Calculus is a branch of mathematics that focuses on limits, functions, derivatives, integrals, and infinite series. This area is a major part of mathematics and forms the basis for many of the equations that describe physics and mechanics. You may need a college-level class to get a good understanding of calculus, but this article can help you get started and keep an eye on the important concepts as well as technical insights.
Steps
Method 1 of 3: Review the Basics of Calculus
Step 1. Keep in mind that calculus is the study of how things change
Calculus is a branch of mathematics that looks at numbers and lines, usually the real world, and describes how they change. While this may not seem helpful at first, calculus is one of the most widely used branches of mathematics in the world. Imagine having the tools to examine how fast your business is growing at any given time or to chart the course of a spaceship and how fast it consumes fuel. Calculus is an important tool in engineering, economics, statistics, chemistry, and physics, and has helped create many real-world inventions and discoveries.
Step 2. Remember that functions are relationships between two numbers and are used to map relationships in the real world
Functions are rules that determine how numbers relate to each other and are used by mathematicians to create graphs. In a function, each entered value produces a result. For example, at y = 2x + 4, each value of "x" gives you a new value of "y". If x = 2, then y = 8; if x = 10, then y = 24. All calculus studies functions to see how they change, using functions to map relationships in the real world.
- Functions are often written as f (x) = x + 3. This means that in the function f (x), 3 is always added to the number you enter for "x". If you want to enter 2, write f (2) = 2 + 3, or f (2) = 5.
- Functions can also map complex movements. For example, NASA has functions that indicate the speed of a rocket based on the amount of fuel it burns, wind resistance, and the weight of the spacecraft.
Step 3. Think about the concept of infinity
Infinity is when a process is repeated over and over. It is not a specific place (you cannot go to infinity), but rather the behavior of a number or an equation does take place forever. This is important for studying change: you may want to know how fast your car is moving at any given time, but does that mean how fast you were in that current second? In that millisecond? In that nanosecond? You could find infinitely smaller amounts of time to be particularly precise and that's where the calculation comes in.
Step 4. Understand the concept of limits
A limit indicates what happens when something is close to infinity. Take the number 1 and divide it by 2. Then continue dividing it by 2 over and over again. 1 would become 1/2, then 1/4, 1/8, 1/16, 1/32, etc. Each time, the number gets smaller and smaller, "getting closer" to zero. But where would it end? How many times do you have to divide by 2 to get zero? In calculus, instead of answering this question, you set a limit. In this case, the limit is equal to 0.
- Limits are easier to see on a graph. For example, they are the points that a graph almost touches, but never does.
- The limits can be a number, nothing, or even infinity. For example, if you add 1 + 2 + 2 + 2 + 2 +… forever, your final number would be infinitely large. The limit would be infinite.
Step 5. Review the essential mathematical concepts of algebra, trigonometry, and precalculus
The calculus is based on many of the forms of mathematics that you have learned for a long time. Knowing these topics completely will allow you to learn and understand calculus much more easily. Some topics to review are:
- Algebra: understand different processes and learn to solve equations and systems of equations for multiple variables. Understand the basics of sets. Learn about graphing equations.
- Geometry: geometry is the study of shapes. Understand the basics of triangles, squares, and circles, and how to calculate things like area and perimeter. Understand angles, lines, and coordinate systems.
- Trigonometry: Trigonometry is a branch of mathematics that deals with the properties of circles and right triangles. Learn how to use trigonometric identities, graphs, functions, and inverse trigonometric functions.
Step 6. Buy a graphing calculator
The calculation is incredibly difficult to understand without seeing what is being done. Graphing calculators take functions and make them visual, allowing you to better understand the work you are going to do. Often times, you can see limits on the screen and calculate derivatives and functions automatically.
Many smartphones and tablets now offer cheap but effective graphing apps if you don't want to buy a full calculator
Method 2 of 3: Understand Derivatives
Step 1. Note that calculus is used to study "instantaneous change."
Knowing why something changes at an exact moment is the heart of the calculation. For example, the calculation indicates not only the speed of a car, but also how much that speed changes at any given moment. This is one of the simplest uses of calculus, but it is incredibly important. Imagine how useful this knowledge would be to the speed of a spaceship trying to reach the moon!
Finding the instantaneous change is called "differentiation." Differential calculus is the first of the two main branches of calculus
Step 2. Use derivatives to understand how things change instantly
"Derivative" is a sophisticated sounding word that inspires anxiety. However, the concept itself is not that difficult to grasp, it just means how quickly something changes. The most common derivatives in daily life are related to speed. You probably don't call it the "derivative of velocity," but you can call it "acceleration."
