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# Calculus

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Calculus is a fundamentally different type of math than other math subjects; calculus is dynamic, whereas other types of math are static. Simply put, calculus is the math of motion, the study of how things change. Or, for a more formal definition:

## Calculus Definition

Calculus is the mathematical study of continuous change. It deals with rates of change and motion and has two branches:

• Differential Calculus
• Deals with rates of change of a function
• Explains a function at a specific point
• Integral Calculus
• Deals with areas under the graph of a function
• Gathers a total quantity of a function over a range

Before the invention of calculus, all math was static and was only really useful in describing objects that weren't moving. That's not very useful, is it? The vast majority of objects are always moving! From the smallest objects – electrons in atoms – to the largest ones, such as planets in the universe, no object is ever always at rest (and in many cases are never at rest). This is where calculus shines. It works in many fields where you wouldn't normally think math would matter. Calculus is used in physics, engineering, statistics, and even in life sciences and economics!

Did you know that...

The Latin word, calculus, means "pebble". Back in Roman times, it was common to use pebbles for simple calculations (like adding and subtracting), so the word calculus developed an association with computation. In fact, the English words calculator and calculation are derived from Latin calculus.

## Where Does Calculus Come From?

So, where does calculus come from? How did early mathematicians come up with these complex ideas?

Calculus was actually invented by two people. Sir Isaac Newton and Gottfried Leibniz, independently of each other, came up with the idea of calculus. While Sir Isaac Newton invented it first, we mainly use Gottfried Leibniz's notation today.

To get an idea of how you could invent calculus, let's start with a seemingly simple problem: to find the area of a circle. Now, we know the formula for the area of a circle:

Graph of a circle

But why is this the case? What kind of thought process leads to this observation? Well, say we don't know this formula. How can we try to find the area of a circle without it? To start, let's try breaking the circle into shapes whose areas are more simple to calculate.

Finding the area of a circle using shapes we know

And after trying to get more and more shapes so that less and less of the circle is left over, let's try a different idea: break the circle up into concentric rings.

Graph of concentric circles

That's great, but now what? Now, let's take just one of these rings, which has a smaller radius, that we will call , that is between 0 and 5.

Graph of concentric circles with one ring highlighted

From here, let's straighten out this ring.

Straightened out ring from concentric circles

With the ring straightened out, now we have a shape whose area is easier to find. But, what shape has an even easier area to find? A rectangle. For simplicity, we can actually approximate the shape of the straightened-out ring as a rectangle.

Approximating the Straightened Out Ring as a Rectangle

This rectangle has a width that is equal to the circumference of the ring, or , and a height of whatever smaller radius of that you chose earlier. Let's rename to , to represent a small difference in radius from one ring of the circle to the next one. So, what do we have now? We have a bunch of rings of the circle approximated as rectangles whose areas we know how to find! And, for smaller and smaller choices of (or breaking the circle into smaller and smaller rings), our approximation of the area of the ring becomes more and more accurate.

Let's go a step further and straighten out all the rings of the circle into rectangles and line them up side by side. Then, placing these rectangles on a graph of the line , we can see that each rectangle extends to the point where it just touches the line.

Rings of concentric circles placed on a graph of the line: y = 2πR

And for smaller and smaller choices of , we can see that the approximation of the total area of the circle becomes more accurate.

Rings of concentric circles placed on a graph of the line: y=2πR with a smaller choice for dr

Now you might notice that as gets smaller, the number of rectangles gets quite large, and won't it be time-consuming to add all their areas together? Take another look at the graph, and you will also notice that the total areas of the rectangles actually look like the area underneath the line, which is a triangle!

Total areas of concentric circles represented as the area under the graph

We know the formula for the area of a triangle:

Which in this case would be:

Which is the formula for the area of a circle!

But wait, how did we get here? Let's take a step back and think about it. We had a problem that could be solved by approximating it with the sum of many smaller numbers, each of which looked like for values of R from 0 to 5. And that small number was our choice of thickness for each ring of the circle. There are two important things to take note of here:

• Not only does play a role in the areas of the rectangles we are adding up, it also represents the spacing between the different values of R.

• The smaller the choice for , the better the approximation. In other words, the smaller we make , the more accurate the answer will be.

By choosing smaller and smaller values for dr to better approximate the original problem, the sum of the total area of the rectangles approaches the area under the graph; and because of that, you can conclude that the answer to the original problem, un-approximated, is equal to the area under this graph.

