AP Calculus Review Average Value of Functions

The average of a set of data is typically defined as the sum of the values divided by the number of data points. But what if you have infinitely many data points? What is the average value of a function? Read on to find out! Defining the average value of a continuous function is not as easy as finding the average of a finite set of data. Average Value of Functions Suppose f is a continuous function defined over an interval [a, b]. In particular f(x) exists at every one of the infinitely-many points x between and including a and b. So, if youre looking for the average value of f on that interval, it wont do any good to try adding up those infinitely-many data points. Instead, the way to tame the infinity is to use calculus. Specifically, we define the average value of a function f as the following definite integral. The Theory Behind the Formula But where does the integral formula for average come from? The key is sampling. If you included enough of the function values, say a thousand, a million, or even more, then that should approximate the average of all infinitely-many points! Lets illustrate with the following example. Estimate the average value of the function f(x) = √(x) + 1 over the interval [1, 3]. Since were just estimating, lets pick four sample points. (The more sample points you pick, the better your estimated average will be.) Divide the interval [1, 3] into four equal subintervals, and lets agree to choose the midpoint of each subinterval. Then plug those midpoints into f to find the sample values. MidpointsHeight: f(x) = (x) + 1 1.25(1.25) + 1 = 2.12 1.75(1.75) + 1 = 2.32 2.25(2.25) + 1 = 2.50 2.75(2.75) + 1 = 2.66 Finally, use the familiar old averaging formula. Add up the data and divide by the number of data points: (2.12 + 2.32 + 2.50 + 2.66)/4 = 2.4 So the (approximate) average of the function is 2.4. Taking it to the Limit However, what weve just done will not give us an exact answer because weve essentially ignored most of the function! What about f(1.3) or f(2.95234)? No matter how many sample points we include, there will always be some missing Unless we can use the magic of calculus to catch them all. The sampling process should remind you of a Riemann Sum. For a quick reminder, feel free to check out AP Calculus Review: Riemann Sums. Midpoint Riemann sum. In the limit as the number of sample points goes to ∞, the Riemann sum becomes a definite integral. Let n be any whole number and xk* stand for the various sample x-values. Then the estimated average is the sum: Next, allow n → ∞ using a limit. We also need to get Δx into the act somehow. The trick is to multiply and divide by (b a). Remember, Δx = (b a)/n. Examples Ok, now that youve seen the theory, lets use the formula in practice! Problem 1 Find the exact average value of f(x) = √(x) + 1 over the interval [1, 3]. Solution Above, we only estimated the average to be 2.4. Now well use the integral formula to determine the average value precisely. (Its interesting to compare our estimate with the exact value above. √(3) + 2/3 ≈ 2.39871747, which means that our estimate of 2.4 was actually pretty good!) Problem 2 The amount of energy associated with a certain chemical reaction is given by E = x ln x, where 1 ≠¤ x ≠¤ e, and x represents the amount of one of the reactants. Find the average energy of the reaction over the range of possible levels of reactant. Solution This problem seems more like chemistry than math! Chemistry can be fun too! But what does this have to do with calculus However, the keyword average tells us that mathematics plays a major role in this problem. In fact, they are simply asking for the average value of f(x) = x ln x, over the interval [1, e]. First set up the integral formula with a = 1 and b = e. Then work out the integration, which involves Integration By Parts in this case. Thus the average energy of the reaction is (e2 + 1)/[ 4(e 1) ], or roughly 1.22. Mean Value Theorem for Integrals Averages are also called means. So you may use the same formula to find the mean value of a function. There is also an important result in calculus that relates the mean value to a particular function value on the given interval. The Mean Value Theorem for Integrals (MVTI). If f is continuous on a closed interval [a, b], then there is at least one point x = c in that interval such that the mean value of the function is equal to f(c). That is, (Caution: There is also a Mean Value Theorem for Derivatives. Its important not to confuse the two.) Problem 3 Let f(x) = 6x2 8x + 1. Determine the value of c at which the mean value of f on [-1, 1] is the same as f(c). Solution According to the Mean Value Theorem for Integrals, there must be at least one such value c. Lets set up the formula and find it! At this point, we will need to solve a quadratic equation. Dont forget your Quadratic Formula! 6c2 8c 2 = 0 c = (2  ± √(7))/3 It seems as though there may be two answers. However, only one lies within the given interval [-1, 1]. c = (2 + √(7))/3 ≈ 1.549, not in the interval. c = (2 √(7))/3 ≈ -0.215, in the interval. Therefore, the only value that satisfies the MVTI is c = (2 √(7))/3. Summary Although average value and the Mean Value Theorem for Integrals are specialized topics and only show up in a few problems on any given AP Calculus test, they are important concepts to master. For one thing, they illustrate how integral calculus can be used in applications. Moreover, working out the average value of a function is no more difficult than computing a definite integral. So now when you see these kinds of problems on the AP Calculus Exam, you can rise to the challenge!

