Calculus begins with two fundamental ideas:

Differentiation measures change.

Integration measures accumulation.

But calculus does not end there.

Once we understand derivatives and integrals, we can begin asking deeper questions:

How can we describe curves that cannot easily be written as ordinary functions?

How can we analyze motion more naturally?

How can we calculate areas involving complicated curves?

How can we work with infinity?

And perhaps most importantly:

How can a computer calculate complicated mathematical functions with remarkable accuracy when computers cannot actually handle infinity or infinitesimals?

These questions lead us to parametric curves, polar coordinates, improper integrals, infinite series, power series, and Taylor series.

This is the subject of Calculus 1C: Coordinate Systems & Infinite Series, the third part of MIT’s three-part introductory calculus sequence.

MIT’s Calculus 1C Course

MIT offers an online course covering these topics through MITx.

Course: Calculus 1C: Coordinate Systems & Infinite Series

The course is organized around three major areas:

Coordinate systems
Understanding infinity
Infinite series

Let’s explore why these subjects are so important.

1. Why Do We Need Different Coordinate Systems?

In elementary calculus, we usually describe a curve using

$y=f(x)$

This is an extremely useful representation.

But it is not always the most convenient one.

Imagine a planet moving around the Sun.

We could describe its horizontal and vertical position using x and y.

But planetary motion naturally involves distance and angle.

A different coordinate system can therefore make the mathematics much simpler.

This is one of the important ideas behind the development and application of calculus:

The right coordinate system can turn a difficult problem into a much simpler one.

2. Parametric Curves

Instead of describing a curve directly as

$y=f(x)$

we can introduce a parameter, usually represented by t:

$x=x(t)$ $y=y(t)$

Together, these equations describe a parametric curve.

The parameter might represent time.

This means that parametric equations can tell us not only what a curve looks like, but also how an object moves along the curve.

For example,

$x=\cos(t)$ $y=\sin(t)$

describes a circle.

As t changes, the point moves around the circle.

This makes parametric equations particularly useful for:

mechanics
physics
robotics
computer graphics
animation
engineering
planetary motion

3. Calculus with Parametric Curves

Once a curve is parameterized, we can apply calculus to it.

$x=x(t)$

and

$y=y(t)$

then

$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$

This allows us to calculate the slope of a parametric curve.

We can also calculate its length.

The arc length from t=a to t=b is

$L=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2},dt$

This gives calculus a way of measuring the length of complicated curved paths.

MIT’s Calculus 1C course specifically includes methods for parameterizing curves and using those parameterizations to solve problems, as well as computing arc length.

4. Polar Coordinates

Cartesian coordinates describe a point using (x,y).

Polar coordinates use a different description (r, θ).

Here:

r is the distance from the origin
θ is the angle

The two coordinate systems are connected by

$x=r\cos(\theta)$

and

$y=r\sin(\theta)$

while

$r^2=x^2+y^2$

Polar coordinates are particularly useful for circular and rotational problems.

For example,

$r=a$

represents a circle centered at the origin.

Some curves that look complicated in Cartesian coordinates become remarkably simple in polar coordinates.

5. Calculus in Polar Coordinates

Changing coordinate systems also changes the way we perform calculus.

For example, the area enclosed by a polar curve can be calculated using

$A=\frac{1}{2}\int_\alpha^\beta r^2,d\theta$

This is another example of an important mathematical principle:

We should choose the representation that matches the structure of the problem.

Instead of forcing every problem into Cartesian coordinates, we can change our perspective.

MIT’s course explicitly covers an introduction to polar coordinates and calculus performed in polar coordinates.

6. Now We Encounter Infinity

The next part of the journey takes us into a fascinating area of calculus:

infinity.

Consider

$\int_1^\infty\frac{1}{x^2},dx$

The interval extends forever.

How can we calculate an area that extends infinitely far?

We don’t treat infinity as an ordinary number.

