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Study Guide

### Introduction to Calculus

Scary. Mysterious. Impossible. These might be words that come to mind when we think about…well, walking a tightrope across Niagara Falls. Calculus used to be described by those words a few hundred years ago, but those were

*literally*the dark ages.Calculus is all about imagining moments in time. That's pretty tough, even for folks here at Shmoop HQ. Back before calculus was invented, talking about calculus and “moments in time” was reason enough to send someone to the stocks. Maybe hearing that makes us wish, a little, that H.G. Wells’

*Time Machine*were non-fiction.Throughout your first calculus class, you’ll see lots of new symbols and notation. Don’t freak. Chances are no one else in your class has seen them before, either. It’s not a totally foreign place, though. There will still be lots of warm and fuzzy addition, subtraction, multiplication, and division to wrap up in. They haven’t changed one bit.

### Why Should I Care?

Calculus is actually pretty cool. It might not be stuff we all use on a day-to-day basis, but it plays a role in most aspects of daily life. From airplanes to cell phones, someone has used calculus to make some portion of the stuff we use all the time.

Hmm…calculus is starting to sound more useful. In algebra, if we add 2 apples to 3 apples, we can see 5 apples. However, concepts in calculus are often things we can’t actually see directly. For example, how fast does an apple hit the ground when dropping it from the Leaning Tower of Pisa? Can we salvage what apple remains?

Maybe calculus can’t solve everything that grandma’s apple sauce recipe can. However, it

*does*come in handy for fields in science such as physics, chemistry, technology, and engineering.### What is Calculus?

Calculus is a pretty big branch on the ol’ mathematics tree. From a big picture standpoint, Omnimax style, calculus is the study of ginormous numbers and super tiny numbers. We’re talking infinite and infinitesimal, here. It’s mind-blowing stuff, from a philosophical point of view. Calculus is a way to describe what happens in an instant in time, or what happens if we looked into the infinite future. It doesn’t always have to do with "time," but you get the point.

It’s abstract, it’s intangible, and it’s useful for those very reasons. It’s good for figuring out physics things like

**acceleration**and**velocity**. It helps the weather man draw out temperature gradients on a map. It’s used for finding the biggest box that can be constructed out of a given amount of cardboard. We want the walls to be tall enough so the puppy can’t jump out, right?From a textbook standpoint, calculus combines all of the stuff we learned in algebra and trig, and throws a bunch of new rules in the mix. There will be many parabolas, there will be new symbols, and there will be pie…no, we mean pi. Not that Pi.

### Basic Elements

What are the nuts and bolts of calculus? What makes it tick? You’ve come to the right place. Most of the basic elements are probably new concepts, and some seem complicated at first glance.

The first basic element is the

**function**, which we've dealing with functions since the good ol' days of Algebra. They usually look something like this:*f*(*x*) =*x*. A function is a rule. When we plug a value into the function, we’ll get another (or the same) value out. In each function, there is also an**independent variable**(the input) and a**dependent variable**(the output) .But the concept that underlies literally

*everything*in calculus is the**limit**. A limit is a value that the dependent variable in a function approaches as the independent variable approaches a given value. As boats approach their docks, functions approach values. One way to find a limit of a function is to plug in values close to the desired value into the independent variable and see what the dependent variable is approaching.The concept of a

**slope**is an idea from algebra. It’s the (change in*y*) divided by the (change in*x*) By finding the slope between two points on a graph, we were able to determine the average rate of change between those points. A**derivative**takes this to next level. It's essentially a slope, but now we'll be able to find the slope at a point on a curvy line, instead of just between two points. Are limits involved here? You better believe it.An

**integral**is a way to find an area. Integrals can be used to find the area of a circle, a square, or an irregular wavy enclosed region (think of a fancy pool). It’s essentially the opposite operation of a derivative, and it’s another way to pull more data from a graph. Computing integrals relies heavily on limits and derivatives. In calculus, the concepts just keep building off of previous ones. Get used to it.### Applications in Science and Engineering

Lots of fields can benefit from the concepts in calculus. In cases where relationships can be graphed, calculus can be used. That's the only prerequisite.

How fast is a diver or a long jumper going upon impact (or at any point during the dive or jump). What path does a gymnast follow when she releases the uneven bars? How long does it take for a car to drive from Point A to Point B?

All of these questions can be answered using calculus.

In a graph of distance vs. time, velocity is

**derivative**. Derivatives describe the rate of change at points on a graph so this shouldn't be too much of a surprise. If we had to describe the rate of change of someone's position, velocity or speed would be a pretty apt description.When chemicals react with one another, calculations about the rates at which they react involve calculus. Engineers might use calculus for optimization problems. For instance, they can find the largest volume that can be held by a soda and/or pop can, while using the smallest possible amount of aluminum. They can also figure out the best size can top and bottom for optimal stacking ability.

Video game engineers might use various forms of calculus to simulate real-life situations. Depending on the angle that a force is applied, where should those angry birds land after sling-shot release? Will the pigs pay? Video games are steeped in calculus simulations.