These are some pretty interesting ideas, right? So now you might be wondering, why go through this effort for something as simple as finding the area of a circle? Well, let's think for a moment... Since we were able to find the area of a circle by reframing the question as finding the area under a graph, could we not also apply that to other, more complex graphs? The answer is, yes, we can! Say, for example, we take the graph of , a parabola.

Graph of a parabola

How could we possibly find the area under a graph like this, say between the values of 0 and 5? This is a much more difficult problem, isn't it? And let's reframe this problem slightly: let's fix the left endpoint at 0 and let the right endpoint vary. Now the question is, can we find a function, let's call it , that gives us the area under the parabola between the left endpoint of 0 and the right endpoint of x?

Area under a parabola

This brings us to the first big topic of calculus: integrals. To use calculus vocabulary, the function we called is known as the integral of the function of the graph. In our case, would be the integral of . Or in a more mathematical notation:

As we progress through calculus, we will discover the tools that will help us find , but for now, what function represents is still a mystery. What we do know is that gives us the area under the parabola from a fixed left endpoint and a variable right endpoint. Now take a moment and think of what else we know about the relationship between and the graph, .

When we increase x by just a tiny bit, say by an amount we will call , we see a resulting change in the area under the graph, which we will call . This tiny difference in area, , can be approximated as a rectangle, just as we were able to approximate as a rectangle in our circle example. The rectangle approximation for , however, has a height of and a width of . And for smaller and smaller choices of , the approximation of the area under the graph becomes more and more accurate, just as with the circle example.

A change in area under a parabola

This gives us a new way to think about how is related to . Changing the output of by is about equal to , where is whatever you choose, times . This relationship can be rearranged to:

And, of course, this relationship becomes more and more accurate as we choose smaller and smaller values for . While the function is still a mystery to us, this relationship is key and, in fact, holds true for much more than just the graph of .

Any function that is defined as the area under some graph has the property that dA divided by dx is approximately equal to the height of the graph at that point. This approximation becomes more accurate for smaller choices of dx.

This brings us to the next big topic of calculus: derivatives. The relationship between , , and the function of the graph, , written as the ratio of divided by is equal to , is called the derivative of A. In mathematical notation:

Now, you may have noticed that the general formulas we've written for the derivative and integral look like they relate to each other. That's because they do! Derivatives and integrals are actually inverses of each other. In other words, a derivative can be used to find an integral and vice versa. The back-and-forth between integrals and derivatives where the derivative of a function for the area under a graph gives the function defining the graph itself is called the Fundamental Theorem of Calculus.

Let's summarize a bit. Generally speaking, a derivative is a measure of how sensitive a function is to small changes in its input, while an integral is a measure of some area under a graph. The Fundamental Theorem of Calculus links the two together and shows how they are inverses of each other.

Now that we have a solid idea of what calculus is and where it comes from let's dig a little deeper. We can gather from our examples in the previous section that there are some main concepts of calculus:

• Calculus is all about approximation or becoming more accurate as some value approaches another value

• There are two types of calculus:

• Calculus that deals with derivatives or differential calculus

• Calculus that deals with integrals, or integral calculus

• There is a fundamental theorem of calculus, and it links differential and integral calculus together

## The Idea of a Limit in Calculus

Before we delve into the types of calculus, let's take a look at what sets calculus apart from other types of math: the idea of a limit. Remember in the previous section when we talked about choosing smaller and smaller values for either or ? When we consider these smaller and smaller values, we are improving the accuracy of our approximations by having or approach zero. Why not just use zero directly? Remember, the formula for the derivative of A is the ratio of divided by , and dividing by zero is impossible! This is where the limit comes in. The limit essentially allows us to see what the answer to a problem (for example, the area under a graph) should be as we get closer and closer to whatever value the limit is. In the case of our examples in the "Where Does Calculus Come From?" section, the limit was zero.

A limit is the value that a function approaches as its independent variable (usually x) approaches a certain value.

## Differential Calculus

Differential calculus is the branch of calculus that deals with the rate of change of one quantity with respect to another quantity. In this branch, we divide things into smaller and smaller sections and study how they change from moment to moment.

Derivatives are how we measure rates of change. Specifically, derivatives measure the instantaneous rate of change of a function at a point, and the instantaneous rate of change of the function at a point is equal to the slope of the tangent line at that point.

## Integral Calculus

When we know the rate of change of a function, integral calculus can be used to find a quantity. In this branch, we sum small sections of things together to discover their overall behavior.

