Keep your introduction short and simple. Generally, your introduction should be between 5 and 10 percent of the overall length of your essay. If you're writing a 10-page paper, your introduction should be approximately 1 page. For shorter essays under 1,000 words, keep your introduction to 1 paragraph, between 100 and 200 words. An introductory paragraph, as the opening of a conventional essay, composition, or report, is designed to grab people's attention. It informs readers about the topic and why they should care about it but also adds enough intrigue to get them to continue to read. In short, the opening paragraph is your chance to make a great first impression. Introduction, foreword, preface refer to material given at the front of a book to explain or introduce it to the reader. A foreword is part of the front matter and is usually written by someone other than the author, often an authority on the subject of the book. A preface is the author's own statement, and often includes acknowledgments.
Calculus is all about changes.
Sam and Alex are traveling in the car ... but the speedometer is broken. |
'Wait a minute ...'
'Well in the last minute we went 1.2 km, so we are going:'
1.2 km per minute x 60 minutes in an hour = 72 km/h
'No, Sam! Not our average for the last minute, or even the last second, I want to know our speed RIGHT NOW.'
'OK, let us measure it up here ... at this road sign... NOW!'
'OK, we were AT the sign for zero seconds, and the distance was ... zero meters!'
The speed is 0m / 0s = 0/0 = I Don't Know!
'I can't calculate it, Alex! I need to know some distance over some time, and you are saying the time should be zero? Can't be done.'
That is pretty amazing ... you'd think it is easy to work out the speed of a car at any point in time, but it isn't.
Even the speedometer of a car just shows us an average of how fast we were going for the last (very short) amount of time.
How About Getting Real Close
But our story is not finished yet!
Sam and Alex get out of the car, because they have arrived on location. Sam is about to do a stunt:
Sam will do a jump off a 20 m building.Alex, as photographer, asks:'How fast will you be falling after 1 second?' |
Sam uses this simplified formula to find the distance fallen:
d = 5t2
- d = distance fallen, in meters
- t = time from jump, in seconds
(Note: the formula is a simpler version of how fast things fall under gravity: d = ½gt2)
Example: at 1 second Sam has fallen
d = 5t2 = 5 × 12 = 5 m
But how fast is that? Speed is distance over time:
Speed = distancetime
So at 1 second:
Speed = 5 m1 second = 5 m/s
'BUT', says Alex, 'again that is an average speed, since you started the jump, ... I want to know the speed at exactly 1 second, so I can set up the camera properly.'
Well ... at exactly 1 second the speed is:
Speed = 5 − 5 m1 − 1 s = 0 m0 s = ???
So again Sam has a problem.
Think about it ... how do we figure out a speed at an exact instant in time?
What is the distance? What is the time difference?
They are both zero, giving us nothing to calculate with!
But Sam has an idea ... invent a time so short it won't matter.
Sam won't even give it a value, and will just call it 'Δt' (called 'delta t').
So Sam works out the difference in distance between t and t+Δt
At 1 second Sam has fallen
5t2 = 5 × (1)2 = 5 m
At (1+Δt) seconds Sam has fallen
5t2 = 5 × (1+Δt)2 m
We can expand(1+Δt)2:
So at (1+Δt) seconds Sam has fallen
In Summary:
So between 1 second and (1+Δt) seconds we get:
Change in distance over time:
So the speed is 10 + 5Δt m/s, and Sam thinks about that Δt value ... he wants Δt to be so small it won't matter ... so he imagines it shrinking towards zero and he gets:
Speed = 10 m/s
Wow! Sam got an answer!
Sam: 'I will be falling at exactly 10 m/s'
Alex: 'I thought you said you couldn't calculate it?'
Sam: 'That was before I used Calculus!'
Yes, indeed, that was Calculus.
The word Calculus comes from Latin meaning 'small stone'.
· Differential Calculus cuts something into small pieces to find how it changes.
· Integral Calculus joins (integrates) the small pieces together to find how much there is.
Sam used Differential Calculus to cut time and distance into such small pieces that a pure answer came out.
And Differential Calculus and Integral Calculus are like inverses of each other, similar to how multiplication and division are inverses, but that is something for us to discover later!
So ... was Sam's result just luck? Does it work for other things?
Let's try doing this for the function y = x3
This will be similar to the previous example, but we will just use a slope on a graph, no one has to jump for this one!
Example: What is the slope of the function y = x3 at x=1 ?
We can expand (1+Δx)3 to 1 + 3Δx + 3(Δx)2 + (Δx)3, and we get:
y = 1 + 3Δx + 3(Δx)2 + (Δx)3
And the difference between the y values from x = 1 to x = 1+Δx is:
Now we can calculate slope:
Once again, as Δx shrinks towards zero we are left with:
Slope = 3
And here we see the graph of y = x3
The slope is continually changing, but at the
point (1, 1) we can draw a line tangent to the curve
and find the slope there really is 3.
(Count the squares if you want!)
Question for you: what is the slope at the point (2, 8)?
Try It Yourself!
Go to the Slope of a Function page, put in the formula 'x^3', then try to find the slope at the point (1, 1).
Zoom in closer and closer and see what value the slope is heading towards.
Conclusion
Calculus is about changes.
Differential calculus cuts something into small pieces to find how it changes.