Instead, we use a limit:

$\int_1^\infty\frac{1}{x^2},dx</h1> <p>\lim_{b\to\infty}\int_1^b\frac{1}{x^2},dx$

If the limit approaches a finite value, the improper integral converges.

If it does not, the integral diverges.

This gives us a rigorous mathematical way of dealing with infinity.

7. Singularities

Infinity can also appear because a function becomes unbounded near a particular point.

Consider

$\int_0^1\frac{1}{\sqrt{x}},dx$

At $x=0$ , the function is not defined.

Yet the total area can still be finite.

We handle this using a limit:

$\int_0^1\frac{1}{\sqrt{x}},dx</h1> <p>\lim_{a\to0^+}\int_a^1\frac{1}{\sqrt{x}},dx$

This illustrates a fascinating feature of calculus:

A function can become arbitrarily large near a point while the total accumulated area remains finite.

MIT’s Calculus 1C syllabus includes both improper integrals and singularities as part of its unit on understanding infinity.

8. Infinite Series

We now arrive at one of the most important ideas for computation.

Suppose we have a complicated function.

Instead of working with the function directly, can we approximate it using simpler functions?

Yes.

One powerful method is an infinite series.

For example:

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots$

There are infinitely many terms.

But we can stop after a finite number of terms:

$e^x\approx1+x+\frac{x^2}{2!}+\frac{x^3}{3!}$

The result is an approximation.

Adding more terms generally improves the approximation within the appropriate range.

9. How Can Computers Calculate So Accurately?

This brings us to the connection between calculus and computing.

A computer cannot literally perform an infinite number of operations.

It must eventually stop.

So how can it calculate values such as

$e^x$

$\sin(x)$

with very high accuracy?

One approach is to use mathematical approximations.

For example:

$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$

A computer can calculate a finite number of these terms.

If the approximation error is sufficiently small, the result can be accurate enough for the application.

This is a fundamental connection between pure mathematics and numerical computing.

MIT itself highlights this motivation in the course description: computers cannot handle infinitesimals or infinity directly, so functions can be approximated using series of simpler functions.

10. Convergence

But there is an important question.

If we keep adding terms to an infinite series, does the result actually approach a finite value?

Consider

$1+\frac12+\frac14+\frac18+\cdots$

The partial sums approach

$2$ .

Therefore, the series converges.

Now consider

$1+1+1+1+\cdots$

The partial sums become

$1,2,3,4,\ldots$

They do not approach a finite number.

The series diverges.

Understanding convergence is therefore essential before we trust an infinite series as a mathematical representation.

MIT’s Calculus 1C includes advanced convergence tests for series as well as power and Taylor series.

11. Power Series

A power series has the general form

$\sum_{n=0}^{\infty}c_n(x-a)^n$

or, written out,

$c_0+c_1(x-a)+c_2(x-a)^2+c_3(x-a)^3+\cdots$

This resembles a polynomial.

The major difference is that a polynomial has a finite number of terms, whereas a power series may have infinitely many.

Power series are extraordinarily useful because they allow complicated functions to be represented in terms of powers of $x$ .

12. Taylor Series

Taylor series take this concept even further.

For a function $f(x)$ around a point $a$ :

$f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\cdots$

The remarkable idea is that the function can be approximated using information about its derivatives at a single point.

For example:

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$

and

$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$

These representations transform complicated functions into combinations of powers that computers can evaluate.

13. Taylor Polynomials

An infinite Taylor series can be truncated to create a finite Taylor polynomial.

For example:

$\sin(x)\approx x-\frac{x^3}{6}+\frac{x^5}{120}$

This is no longer infinite.

It is an ordinary polynomial.

But near the point around which the Taylor expansion was constructed, it can provide an excellent approximation.

This is one of the central ideas behind numerical mathematics:

Replace a complicated problem with a simpler problem whose answer is sufficiently close.

14. Why Approximation Is So Important

In the real world, perfect precision is rarely available.

Measurements have limited accuracy.

Physical models are approximations.

Sensors have finite resolution.