Integration is the method we use in calculus to find the area either underneath a graph, or in between graphs.

## The Fundamental Theorem of Calculus

The fundamental theorem of calculus links differential and integral calculus by stating that differentiation and integration are inverses of each other and is divided into two parts:

• Part 1 – shows the relationship between derivatives and integrals

• Part 2 – uses the relationship established in part 1 to show how to calculate an integral on a specific range

The definitions for the fundamental theorem of calculus are as follows:

[1] Part 1 of the fundamental theorem of calculus states that:

If a function, that we will call , is continuous on an interval of , and another function, that we will call , is defined as:

Then, on the same interval of .

[2] Part 2 of the fundamental theorem of calculus states that:

If a function, that we will call , is continuous on an interval of , and another function, that we will call , is any antiderivative of , then:

## Practical Applications of Calculus

Calculus has a wide variety, and a long history, of useful applications. In general, calculus is used in STEM (Science Technology Engineering Math) applications as well as in medicine, economics, and construction, just to name a few. A form of calculus was used back in ancient Egypt to build the pyramids! But the calculus we are learning today is the calculus that Sir Isaac Newton and Gottfried Leibniz developed in the seventeenth century.

## AP Calculus: AB and BC

AP Calculus is broken down into two courses, AP Calculus AB and AP Calculus BC. The difference between these two courses is that AP Calculus BC covers everything that AP Calculus AB covers, plus a couple of extra topics. Please have a look at our articles on each topic for a full study of AP Calculus!

### AP Calculus AB

The AP Calculus AB course covers many topics of calculus. A brief overview of them is listed below:

### AP Calculus BC

The AP Calculus BC course covers everything that AP Calculus AB does, plus these extra topics:

## Calculus - Key takeaways

• Calculus is the study of how things change - it deals with rates and changes of motion.
• There are two types of calculus - and they are inverses (or opposites) of each other:
• Differential Calculus
• Integral Calculus
• Differential calculus uses derivatives and is used to determine the rate of change of a quantity.
• Integral calculus uses integrals and is used to determine the quantity where the rate of change is known.
• The Fundamental Theorem of Calculus relates differential calculus to integral calculus as inverses of each other.
• The idea of a limit is what sets calculus apart from other areas of math.
• Calculus has many practical applications!
• AP Calculus is broken down into two courses:
• AP Calculus AB
• AP Calculus BC

Calculus is the mathematical study of continuous change. It deals with rates of change and motion and has two branches:

• Differential Calculus
• Deals with rates of change of a function
• Explains a function at a specific point
• Integral Calculus
• Deals with areas under the graph of a function
• Gathers a total quantity of a function over a range

Derivatives are how we measure rates of change. Specifically, derivatives measure the instantaneous rate of change of a function at a point, and the instantaneous rate of change of the function at a point is equal to the slope of the tangent line at that point.

A limit in calculus is the value that a function approaches as its independent variable (usually x) approaches a certain value.

Calculus has a wide variety, and a long history, of useful applications. In general, calculus is used in STEM (Science Technology Engineering Math) applications as well as in medicine, economics, and construction, just to name a few. A form of calculus was used back in Ancient Egypt to build the pyramids! But the calculus we are learning today is the calculus that Sir Isaac Newton and Gottfried Leibniz developed in the seventeenth century.

Integration is the method we use in calculus to find the area underneath a graph, or in between graphs.

## Final Calculus Quiz

Question

What is the product rule, in words?

In words, the product rule says that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

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Question

What is a common mistake when using the product rule?

A common mistake when using the product rule is assuming the derivative of a product of two functions is the product of their derivatives.

This is incorrect!

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Question

How can The Product Rule be proved?

The product rule can be proved by using limits and some simple algebraic manipulation.

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Question

How can the product rule be proved?

The product rule can be proved using limits and algebraic manipulation.

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Question

What are some other use cases for the product rule?

• The Product Rule of Exponents
• The Product Rule of Logarithms
• The Zero Product Rule

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Question

What is AP Calculus?

Calculus is the mathematical study of continuous change. It deals with rates of change and motion, and has two branches:

• Differential Calculus
• Deals with rates of change of a function
• Explains a function at a specific point
• Integral Calculus
• Deals with areas under the graph of a function
• Gathers a total quantity of a function over a range

Show question

Question

How is AP Calculus different from other types of math?

Calculus is dynamic, where other types of math are static. Simply put, calculus is the math of motion, the study of how things change.