Describe The Aspects Of Nursing Education And Practice - 3575 Words

Outline The Foundations Of Nursing Within The Aspects Of Nursing Education And Practice (Research Paper Sample) Content: Foundations of Nursing NameInstitutionDate The relevance and existence of associate degree programs in nursing education has attracted intensive debates in the nursing professional. The course was introduced over five decades ago due address the losses and issues associated with shortage of nurses around the globe. Shortage of nurses in the global healthcare systems was significantly influenced by government and population demand for nurses and medical services, as well as increase in global populations. Nurse educators and curriculum developers argue that initiating a bachelors of science in nursing would create sufficiency and efficiency in healthcare systems, as well as increase and engage professionalism in nursing practice (Keating, 2014). BSN program is associated with specialization, effectiveness and professionalism. However, associate nursing program are highly preferred in clinical practice since they are associated with immense practical and technical know ledge in clinical and ethical practice. In the American healthcare system, associate nurses continue to express and present contemporary efficacy and rich clinical practice heritage. Despite the technical efficiency associated with the nursing program, the modern healthcare system requires that students should begin the nursing career from a BSN graduate program. This would provide technical and professional knowledge to students hence makes them more dynamic and adaptable to changes in nursing practice (Mashhad BehNashr, 2011). They are adequately trained and prepared to face the modern clinical complications. The urgency and need for BSN nurses is associated with diverse internal and external factors. This study will focus on revealing the need for a BSN program in Hartford community college in relation to social and institutional institution factors. Social factorsSocial factors in healthcare describe the relationship and well as the association between a nursing education insti tution and the global society. The need for a BSN program is critically related to the global societys need for quality healthcare. The modern social societies are facing complex health complications that require intensive, advanced, and informed decision making capabilities. Also, increase in global populations prompt the need for advanced nurses in clinical practice. Graduate BSNs are able to work independently as both nurses and physicians due to their immense knowledge in disease diagnosis, treatment, management and prevention. The Howard community college should adapt a BSN program; produce competent and skilled nurses who are well equipped and talented in patient care and disease management. Advancement in disease diagnosis, treatment, and management of disease condition calls for advancement in knowledge, skills and treatment approach (Keating, 2014). Institutional factorsInstitutions operated and managed by BSN nurses is associated with proficiency and efficacy in patient ca re. This is reflected in the low number of patient deaths in the institutions. Nurse education should be well equipped and facilitated to ensure that it offers quality education that meets the needs and requirements of the global nursing job market. The diversification and evolutional practices experienced in the nursing clinical setting should be addressed in nursing education institutions hence ensure production of prepared professional. For instance, Elements of Proliferation and advancement of technology in clinical and healthcare systems are being experience on a daily basis. Incorporation of technology in nursing practice and research has facilitated notable advancements, proficiency, efficiency, and effectiveness. These advancements and implementation of technology in nursing curriculum has prompted and necessitated diversity in nursing curriculum. This includes the need to incorporate nursing technology education in the nursing educat5ion curriculum hence ensure that stude nts and novice staff joining the nursing profession are able and prepared to implement these technologies in daily practice routines. Quality and safety decision making on regulations and accreditationThe element of substituting associate degree nursing programs with BSN programs has generated wide and heated debates. These debates are based on the comparison of competency outcomes between the two programs. However, Nursing education institutions and environments must address the aspect of concept understanding in both practical and theoretical concepts. Diverse institutions and organizational reports have proposed that all associate degree graduates enroll for further studies hence advance to BSN so as to remain valid and updated on clinical practices. The institute of medicine play critical role in designing and developing the BSN program curriculum. The