Acceleration is a derivative. This indicates how fast something increases or decreases speed, or how it changes
Step 3. Note that the rate of change is the slope between two points
This is one of the key findings of the calculation. The rate of change between two points is equal to the slope of the line connecting them. Think of a basic line, such as the equation y = 3x. The slope of the line is 3, which means that for each new value of "x", "y" is multiplied by 3. The slope is the same as the rate of change: a slope of 3 means that the line changes by a factor of 3 for each change in "x". When x = 2, y = 6; when x = 3, y = 9.
- The slope of a line is the change in "y" divided by the change in "x".
- The larger the slope, the steeper the line will be. It can be said that steep lines change very quickly.
- Review how to find the slope of a line if you don't remember well.
Step 4. Note that you can find the slope of curved lines
Finding the slope of a straight line is pretty simple: how much does "y" change for each value of "x"? But complex equations with curves, like y = x2, they are much more difficult to find. However, you can still find the rate of change between any two points. Just draw a line between them and calculate the slope to find the rate of change.
- For example, at y = x2, you can take any two points and get the slope. Take (1, 1) and (2, 4). The slope between them would be equal to (4-1) / (2-1) = 4/2 = 2. This means that the rate of change between x = 1 and x = 2 is 2.
Step 5. Bring the points closer together to get a more accurate rate of change
The closer the two points are, the more likely you are to have an accurate answer. Let's say you want to know how fast your car accelerates right when you hit the gas pedal. You shouldn't measure the change in speed between your house and the supermarket, you should measure the change in speed the second after you hit the accelerator. The closer your measurement is to that instant, the more accurate your reading will be.
For example, scientists study the rate at which some species go extinct to try to save them. However, more animals often die in the winter than in the summer, so studying the rate of change throughout the year is not as helpful. Scientists would find the rate of change between closest points, such as from July 1 to August 1
Step 6. Use infinitely small lines to find the "instantaneous rate of change" or the derivative
This is where the calculation often gets confusing, but this is actually the result of two simple facts. First, you know that the slope of a line is equal to how fast it changes. Second, you know that the closer the points are to the line, the more accurate the reading will be. But how can you find the rate of change at a point if the slope is the relationship between two points? The calculation answer: you have to choose two points infinitely close to each other.
Think of the example where you kept dividing 1 by 2 over and over again, getting 1/2, 1/4, 1/8, etc. In the long run, you get so close to zero that the answer is practically zero. Here, the points are so close to each other that they are "practically instantaneous". This is the nature of derivatives
Step 7. Learn to calculate a variety of derivatives
There are many different techniques for finding a derivative depending on the equation, but most make sense if you remember the basic principles of derivatives outlined above. Derivatives are basically a way of finding the slope of an "infinitely small" line. Now that you know the theory of derivatives, a large part of the work is to find the answers.
Step 8. Look for derived equations to predict the rate of change at any moment
Using derivatives to find the rate of change at a point is helpful, but the beauty of calculus is that it allows you to create a new model for each function. For example, the derivative of y = x2 is Yl = 2x. This means that you can find the derivative for each point on the graph y = x2 simply by substituting it into the derived equation. At the point (2, 4), where x = 2 and y = 4, the derivative is 4, since Yl = 2*(2).
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There are different notations for derivatives. In the previous step, the derivatives were marked with a prime symbol: for the derivative of y, { displaystyle y,}
, hay que escribir y′.{displaystyle y^{prime }.}
This is called" de=" />
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There is also another popular way of writing derivatives. Instead of using the prime symbol, we write ddx. { Displaystyle { frac { mathrm {d}} { mathrm {d} x}}.}
Recuerda que la función y=x2{displaystyle y=x^{2}}
depende de la variable x.{displaystyle x.}
Entonces, escribimos la derivada como dydx{displaystyle {frac {mathrm {d} y}{mathrm {d} x}}}
, la derivada de y{displaystyle y}
con respecto a x.{displaystyle x.}
This is called" de=" />
Step 9. Recall examples of derivatives from the real world in case you still have difficulty understanding
The easiest example is based on speed, which offers many different derivatives that we see every day. Remember: a derivative is a measure of how fast something is changing. Think of the basic experiment of rolling a marble on a table, in which you measure how far and how fast it moves each time. Now imagine that the marble draws a line on a graph. Here you must use derivatives to measure instantaneous changes at any point on that line.
- How fast does the marble change location? What is the rate of change, or the derivative, of the movement of the marble? This derivative is what we call "velocity".