Show question

Question

Explain where calculus comes from by describing how to find the area of a circle without using the formula.

1. Break the circle into rings

2. Straighten those rings out

3. Approximate the rings as rectangles

4. Line them up side by side under the graph of the line of the circumference of the circle

5. Realize that this can be represented by the area of a triangle

6. Calculate the area of that triangle

7. The area of that triangle is the area of the circle!

Show question

Question

What is the key to why finding the area of a circle can be done by finding the area of a triangle, whose graph is the circumference of the circle?

By choosing smaller and smaller values for dr to better and better approximate the original problem, the sum of the total area of the rectangles approaches the area under the graph; and because of that, you can then conclude that the answer to the original problem, un-approximated, is equal to the area under this graph.

Show question

Question

What is an integral?

• An integral is a function that gives the area under a graph.
• The function we called $$A(x)$$ is known as the integral of the function of the graph. In a general case, $$A(x)$$ would be the integral of $$f(x)$$.

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Question

What is true about any integral?

Any function that is defined as the area under some graph has the property that its difference in area, $$\mathrm{d}A$$, divided by a difference in input, $$\mathrm{d}x$$, is approximately equal to the height of the graph at that point. This approximation becomes more accurate for smaller choices of $$\mathrm{d}x$$.

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Question

What is a derivative?

• Generally speaking, a derivative is a measure of how sensitive a function is to small changes in its input.
• The relationship between $$\mathrm{d}A$$, $$\mathrm{d}x$$, and the function of the graph, $$f(x)$$, written as the ratio of $$\mathrm{d}A$$ divided by $$\mathrm{d}x$$ is equal to $$f(x)$$, is called the derivative of $$A$$.

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Question

What is the fundamental theorem of calculus?

The fundamental theorem of calculus links differential and integral calculus by stating that differentiation and integration are inverses of each other, and is divided into two parts:

• Part 1 - shows the relationship between derivatives and integrals

• Part 2 - uses the relationship established in part 1 to show how to calculate an integral on a specific range

Show question

Question

What notion sets calculus apart from other types of math?

The idea of a limit.

Show question

Question

What is the idea of a limit?

When we consider smaller and smaller values, we are improving the accuracy of our approximations by having $$\mathrm{d}r$$ or $$\mathrm{d}x$$ approach zero. Why not just use zero directly? Remember, the formula for the derivative of $$A$$ is the ratio of $$\mathrm{d}A$$ divided by $$\mathrm{d}x$$, and dividing by zero is impossible! This is where the limit comes in. The limit essentially allows us to see what the answer to a problem (for example, the area under a graph) should be as we get closer and closer to whatever value the limit is.

Show question

Question

What is differential calculus?

Differential calculus is the branch of calculus that deals with the rate of change of one quantity with respect to another quantity. In this branch, we divide things into smaller and smaller sections and study how they change from moment to moment.

Show question

Question

What is integral calculus?

When we know the rate of change of a function, integral calculus can be used to find a quantity. In this branch, we sum small sections of things together to discover their overall behavior.

Show question

Question

What are some practical applications of calculus?

• Sciences - like biology, physics, and chemistry
• Technology - like computer science
• Engineering - like mechanical engineering
• Math - like statistics
• Medicine
• Economics
• Construction

Show question

Question

What is the absolute maximum of a function?

The absolute maximum of a function is the greatest output in its range.

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Question

What is the absolute minimum of a function?

The absolute minimum of a function is the least output in its range.

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Question

What is a relative maximum?

A relative maximum of a function is an output that is greater than the outputs next to it.

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Question

What is a relative minimum?

A relative minimum of a function is an output that is less than the outputs next to it.

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Question

Do all functions have an absolute maximum and an absolute minimum?

No. A function may keep increasing or decreasing so no absolute maximum or minimum is reached.

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Question

Where can you find the absolute maximum or the absolute minimum of a parabola?

At its vertex. If the parabola opens upwards it is a minimum. If a parabola opens downwards it is a maximum.

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Question

How can you identify relative minima and maxima in a graph?

The peaks of the graph are the relative maxima. The valleys are the relative minima.

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Question

What is a stationary point?

A point where the derivative (or the slope) of a function is equal to zero.

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Question

Apart from using the definition of a derivative, how can you prove the derivative of the tangent function?

Using the quotient rule and the derivatives of sine and cosine functions.

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Question

Can the denominator of a fraction be zero?

No, if the denominator of a fraction is zero, then it is undefined.

