homework solutions calculus

homework solutions calculus

Homework Solutions in Calculus

1. Introduction to Calculus

Calculus is a branch of mathematics that attempts to understand the behavior of functions. It deals with the mathematics of change and is used to solve a wide variety of problems in science and engineering. There are essentially two main topics that form the foundation of calculus: differential calculus and integral calculus. The first half of the course deals with differential calculus, the study of how things change. The second half of the course focuses on integral calculus, the study of the accumulation of stuff. What both differential and integral calculus have in common is that they are both subfields of the same subject. That is, we break up the study of calculus into different topics based on the type of function we are dealing with. For example, if we have a function that changes direction, we might use differential calculus to find at what x values that change occurs. On the other hand, if given a rate of change of a function, we might use integral calculus to find the actual function. Overall, calculus is the hallmark of modern mathematics and has been pivotal in advancing human knowledge throughout the ages. From Newton’s analysis of planetary motion to the elucidation of the Higgs particle, calculus has left a major footprint on human understanding of the natural world. What all students find within their first few days of meeting calculus is that it is not like the calculus that they were taught in high school. In fact, the main thing that most people remember from this encounter is the debacle of limits, which is a process that allows us to begin to understand what derivatives really are. Yet, after analyzing limits and learning how derivatives work, many students find that beginning calculus students perform a very important skill that will be needed in further math and science studies. This important skill is following directions carefully and taking one step at a time. By practicing this skill as they master calculus, students will find that calculus is not impossible. Instead, it is an interesting and challenging study that has intriguing applications in many facets of modern life.

2. Basic Differentiation

When we differentiate a function, we are really investigating the rate of change of the function at any point. From examples at GCSE, you may have been given a function such as \(y = x^2 + 3x + 2\) and had to find the value of \(\frac{dy}{dx}\). We’re going to follow a process to calculate the differentiation of any function, even if we don’t know its formula. So, here’s the process: 1. Choose a function to differentiate, such as \(y = x^2 + 3x + 2\). 2. Introduce a very small change in x and find out the corresponding change in y. The change in y over the change in x is known as the rate of change of y with respect to x and is denoted \(\frac{dy}{dx}\). 3. So, \(\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\Delta y}{\Delta x}\). This means find out what happens to the rate of change of y with respect to x as \(\delta x\) tends to zero. By the way, \(\delta x\) means a very small change in x. 4. For our function, \(y = x^2 + 3x + 2\), find \(\frac{\Delta y}{\Delta x}\) and then make \(\delta x\) very small (try 0.1 and 0.01) and keep finding \(\frac{\Delta y}{\Delta x}\) until you can deduce the rest of the function. Well, we’ll investigate a few more examples later but you are now able to differentiate any function using this process. Try to differentiate \(y = 5x^2 + 2x + 1\) using the process and then check your answer by using the rules. Also, if you have a negative \(x^n\) term, the process is the same but bring the power of \(x\) in front of the \(x\) and then reduce the power by 1. For the number term, the rate of change is zero, because there’s no \(x\) to change. So, the rule is: \(y = x^n\) then \(\frac{dy}{dx} = nx^{(n-1)}\) and \(y = v\) where v is a constant, then \(\frac{dy}{dx} = 0\). Do check you use the rules, because you’ll often find the process takes a long time and you’ll probably make a mistake, so the rules can be used as a check!

3. Integration Techniques

The next section focuses on integration techniques, including methods such as substitution and integration by parts. For example, when solving the integral of the product of a trigonometric function and a polynomial, a useful method is to substitute a variable for the inner function. This is known as the substitution method. A substitution is made by expressing some part of the integrand in terms of a new variable. This is the most widely applicable of all methods. On the other hand, when attempting to compute the integral of a product involving algebraic or logarithmic functions, integration by parts is the method to use. Integration by parts is a “fancy” way of doing the product rule for differentiation. The formula comes from the product rule, and when one side of the formula is used, integration by parts is the result – basically, it helps to solve problems that have one function of a function and a derivative beside it. This method is generally useful when the functions involved have terms that get simpler (either become a constant or vanish completely) as the process is repeated. The aim here is to learn to recognize the form of the integral you are faced with and choose an appropriate strategy for the particular functions present. These sections clearly explain the methodologies in a detailed yet understandable manner, making the material accessible to new learners. One of the most difficult areas in learning calculus is the need to understand the weaker functions that define an integral, not just the ‘answer’ when an integral is computed. The book always emphasizes understanding what is meant by an integral as a process that progressively accumulates an overall effect. This helps to give a deeper understanding both of how the material in a course on calculus holds together and also of the student’s own progress. As an integral represents the accumulation of an infinite number of infinitesimal quantities, it requires the use of some form of approximation. The most common methods of approximation are to use numerical methods or calculus. The trapezium rule and the method of determining are given context in ‘real life’ applications of calculating the volumes of irregular 3D shapes. In order to do this, the basic ideas of how the computer performs such calculations are outlined and provided in a way that students can practically apply the knowledge they have learned. Since this is especially critical in the development of the education system, with less and less time to detail separate sections for learning in a hypothetical sense and for developing skills that can be used practically in the world. The capacity for calculation with volume works in the examples, but also the ability to write a program that allows the computer to calculate volumes of objects given their equations is huge. At a more basic level, technology in the form of graphical calculators affords the use of simplified software to demonstrate the meaning of integrals when used to calculate areas underneath functions. Modernizing the learning process in this fashion is not only enthusiastic and interesting to students, but also reflects that teaching methods have changed and improved. By embracing new technology to demonstrate more advanced principles, we encourage students to see the value of programming and even computational mathematics.