- Roll the marble down a slope and watch how quickly it picks up speed. What is the rate of change, or the derivative, of the speed of the marble? This derivative is what we call "acceleration".
- Roll the marble down a path that goes up and down, like a roller coaster. How fast do you speed downhill and how fast do you lose speed uphill? How fast is it moving exactly halfway up the first slope? This would be the instantaneous rate of change, or the derivative, of that marble at that specific point.
Method 3 of 3: Understand the integrals
Step 1. Keep in mind that you must use calculus to find complex areas and volumes
Calculus allows you to measure complex shapes that are usually too difficult. For example, consider trying to find out how much water is in a long lake in a peculiar shape. It would be impossible to measure each gallon of water separately or to use a ruler to measure the shape of the lake. The calculation allows you to study how the edges of the lake change and use that information to find out how much water is in it.
- Making geographic models and studying volume is to use the integration. Integration is the second main branch of calculus.
Step 2. Understand that integration finds the area under a graph
Integration is used to measure the space under any line, which allows you to find the area of odd or irregular shapes. Consider the equation y = x2, which looks like a giant U. You may want to find out how much space is under the U and you can use integration to find it. While this may seem pointless, think about the uses in manufacturing. You can make a function that looks like a new part and use integration to find out the area of that part, which will help you order the correct amount of that material.
Step 3. Keep in mind that it is necessary to select an area to integrate
You can't just integrate an entire function. For example, y = x is a diagonal line that goes on infinitely and you can't integrate all of it because it would never end. When integrating functions, you have to choose an area, like all points between x = 2 and x = 5.
Step 4. Remember how to find the area of a rectangle
Suppose you have a flat line on a graph, such as y = 4. To find the area below it, you would find the area of a rectangle between y = 0 and y = 4. This is easy to measure, but would never work for curved lines that they cannot easily be converted to rectangles.
Step 5. Note that the integration adds many small rectangles to find the area
If you focus too closely on a curve, it will look flat. This happens every day: you cannot see the curve of the Earth because we are so close to its surface. Integration creates an infinite number of rectangles under a curve that are so small that they are practically flat, allowing them to be measured. Add them together to get the area under a curve.
Imagine that you are adding many small pieces below the graph and the width of each piece is almost zero
Step 6. Find out how to read and write integrals correctly
Integrals have 4 parts. A typical integral looks like this:
∫f (x) dx { displaystyle \ int f (x) mathrm {d} x}
- El primer símbolo, ∫, {displaystyle \int, }
- La segunda parte, f(x), {displaystyle f(x), }
-
Finally, the dx { displaystyle \ mathrm {d} x}
al final te dirá con respecto a qué variable vas a integrar. Como la función f(x){displaystyle f(x)}
depende de x, {displaystyle x, }
debes integrarla con respecto a ella.
- Recuerda que la variable que vas a integrar no siempre va a ser x, {displaystyle x, }
es el símbolo de la integración (en realidad es una S alargada).
is the function. When it is inside the integral, it is called
, así que ten cuidado con lo que escribas.
Step 7. Learn to find integrals
Integration comes in many forms and you will need to learn many different formulas to integrate each function. However, they all follow the principles outlined above: integration finds an infinite number of rectangles to add. These are the different ways to find integrals:
- Integrate by substitution.
- Integrate indefinite integrals.
- Integrate by parts.
Step 8. Note that integration reverses differentiation
This is an unquestionable rule of thumb and the one that has led to many scientific and technological advances. Because integration and differentiation are so closely related, a combination of the two can be used to find rate of change, acceleration, speed, location, motion, etc., regardless of what information you have.
For example, remember that the derivative of the velocity is the acceleration, so you can use the velocity to find the acceleration. But, if you only know the acceleration of something (like objects falling due to gravity), you can integrate it to find the velocity. So no matter what data you have, you can use integration and differentiation to find out more
Step 9. Note that integration can also find the volume of three-dimensional objects
Rotating a planar shape is one way to create three-dimensional solids. Imagine spinning a coin on the table in front of you - see how it appears to form a sphere when you spin. You can use this concept to find volume in a process known as "volume per rotation."
With this you can find the volume of any solid in the world, as long as you have a function that reflects it. For example, you can create a function that plots the bottom of a lake and then use it to find the volume of the lake or how much water it holds
Advice
- Clear up your problems by consulting your teacher.
- Start from the basics.
- Pay attention in class.
- Practice makes perfect, so do the practice problems in your textbook and review the answers to help you understand the concepts.