• BUT, the numerator of a fraction can be zero.

• Any fraction with a zero in the numerator is equal to zero, as long as zero is not in the denominator.

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Question

What is the key to adding/subtracting fractions?

When adding or subtracting fractions, the denominators must be equal.

• To add or subtract fractions, the Least Common Denominator (LCD) must be found.

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Question

What happens when a fraction is multiplied by its reciprocal?

Multiplying a fraction by its reciprocal always gives us 1, provided the numerator of the original fraction is not zero.

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Question

What does it mean to take an absolute value of a number or variable?

Taking the absolute value of a number (or variable) means turning a negative number (or variable) positive, and doing nothing to a positive number (or variable) or zero.

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Question

What does the endpoint of an angle on the unit circle give us?

The endpoint of an angle on the unit circle gives us, in order, the angle's cosine and sine values. The x-coordinate is the cosine value, and the y-coordinate is the sine value. We can remember the order by remembering x and y are in alphabetical order, just like cosine and sine.

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Question

What is a helpful mnemonic device to remember which functions are positive in what quadrants of the unit circle?

All Students Take Calculus

• All - All 6 trig functions are positive in the first quadrant.

• Students - only Sine (and its reciprocal, cosecant) are positive in the second quadrant.

• Take - only Tangent (and its reciprocal, cotangent) are positive in the third quadrant.

• Calculus - only Cosine (and its reciprocal, secant) are positive in the fourth quadrant.

Show question

Question

What are the 3 main ways to manipulate functions?

1. Function composition
2. Function transformations
3. Algebraic manipulation

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Question

What does function composition involve?

Function composition involves taking one function, plugging it into another function, and then solving, usually for a value of x.

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Question

How can a graph be transformed?

Horizontally and Vertically

Show question

Question

What are the 4 ways a graph can be horizontally and/or vertically transformed?

1. Shifts
2. Shrinks
3. Stretches
4. Reflections

Show question

Question

How do shifts, shrinks, stretches, and reflections work for horizontal transformations?

• Shifts - Adding a number to x shifts the function to the left, subtracting shifts it to the right.

• Shrinks - Multiplying x by a number greater than 1 shrinks the function.

• Stretches - Multiplying x by a number less than 1 stretches the function.

• Reflections - Multiplying x by -1 reflects the function horizontally (over the y-axis).

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Question

How do shifts, shrinks, stretches, and reflections work for vertical transformations?

• Shifts - Adding a number to the entire function shifts it up, subtracting shifts it down.

• Shrinks - Multiplying the entire function by a number less than 1 shrinks the function.

• Stretches - Multiplying the entire function by a number greater than 1 stretches the function.

• Reflections - Multiplying the entire function by -1 reflects it vertically (over the x-axis).

Show question

Question

How can functions be manipulated using algebraic manipulation?

Functions can be algebraically manipulated by:

• Adding, subtracting, multiplying, and/or dividing a value from a function.
• Adding, subtracting, multiplying, and/or dividing two or more functions.
• Taking a function to a power, root, log, etc.
• Simplifying an expression, equation, or function.

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Question

Which functions are inverses of logarithmic functions?

Exponential functions.

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Question

What is the differentiation rule of the natural logarithmic function $$\ln{x}$$?

$\frac{\mathrm{d}}{\mathrm{d}x} \ln{x} = \frac{1}{x}.$

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Question

What better describes a logarithmic function?

It is a slowly increasing function defined over the positive numbers.

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Question

Find the derivative of $$g(x) = \ln{\sqrt{x}}.$$

$$g'(x)= \frac{1}{2x}.$$

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Question

Which of the following is the derivative of $$f(x)= \log_{5}{x}$$?

$f'(x) = \left( \frac{1}{\ln{5}} \right) \left( \frac{1}{x} \right).$

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Question

How can you prove the derivative of the natural logarithmic function?

By using implicit differentiation and the differentiation rule for the exponential function.

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Question

Find the derivative of $$f(x)= \ln{x^3}.$$

$$f'(x) = \frac{3}{x}.$$

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Question

Find the derivative of $$g(x) = e^{\ln{x}}.$$

$$g'(x) = 1.$$

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Question

Which of the following is the derivative of $$h(x)= \log_{2}{x}$$ ?

$$h'(x) = \frac{1}{\ln{2}}\frac{1}{x}.$$

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Question

What is Logarithmic Differentiation?

Logarithmic Differentiation is a method used to find the derivative of a function using the properties of logarithms.

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