called the upper limit of integration A function is said to be integrable on a b if it is continuous on the interval a b The de nite integral of a nonnegative function The de nite integral of a general function From this section you should be able to Explain the procedure us ed to approximate area under a curve Use Riemann sums to approximate the area under a curve using right endpoints left endpoints or midpoints Explain what we mean by de nite integral of a nonnegative function or a general function 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 04x2 Find the actual cost of producing the 101st item The cost of producing the 101St item can be found by computing the average rate of C101 C100 chan e that is b com utin g y p g 101 100 Note that wwwherex100andh1 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 However it is often inconvenient 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 find 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 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 ll define the marginal cost function as the derivative of the total cost function You will find 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 003x2 l35x 15000 Find the marginal cost function Use the marginal cost function to approximate the actual cost of producing the 2151 and 181st headsets Average Cost and Marginal Average Cost Suppose C x 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 this function by C x Then we can express the average cost function as C x Q The der1vat1ve of the average cost functlon 1s called the marginal average x 5x 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 C x 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 04x 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 final function of interest is the profit function This function can be expressed as Px Rx C x where Rxis the revenue function and C x is the cost function As before we will find 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 profit 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 008x2 500x 250000 and the demand function is given by p 600 08x Find the profit function Then find the marginal profit function Use the marginal profit function to compute the actual profit 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 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 what it is used for Find a marginal cost function and use it to approximate the cost of producing the x lst item Find an average cost function Find Min C x and explain what it means Find a revenue function Find a marginal revenue function and use it to approximate the revenue realized on the sale ofthe x 1 item Find a profit function Find a marginal profit function and use it to approximate the profit realized on the sale ofthe x lst item Math 13 14 Abs olute Extrema In earlier lessons you learned how to find relative local extrema These points were the high points and low points relative to the other points around them In this lesson 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 specific closed interval Absolute Extrema on the Domain off Definition If f x g f c for all x in the domain off the fc is called the absolute maximum value off If fx 2 fc for all x in the domain off the fc is called the absolute minimum value of Sometimes you will be asked to find the absolute extrema over the interval 00 00 Example 1 State the absolute maximum andor absolute minimum values Example 2 State the absolute maximum andor absolute minimum values 1 mwAmm7loa a 1234567891
Are you sure you want to buy this material for
You're already Subscribed!
Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'