4. Applications of Calculus

Real-world problems are not limited to average rate of change and area calculation. Hence, the knowledge of rate of change that we learned in differential calculus can be greatly extended to solve various problems in different applications. As we have discussed in the introduction section, one of the amazing applications of calculus is to optimize a given function. We can even apply the concept of rate of change to find the solution for many real-life problems, such as those in economics. For example, calculus is referred to as a “big gun” in solving economics problems. Anyone who has ever been frustrated by the high price of gas or by the complex rules of our economy will find that the study of economics is highly rewarding. Calculus is especially helpful in cases of growth and decay. If we have a concrete object, its mass or how much of a material is disintegrated extends over time, we can apply the concept of calculus to find the rate of decay of the object. Take a look at these two rates: for any time of “t”, we call it the “instantaneous rate of change” or simply the “rate of change”. On the other hand, if the rate of change of “q” with respect to “t” is given, we call it the “derivative of q with respect to t” and we will use the notation “dq/dt”. With “dq/dt”, now we have a new function to represent the rate of change of “q”. In this course, we are mainly focusing on how to apply the “rate of change” in solving various real-life problems. Thanks to the great invention of Newton and Leibniz, calculus has found its way to solve many practical problems. For example, it can be used as a powerful tool to find the air pressure rushing past an aircraft wing so the aircraft can stay level. It can also be utilized to discover how the profit of a company can be maximized. As for material science, it can provide a way to compute the total force that a spring applies. These are just a few examples to show the great achievements or solutions that have been discovered by this wonderful subject. Hence, as a learner, I hope you can appreciate the beauty of calculus and be able to apply the knowledge in solving problems. Last but not least, nowadays calculus is widely used not only in solving problems in science and engineering but also in medical, economics, the stock market, and many other areas. We can see how powerful a subdiscipline can grow and how much impact it has on different domains as a whole. I hope you enjoy the course and can discover the miracle of the “calculus” world.

5. Advanced Topics in Calculus

The purpose of this chapter is to introduce the student to more complex and interdisciplinary methods of mathematical analysis. Specifically, this chapter will cover techniques of mathematical analysis which rely on concepts from both calculus and linear algebra. Alongside the standard “process” of doing calculus (deriving and integrating functions), the student will also study “vectors”, mathematical objects which generalize the concepts of “distance”, “direction” and “linearity” in various dimensions. Vectors are central to the study of “multivariable calculus”, which is the focus of this chapter and the following. We will introduce vectors and the arithmetic, algebra and calculus of functions of several variables. Such “n-dimensional” analysis is the base for many engineers and indeed for those who specialize in calculus itself: the “real number line” (which uses one variable) is a single dimension example and this new, extended study generalizes much of the traditional concepts used in the subject. Secondly, by studying calculus in three dimensions and beyond, a new range of geometric, quadratic and even trigonometric applications become possible. Some theorists, particularly in the modern fields of “mathematical biology” and “complex system modeling”, make use of vectors and other “higher order” techniques to understand systems that have a lot of variables and are often dynamic. These can include models which describe the interactions of many different agents (e.g. the cells in a human body) and general phenomena such as the changing spread of diseases or the response of physical matter to differing electromagnetic fields. So, by studying the modules in this book, a student is reaching for a new level of understanding and indeed the next steps in modern calculus. We will start by discussing vectors and describing the world of three dimensions but even further worlds may become clearer as one delves deeper into the study of calculus and its applications.