the homework solutions calculus

the homework solutions calculus

Homework Solutions for Calculus

1. Introduction to Calculus

Okay, so in this section we are going to take a look at the basic idea behind calculus and we are going to start to learn about the two main components of calculus. The first main idea is something called a derivative, which is a way to measure how a quantity is changing. The second main idea of calculus is something called an integral. Maybe you have heard a little bit about calculus before and maybe you have an idea in your head about what calculus is. But for the moment, I would like to start off assuming that you do not know anything about calculus. So what is calculus? Well, calculus is a big field of mathematics that is used to describe, model and analyze things that change. So for example, if you throw a ball up into the air, every second the height of the ball changes and you could use calculus to model and analyze that motion. Or maybe you have seen graphs of functions like y equals x squared or y equals sine of x. Well, calculus is the mathematics that helps us understand where these graphs come from, what they mean and what kind of things we can do with them. Calculus can be used to calculate the properties of curves and it can be used to determine the way in which different quantities can change with respect to each other. And it’s also used in lots of other areas too; physics, engineering, computer science and lots of other fields all make use of calculus. So hopefully by the end of this section you’ll start to see how calculus can be useful and you will have a better understanding of what calculus is all about.

2. Differentiation Techniques

Chain Rule: The chain rule allows differentiation of composite functions. Given function y = f(g(x)), where u = g(x) and y = f(u), the derivative is given by dy/dx = dy/du * du/dx.

Quotient Rule: If y = u / v, the derivative is given by (v * du/dx – u * dv/dx) / v^2.

Product Rule: In the event that we need to find the derivative of a function in the form of f(x) = u(x)v(x), where u(x) and v(x) are both functions of x, then the derivative is u'(x)v(x) + v'(x)u(x).

Difference Rule: It is similar to the sum rule. If f(x) can be expressed as the difference between two other functions u(x) and v(x), then the derivative of f(x) is the difference between their derivatives. Therefore, if f(x) = u(x) – v(x), then f'(x) = u'(x) – v'(x). For instance, if f(x) = 3x^2 – x, then f'(x) = 6x – 1.

Sum Rule: When a function f(x) can be expressed as the sum of two other functions u(x) and v(x), the derivative of f(x) is the sum of their derivatives. Therefore, if f(x) = u(x) + v(x), then f'(x) = u'(x) + v'(x). For example, if f(x) = 3x^2 + 2x + 7, the derivative f'(x) = 6x + 2.

Constant Rule: For any constant, c, the derivative of c with respect to x is always equal to 0. It doesn’t matter what the value of the constant is. For instance, the derivative of 10 with respect to x is equal to 0. However, this is only true for constants in differentiation.

Power Rule: Differentiate a function in the form of f(x) = x^n with respect to x, the derivative is given by f'(x) = nx^(n-1). For example, if f(x) = 3x^2, the derivative f'(x) = 6x.

Differentiation is a process of finding the derivative of a function. The derivative of a function y = f(x) with respect to x is written as f'(x). Given f(x), we can find f'(x) using a set of rules and formulas. Differentiation techniques are a core component of calculus and are useful no matter what is being considered in the area of mathematics, whether it is related to mathematics or its applications to physics. Here are the various rules and formulas you need to use in order to apply differentiation successfully.

3. Integration Methods

Remember in calculus, the antiderivative, or indefinite integral, of a function f is a function whose derivative is f. The indefinite integral of f(x) with respect to x is written as ∫f(x)dx. This is only our first lesson in integration methods. We have several methods to study and it requires lots of practice to familiarize ourselves with the methods and uses. I encourage all of my class members to complete the homework with the assistance from this book and discuss any problems with me if necessary. Also, I pointed out that do not simply let the computer do all the job for us. One of the purposes to introduce the computer algebra program is to compare the results with those obtained from using the methods. Always make sure you get a sensible answer by yourself! Also, remind the class to do the problems step by step and show all necessary workings. This is to make sure the reduction or evaluation of some ∫ does not lose marks when we put down the final answer. Last but not least, I like to use the example to demonstrate how the knowledge learned from calculus can be applied to solve a real-life problem. This not only provides interests to the students but also reinforces their understandings. Work and explanations are given. We just need to follow the steps to write down U and dU and then finish the rest. Completion of square, we can say we complete the square of x^2+x+1. Because the part x^2+x can be written as (x+1/2)^2-1/4. This step was usually used to help us evaluate the integrals when the integrals cannot be directly integrated. Generally, a rule in place of using ∫x^ndx=n/(n+1)x^(n+1)+c, where c is a constant. This would be useful if the integrand contains x^n when n is a positive integer. Some examples are done. Students are required to learn the formula and then give the answer. It is a good practice for students when integrating questions. Later, do some advanced examples that method shown in point 8 reduction formula is used. This is to give the students a feel and get used to the methods of solving it. Students are told that using the reduction formula which is not shown in the textbook. This is to expose the students to varieties of methods and that means different methods can be used to solve the integrals.