### What is Calculus?

Calculus is a pretty big branch on the ol’ mathematics tree. From a big picture standpoint, Omnimax style, calculus is the study of ginormous numbers and super tiny numbers. We’re talking infinite and infinitesimal, here. It’s mind-blowing stuff, from a philosophical point of view. Calculus is a way to describe what happens in an instant in time, or what happens if we looked into the infinite future. It doesn’t always have to do with "time," but you get the point.

It’s abstract, it’s intangible, and it’s useful for those very reasons. It’s good for figuring out physics things like

**acceleration**and**velocity**. It helps the weather man draw out temperature gradients on a map. It’s used for finding the biggest box that can be constructed out of a given amount of cardboard. We want the walls to be tall enough so the puppy can’t jump out, right?From a textbook standpoint, calculus combines all of the stuff we learned in algebra and trig, and throws a bunch of new rules in the mix. There will be many parabolas, there will be new symbols, and there will be pie…no, we mean pi. Not that Pi.

### Basic Elements

What are the nuts and bolts of calculus? What makes it tick? You’ve come to the right place. Most of the basic elements are probably new concepts, and some seem complicated at first glance.

The first basic element is the

**function**, which we've dealing with functions since the good ol' days of Algebra. They usually look something like this:*f*(*x*) =*x*. A function is a rule. When we plug a value into the function, we’ll get another (or the same) value out. In each function, there is also an**independent variable**(the input) and a**dependent variable**(the output) .But the concept that underlies literally

*everything*in calculus is the**limit**. A limit is a value that the dependent variable in a function approaches as the independent variable approaches a given value. As boats approach their docks, functions approach values. One way to find a limit of a function is to plug in values close to the desired value into the independent variable and see what the dependent variable is approaching.The concept of a

**slope**is an idea from algebra. It’s the (change in*y*) divided by the (change in*x*) By finding the slope between two points on a graph, we were able to determine the average rate of change between those points. A**derivative**takes this to next level. It's essentially a slope, but now we'll be able to find the slope at a point on a curvy line, instead of just between two points. Are limits involved here? You better believe it.An

**integral**is a way to find an area. Integrals can be used to find the area of a circle, a square, or an irregular wavy enclosed region (think of a fancy pool). It’s essentially the opposite operation of a derivative, and it’s another way to pull more data from a graph. Computing integrals relies heavily on limits and derivatives. In calculus, the concepts just keep building off of previous ones. Get used to it.### Applications in Science and Engineering

Lots of fields can benefit from the concepts in calculus. In cases where relationships can be graphed, calculus can be used. That's the only prerequisite.

How fast is a diver or a long jumper going upon impact (or at any point during the dive or jump). What path does a gymnast follow when she releases the uneven bars? How long does it take for a car to drive from Point A to Point B?

All of these questions can be answered using calculus.

In a graph of distance vs. time, velocity is

**derivative**. Derivatives describe the rate of change at points on a graph so this shouldn't be too much of a surprise. If we had to describe the rate of change of someone's position, velocity or speed would be a pretty apt description.When chemicals react with one another, calculations about the rates at which they react involve calculus. Engineers might use calculus for optimization problems. For instance, they can find the largest volume that can be held by a soda and/or pop can, while using the smallest possible amount of aluminum. They can also figure out the best size can top and bottom for optimal stacking ability.

Video game engineers might use various forms of calculus to simulate real-life situations. Depending on the angle that a force is applied, where should those angry birds land after sling-shot release? Will the pigs pay? Video games are steeped in calculus simulations.

### Major Calculus Themes

There will be several themes that will pop up again and again throughout this calculus guide. Some of them—actually, most of them you’ve seen before.

*Deep exhale of relief*. The only difference is, you will get to know them inside and out. You will get a full, 360° panoramic view. You will probably start to dream about them.### Graphs

Here’s an old friend. We met her back in the days of algebra. In just about every part of this calculus guide, there will be a graph to analyze. Eventually, we’ll learn to love them. We’ll also learn to draw them; not just straight lines, but curves. Don your fanciest beret and grab your best No. 2’s. We prefer canvas and paint brushes, but that might be a bit much for homework.

### Functions

These are the rules of the game.

When we plug the

*independent*variable into a function, we get the*dependent*variable. To jog your memory, functions generally have the form*f*(*x*). Calculus is all about learning how to pull information out of different functions. We’ll take derivatives. We’ll take integrals. We’ll take limits. We'll even take a few (*several*) snack breaks. This calculus business is some pretty hunger-inducing stuff.Functions are super important for constructing graphs; they set the rules for plotting...just not the Voldemort kind of plotting.

### Areas

We’ve been dealing with areas and volumes since geometry. Actually, we’ve been dealing with them since we figured out a square peg doesn’t fit in a round hole. Ever wonder how the math wizards figured out those volume equations? That’s right, they used calculus.

We’ll use calculus to find the area created by a graph. Later, we’ll add a third dimension to find the volume of shapes. Not just uniform shapes like cylinders, but wiggly vase-like shapes. Maybe we’ll throw in a T-Rex shaped volume, for a challenge. Who knew paleontologists use calculus, too?

