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# Calc for BusLife Sciences MATH 1314

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This 34 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 Mary Flagg in Fall. Since its upload, it has received 24 views. For similar materials see /class/208403/math-1314-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

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 y 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 T 1000 2000 30 Math 1314 Lesson 21 Page 1 of6 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 x2 2 and gx x lfromx0tox2 v uswza v v 7x71 Example 2 Find the area between the function fx x3 x and the x aXis from x l to x 1 Math 1314 Lesson 21 Page 2 of6 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 Math 1314 Lesson 21 Page 3 of6 Example 5 Without any effort to curb population growth a government estimates that its population will grow at the rate of 60a 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 Math 1314 Lesson 21 Page 4 of6 Example 6 The management of a hotel chain expects its profits to grow at the rate of 11E million dollars per year lyears from now If the renovate some of their existing hotels and acquire some new ones their profits would grow at the rate of If 21 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 Math 1314 Lesson 21 Page 5 of6 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 21 Page 6 of6 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 39x f x for all x in I The process of nding an antiderivative is called antidifferentiation or nding an indefinite integral l 3 3 2 Example 1 Determ1ne if F 1s an antiderivatlve of f if F x 3x Ex 2x 5 and fxx2 3x2 Example 2 Suppose Hx x3 10 and Kx x3 27 If fx 3x2 show that each ofH and K is an antiderivative off and draw a conclusion Math 1314 Lesson 16 Page 1 of9 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 jkdx kxC Example 3 Jde Rule 2 The Power Rule n1 x Jxndxznl Cn l Example 4 1x4 dx Example 5 J wde Math 1314 Lesson 16 Page 2 of9 Example 6 lisdx x2 Rule 3 The Inde nite Integral of a Constant Multiple of a Function Icfxdx 0J fxdx Example 7 I4dex Example 8 I dx x Math 1314 Lesson 16 Page 3 of9 Rule 4 The Sum Difference Rule H x i gxdx j fxdx i jgmdx Example 9 sz 5x 1dx Rule 5 The Indefinite Integral of the Exponential Function Ie dx ex C Example 10 156 4x3 dx Math 1314 Lesson 16 Page 4 of9 Rule 6 The Inde nite Integral 0f the Function f x l x 1 j dx1nxc x 0 x 5 2 Example 11 I 3x Tjdx x x Applying the Rules 3x 4x2 5x3 dx Example 12 J x Math 1314 Lesson 16 Page 5 of9 Example 13 1x2 dx x x 12 Example 14 IN x3 76 dx x Math 1314 Lesson 16 Page 6 of9 Differential Equations A differential equation is an equation that involves the derivative or differential of some function So if we write f 39x 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 3x2 5x 3 1s a solutlon of the d1fferent1a1 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 nd a particular solution that is we can nd C If we know that f 2 1 we can substitute this information into our general solution and solve for C f 2 1 is called an initial condition Math 1314 Lesson 16 Page 7 of9 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 nd the function f by rst nding the general solution and then nding the value of C that satis es the initial condition Example 15 Solve the initial value problem f x2x 5 f23 Example 16 Solve the initial value problem f x 3e 2x f 0 7 Math 1314 Lesson 16 Page 8 of9 Example 17 A cable television provider estimates that the number of its subscribers will grow at the 3 rate of 100 210thew 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 inde nite 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 1314 Lesson 16 Page 9 of9 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 rst 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 nd the ratlonal zeros of a functlon by factor1ng Example 1 Find the rational zeros f x x3 16x Example 2 Find the rational zeros fx x4 1 1x2 18 Math 1314 Lesson 12 Page 1 of 13 Note that zeros that are square roots are NOT rational roots Imaginary solutions to the equation f x 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 anilxx391 a0 where an at 0 and do at 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 Math 1314 Lesson 12 Page 2 of 13 Example 5 Find all rational zeros of f x x3 7x2 16x 12 or state that there are none Example 6 Find all rational zeros of fx x4 4x3 3x2 4x 4or state that there are none Math 1314 Lesson 12 Page 3 of 13 Example 7 Find all rational zeros of f x x3 3x2 1 or state that there are none Example 8 Find all rational zeros of f x x4 5x3 4x2 or state that there are none Math 1314 Lesson 12 Page 4 of 13 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 Math 1314 Lesson 12 Page 5 of 13 Recall the generalizations about end behavior of a polynomial function from College Algebra Example 10 Use the guide to curve sketching to sketch f x x4 4x3 Math 1314 Lesson 12 Page 6 of 13 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 15x3 Note the approximate zeros of the function are 022 172 and 7 94 Math 1314 Lesson 12 Page 7 of 13 Example 12 Use the guide to curve sketching to sketch fx x3 7x2 16x 12 Note we found the rational zeros in example 5 Math 1314 Lesson 12 Page 8 of 13 Example 13 Use the guide to curve sketching to sketch fx x3 8x2 19x 12 Math 1314 Lesson 12 Page 9 of 13 Example 14 Use the guide to curve sketching to sketch f x xe Math 1314 Lesson 12 Page 10 of 13 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 of in ections concave concave Math 1314 Lesson 12 Page 11 of 13 Example 16 Sketch the function if you are given the following information end behavior 164061 and 00 00 164 and 061 concave oo 027 00 concave 123 027 Math 1314 Lesson 12 Page 12 of 13 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 Math 1314 Lesson 12 Page 13 of 13 Page 1 of6 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 finding such an area Example 2 Here we ll draw some rectangles to approximate the area under the curve We can find the area of each rectangle then add up the areas to approximate the area under the curve Math 1314 Lesson 18 Page 1 of6 Page 2 of 6 Example 3 Next We ll increase the number of rectangles Example 4 And We ll increase the number of rectangles again Math 1314 Lesson 18 Page 2 of6 Page 3 of6 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 61 b Then the area of the region under the graph of f is given by A1iIgfxlfxZfxnAx b where x1 x2 x are arbitrary pomts 1n the interval a b of equal Width Ax 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 f x 2x2 l on the interval 0 2 Math 1314 Lesson 18 Page 3 of6 Page 4 of6 Example 5 Use right endpoints and 4 subdivisions of the interval to approximate the area under f x 2x2 1 on the interval0 2 Example 6 Use midpoints and 4 subdivisions of the interval to approximate the area under f x 2x2 1 on the interval 0 2 Math 1314 Lesson 18 Page 4 of6 Page 5 of6 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 r7 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 Math 1314 Lesson 18 Page 5 of6 Page 6 of6 The de nite integral ofa nonnegative function The de nite integral of a general function From this section you should be able to Explain the procedure used 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 18 Page 6 of6

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