– Substitution – Integration by parts – Trigonometric substitution – Partial fractions – Use of computer algebra – Using difference of squares – Using completing the square – Using the variation of parameter – Using reduction formula

The book then delves into differentiation techniques, exploring various methods for finding derivatives. It also covers integration methods, discussing different approaches for finding antiderivatives and solving definite integrals. The book then moves on to applications of calculus, demonstrating how calculus can be used to solve real-world problems. Great! We know now the first two chapters of this book deal with basic concepts of calculus. The third chapter covers integration methods. This chapter is important to us because the new thing in this book is to introduce how integration methods can be used in real life. Other two chapters deal with applications of calculus and advanced topics in calculus. This is also our first lesson in integration. Before studying the details, we summarize the methods to be introduced. There are nine methods we will learn to solve problems. They are:

4. Applications of Calculus

Through applications, it is possible to understand the complete and partial behaviors of certain functions and, in the processes, to solve a variety of real-world problems. This is the case not only in the applications of the derivative and the definite integral, but also when we are dealing with a function in general. In this section, we will study three important applications of calculus: the solution of optimization problems, the determination of lengths of plane curves and the determination of areas of plane regions, all of them studied in great detail. These applications have numerous practical advantages. For example, the solution of optimization problems, a process used many times in economics and business to find maximum profit or minimum cost, can be applied in agriculture in order to maximize the revenue from a cultivated field, as we will see in the study of the first derivative and its applications. In reality, whenever we start a calculus subject we can start by saying that calculus is the mathematics of change and there is something we called ‘calculus with a vengeance’, that is the symbolic manipulative calculus. This type of calculus starts with the definition of two major concepts, that is the derivative and the integral. And when we integrate second and higher derivatives, we call it ‘the calculus of variations’. This part of calculus deals with the maximum and minimum as well as with finding lengths of arcs or surfaces of solids, it’s a various discipline. We shall start our studies of the applications of the derivative by studying the term ‘rate of change’. A rate is a quantity divided by the time when it is affected. For example, if a car travels a distance of 5 km for 3 hours, then the rate of its change of distance is 5/3 km per hour.

5. Advanced Topics in Calculus

Advanced topics in calculus cover three main fields: polar coordinates, parametric equations, and infinite series, each of which is entirely different from the previous sections. Type 1 polar equations are essentially circles, albeit in a slightly different form. They are somewhat like a more complicated version of normal circles because in this case the center and radius are given in terms of x and y. So there is a simple way to identify a polar equation for a circle from a regular Cartesian circle. Well, in a regular x-y coordinate system, a circle of radius r and center (h, k) satisfies the equation: (x-h)^2 + (y-k)^2 = r^2, whereas in type 1 form we have R = acosθ or R = asinθ or R = acosecθ or R = asecθ. When looking at how Graphmatica arranged things in comparison to the polar equations worksheet, it was simple to see how the first branch opens to the right because sine is positive and the second branch opens up making a clockwise circle when cosine is negative. Even though the “computer software” does make it easy to determine which branch opens in which direction, something that I did not know before. Overall, it was extremely useful to use both manual and digital forms of graphing polar equations because the process really required a thorough understanding of how the graph works which the worksheet can really provide. Also, the digital form offered a nice way to check if I implemented it correctly. However, to be successful, it would be a confusing area if we only know about Graphmatica, without any idea about how we draw the graph. I am confident with the concept of the polar equations and the tries to make the graphs manually because this type of concept would definitely come out in IB exam. Also, it surprises me how understandable the content in the polar equations section is. For example, the very first slide assists me to understand what the polar curve looks like when the radius, R is equal to acosθ. And we also go through it clearly with good examples, so it boosted my understanding of the polar curve and the tries to draw the ice-cream shape on a circle line in the first slide.