- Learn more at Introduction to Derivatives
Integral calculus joins (integrates) the small pieces together to find how much there is.
- Learn more at Introduction to Integration
No matter what type of writing you do, a strong introduction is important for setting the tone for your work. From blog posts to high school essays, the right introduction can pique a reader's interest and draw them in to keep reading. Use these strong introduction examples to inspire you as you write.
Strong Introductions for Essays
Whether you're writing an essay for class, answering an essay question on a test, crafting a college application, or writing any other type of essay-style work, your introduction paragraph is one of the most important you'll write.
Use a Surprising Fact
You can capture the reader's attention with a surprising fact or statement. Then, state your thesis clearly and succinctly, as in this example:
With recent studies showing that, on average, dog owners live up to 24% longer than those without canine companions, there's no doubt that a dog can have a range of physical and mental health benefits for its owner. These include a more active lifestyle, a social network of other pet owners, companionship, and a sense of purpose.
Pose a Question
Another way to engage your reader in your essay introduction is to ask a rhetorical question. This is a good way to start a persuasive essay or even a newspaper editorial. This example shows how you must carefully choose this question to support your thesis:
Global climate change is a crisis that affects everyone, rich and poor, young and old. From rising ocean levels to increased temperature extremes, the world is changing for everyone. How will your life be different in the coming years?
Start With an Anecdote
A short anecdote is a good way to start a personal essay, especially if it fits with your overall theme. Consider this example:
When I got home from school that day, my grandmother greeted me with a plate of cookies and a worried expression. I hadn't received the scholarship I needed to go to ballet camp, and we'd need to find another way to earn the money. That's when I started my business giving dance lessons to preschoolers, and it's taught me a lot about how to solve problems on my own.
Great Report Introductions
Reports aren't always the most fascinating kind of writing to read, so a really great introduction can make a huge difference in your reader engagement. You need to be clear about what you're discussing and offer any background information your reader might need.
Set the Stage
Give the reader important background information with an introduction like this:
With our sales up 25% and more than 2,500 positive reviews online, it's safe to say our product is embraced by consumers. We're using this information to project growth for the coming year.
State Your Point Clearly
Reports are about facts, but readers may not want to wade through all the data to get to your point. A strong introduction can give them a preview of what they'll find inside:
The water quality in Smith County has improved from a century ago, but there are still a number of contaminants of concern. Storm water run-off has been the major contributor to water pollution in the area.
Strong News and Feature Article Introductions
When you introduce an article, either online or in a more traditional newspaper, it's essential to have a 'hook.' Writing a hook takes a little extra thought, but it will make the reader continue past your introduction and into the body of your article.
Start With Something Shocking
People naturally want to read things that are shocking or surprising. Offer up a little drama for a strong introduction paragraph:
When Charles Jones went into the gas station Tuesday night, he did not expect to be caught in the middle of an armed robbery. However, that is exactly what happened.
Use a Statistic
You can start with a good statistic and hint at more information to come. People will read on because they want to know the rest of the story.
While for 60% of people, milk is the perfect beverage to go with a cookie, that's not true for everyone. A new study shows that up to 40% of people prefer something different with their bedtime snack.
Creative Introductions for Blog Posts
When you write a blog post, you need to engage your reader right away with a creative introduction. In the world of online reading, it's easy to navigate away to something else. The introduction is your chance to keep your reader on your blog.
Get Personal
Depending on the type of blog post you're writing, it can make sense to offer some personal perspective on your topic right away. Your reader is here for your voice, so you need to use it. This example will show you one way to do that:
My sister thought I was crazy to try to remake some thrift store jeans into an awesome maxi skirt. She told me she'd personally model the skirt for the blog if it worked. Well, when you see how fabulous she looks, you'll need to make one too.
Use Humor
You can also engage your audience right away with a humorous blog post introduction. Try a funny personal anecdote:
Introduction To Psychology
I'm not going to lie. The first time I made an art doll out of clay, it was a total nightmare. I literally had bad dreams about that thing for a week. But I'll save you the terror and show you how to create a not-so-scary doll with these techniques.
Great Short Story Introductions
If you're writing a short story, you need a creative introduction to get your reader engaged right away. In a novel, you have the whole first chapter, but because the length of your short story is limited, your introduction needs to suck your reader in within just a few sentences.
Bring in Vivid Imagery
Make your reader experience the world of your story by adding vivid sensory details. Using imagery that involves the senses lets your reader feel like he or she is in the story.
Even before I opened my eyes, the scent of lilacs told me I had a visitor. Here in the nursing home, everything has the same smell - a mixture of cafeteria food, cleaning products, and old people odors. A sudden scent of lilacs meant someone was here to see me.
Start in the Middle of Things
When you're writing fiction, you can start your story at the last possible moment to get your reader engaged. This is called 'in medias res' (in the middle of things). Because the action is already happening, your reader won't be able to stop reading. Try something like this creative introduction example:
I hit the water with a slap that knocked the wind out of me. For a moment, I could hear my sister screaming from the deck of the ship, but then everything went quiet as my ears went under water.
Engage the Reader With a Strong Introduction
No matter what type of writing you're doing, a strong introduction is one way to get your audience to keep reading. After you've captured their attention, get more tips to engage the reader in the rest of your piece. An engaged reader means your writing is effective and powerful.
Introduction Speech
B.A. English