institute has drafted vast reports on the effectiveness of BSN program in the corporate world in comparison to associate degree program. The institute of medicine calls for diverse reforms in nursing education that facilitate interdisciplinary and inter-professional education and competency. In this regard, the IOM states that inter-professional education of nurses provides dynamic skills hence nurses can suit and work in diverse professional fields. BSN program provides all rounded skills and competencies hence graduates can work in diversified career platforms. BSN program provides a systemic arena where nurses can evaluate and improve their competencies over time (Mashhad BehNashr, 2011). IOM plays a key role in influencing the development and delivery of BSN program at Hartford community college. Quality and safety decision making on regulations and accreditationAmerican Association of Colleges of Nursing (AACN) has diverse policies governing safety operation of a given entity. This institution plays a vital role in the issuance and assurance of quality and safety healthcare protocols. Concepts of regu lations are diverse and differ from aspects of accreditation. Regulation of healthcare facilities entails governing and monitoring healthcare facilities and ensuring that regional rules and policies on safety and quality services are followed and implemented. The practice of accreditation revolves around the issuance of certificates of compliance and acknowledgement that the facility meets the regulatory standards and requirements of healthcare practicing. Healthcare regulatory and accreditation ensures that healthcare facilities operate within their means as well as within regional healthcare laws and policies. In this case, The Hartford Community College facility adequately met the required standards of mission, organization, administration, safety and quality. American Association of Colleges of Nursing (AACN) assessed and evaluated the operations and procedures at Hartford Community College and medical facility and issued a baccalaureate nurse residency accreditation. This meant that the regulatory agency approved that the facility met all the obligations and requirements hence accreditation (Douglas College, 2016). However, Hartford Community College may experience financial limitations and difficulties when implementing this agenda. This is because BSN program require massive funding to facilitate students clinical and in class learning facilities. Application of learning theories in nursing educationLearning theories provide critical guidelines and policies to be implemented in education setups to enhance teaching, learning and understanding. Learning theories play a critical role in nursing education since they facilitate conceptual understanding among students and teaching among educators (Sobhaninejad, 2015). Educators implement aspects of these theories in nursing education to facilitate practical and theoretical learning that suit diverse learning situations. The Hartford Community College seeks to implement the humanist learning theory in providin g BSN education. The humanist learning theory and concepts states that in order for student to achieve optimum self actualization, they must be satisfied and in congruence with the diverse levels of BSN entry requirements. Sobhaninejad (2015) states that these concepts of learning focus in developing a student centered perspective in knowledge delivery. The humanist theory calls for theoretical and experiential learning protocols. The perspective concepts enabled students to develop personal decision making capabilities. This is done and facilitated through where the nursing educators give students a chance to learn through curiosity. However, learning institutions provide clear guidelines in the implementation of this learning theory in relation to the BSN curriculum. Advantages of implementing humanist theory inn teaching BSN programHumanist learning perspective in nursing education is the fact that it facilitates fast and sound decision making.Independent learning facilitates un derstanding of nursing concepts. this is achieved through a mutual educator- student understanding relationship.Humanist perspective plays a critical role in experiential learning since learning is based on students personal needs, interest and commitment (Sobhaninejad, 2015). However,Disadvantages of implementing humanist theory inn teaching BSN programUtilization of humanist theory and perspectives in teaching and learning BSN program causes compatibility and adaptability limitations to students and nursing educators who are used to traditional learning and teaching methods. Inability to adapt may be reflected in poor clinical and in class performance. Humanist model may bring about laziness, and dissatisfaction among unfocused students (Sobhaninejad, 2015). The model works exemplary well on self-motivated students and nursing educators. Adult education models in developing BSN programThe tradit...