### Tangents

Imagine flinging a yo-yo in a circle around your head. You let go, and it flies off in a straight line towards your mom’s favorite ceramic figurine.

Oops.

Tangent lines have caused you grief, yet again. In this case, the tangent is the line formed from the yo-yo's point of release to the point of impact. A tangent is a straight line that touches a point on a graph without actually crossing the graph at that point. They’ll show up most often when we talk about slopes and derivatives. Yes, you also saw these in trigonometry and geometry. They're pretty useful, but not for reconstructing ceramic figurines. Sigh...

### Limits

When driving, we try to approach and stay near the speed limit. Well, some of us do, anyway.

Limits in calculus have a similar idea; we’re trying to approach

*some number*. We actually pick numbers that are close to*some number*. Infinitesimally close. Maybe too close, like a close talker. Limits are useful because they get us close to an instant, which is an abstract idea. An instant is difficult to wrap your head around. A limit is much more concrete and mathematical. It brings a sense of peace and justice to a chaotic world. Or is that just us?

### Graphs

Here’s an old friend. We met her back in the days of algebra. In just about every part of this calculus guide, there will be a graph to analyze. Eventually, we’ll learn to love them. We’ll also learn to draw them; not just straight lines, but curves. Don your fanciest beret and grab your best No. 2’s. We prefer canvas and paint brushes, but that might be a bit much for homework.

### Functions

These are the rules of the game.

When we plug the

*independent*variable into a function, we get the*dependent*variable. To jog your memory, functions generally have the form*f*(*x*). Calculus is all about learning how to pull information out of different functions. We’ll take derivatives. We’ll take integrals. We’ll take limits. We'll even take a few (*several*) snack breaks. This calculus business is some pretty hunger-inducing stuff.Functions are super important for constructing graphs; they set the rules for plotting...just not the Voldemort kind of plotting.

### Areas

We’ve been dealing with areas and volumes since geometry. Actually, we’ve been dealing with them since we figured out a square peg doesn’t fit in a round hole. Ever wonder how the math wizards figured out those volume equations? That’s right, they used calculus.

We’ll use calculus to find the area created by a graph. Later, we’ll add a third dimension to find the volume of shapes. Not just uniform shapes like cylinders, but wiggly vase-like shapes. Maybe we’ll throw in a T-Rex shaped volume, for a challenge. Who knew paleontologists use calculus, too?

### Tangents

Imagine flinging a yo-yo in a circle around your head. You let go, and it flies off in a straight line towards your mom’s favorite ceramic figurine.

Oops.

Tangent lines have caused you grief, yet again. In this case, the tangent is the line formed from the yo-yo's point of release to the point of impact. A tangent is a straight line that touches a point on a graph without actually crossing the graph at that point. They’ll show up most often when we talk about slopes and derivatives. Yes, you also saw these in trigonometry and geometry. They're pretty useful, but not for reconstructing ceramic figurines. Sigh...

### Limits

When driving, we try to approach and stay near the speed limit. Well, some of us do, anyway.

Limits in calculus have a similar idea; we’re trying to approach

*some number*. We actually pick numbers that are close to*some number*. Infinitesimally close. Maybe too close, like a close talker. Limits are useful because they get us close to an instant, which is an abstract idea. An instant is difficult to wrap your head around. A limit is much more concrete and mathematical. It brings a sense of peace and justice to a chaotic world. Or is that just us?### Key Skills

There are some things we need to know before diving head first into calculus. Think of them as your parachute, base jumping suit, or cat-like skills. They are necessary for survival.

### Algebra

Before starting calculus, we need to know the rules of algebra. We’re talking all of the rules. Even those log ones.

If you

*accidentally*tossed out your notes from algebra class, you can always check out the Shmoop Algebra Guide to review. We'll run into polynomials, natural numbers, square roots, and inequalities, plus all the other stuff. We’ve seen them all before, so that might actually be welcomed news. It’s not all limits all the time.### Trigonometry

We will see radians,

*θ*’s and*π’*s from time to time in calculus. Remember the unit circle, special right triangles, and all that jazz? Believe us—these will come in handy. Sines (sin), cosines (cos), and tangents (tan) will be regulars, as will be their inverses. Feel free to review them if you need a refresher.We will also learn how to switch from polar to Cartesian coordinates. Pythagoras will suddenly become a very useful guy to know.

### Algebra

Before starting calculus, we need to know the rules of algebra. We’re talking all of the rules. Even those log ones.

If you

*accidentally*tossed out your notes from algebra class, you can always check out the Shmoop Algebra Guide to review. We'll run into polynomials, natural numbers, square roots, and inequalities, plus all the other stuff. We’ve seen them all before, so that might actually be welcomed news. It’s not all limits all the time.### Trigonometry

We will see radians,

*θ*’s and*π’*s from time to time in calculus. Remember the unit circle, special right triangles, and all that jazz? Believe us—these will come in handy. Sines (sin), cosines (cos), and tangents (tan) will be regulars, as will be their inverses. Feel free to review them if you need a refresher.We will also learn how to switch from polar to Cartesian coordinates. Pythagoras will suddenly become a very useful guy to know.

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