Computers use finite numerical representations.

Therefore, practical mathematics is often about controlling approximation error rather than achieving absolute perfection.

Infinite series provide one of the most elegant mathematical frameworks for doing this.

We can decide how many terms are needed to achieve a desired level of accuracy.

15. Calculus Becomes a Language of Approximation

This gives us a broader understanding of calculus.

Differentiation can approximate a function locally using its tangent line:

$f(x)\approx f(a)+f'(a)(x-a)$

Taylor series extend this idea.

Instead of using only the first derivative, we can incorporate higher derivatives:

$f(x)\approx f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots$

So Taylor series can be viewed as a systematic extension of the local approximation idea introduced by derivatives.

16. The Three-Part Calculus Journey

The three courses now form a coherent story.

Calculus 1A — Differentiation

We study change.

$f'(x)$

Calculus 1B — Integration

We study accumulation.

$\int f(x),dx$

Calculus 1C — Coordinate Systems & Infinite Series

We learn how to describe complicated curves and approximate complicated functions.

$\sum_{n=0}^{\infty}a_n$

The progression is powerful.

First we learn to understand change.

Then we learn to understand accumulation.

Finally, we learn to change our mathematical perspective and approximate complicated objects.

17. From Newton to Modern Computing

The historical connection is particularly interesting.

Newton developed calculus partly in response to problems involving motion and planetary orbits.

Today, the same mathematical ideas have evolved into tools used throughout science and engineering.

Calculus is used in:

physics
astronomy
engineering
economics
computer graphics
machine learning
statistics
robotics
scientific computing
numerical simulation

The mathematical problems may have changed dramatically, but the underlying ideas remain remarkably similar.

18. The Most Important Lesson

Perhaps the most valuable lesson of Calculus 1C is not any individual formula.

It is the ability to change perspective.

A difficult curve may become simple after changing coordinates.

An infinite area may become manageable through a limit.

A complicated function may become computable through a series.

A difficult numerical problem may become easy after replacing it with a sufficiently accurate approximation.

In other words:

Mathematics becomes powerful when we choose the right representation of a problem.

That is the deeper purpose of coordinate systems and infinite series.

Explore the MITx Course

If you want to study these ideas systematically, MIT’s official Calculus 1C: Coordinate Systems & Infinite Series course is an excellent next step.

The course covers:

parametric curves
arc length
polar coordinates
calculus in polar coordinates
improper integrals
singularities
infinite series
convergence tests
power series
Taylor series
manipulation of power series

MIT lists Calculus 1A: Differentiation and Calculus 1B: Integration as prerequisites.

And it leads to one of the most beautiful ideas in mathematics:

Even when exact answers are difficult—or impossible—we can often construct approximations that are accurate enough to understand and work with the real world.

Calculus 1C: Coordinate Systems & Infinite Series — From Curves to Infinity

MIT’s Calculus 1C Course

1. Why Do We Need Different Coordinate Systems?

2. Parametric Curves

3. Calculus with Parametric Curves

4. Polar Coordinates

5. Calculus in Polar Coordinates

6. Now We Encounter Infinity

7. Singularities

8. Infinite Series

9. How Can Computers Calculate So Accurately?

10. Convergence

11. Power Series

12. Taylor Series

13. Taylor Polynomials

14. Why Approximation Is So Important

15. Calculus Becomes a Language of Approximation

16. The Three-Part Calculus Journey

Calculus 1A — Differentiation

Calculus 1B — Integration

Calculus 1C — Coordinate Systems & Infinite Series

17. From Newton to Modern Computing

18. The Most Important Lesson

Explore the MITx Course

Like this:

DavidsonNext: AP® Calculus: Challenging Concepts from Calculus AB & Calculus BC

Calculus 1A: Differentiation

Calculus 1B: Integration

Calculus 1C: Coordinate Systems & Infinite Series

MIT’s Calculus 1C Course

1. Why Do We Need Different Coordinate Systems?

2. Parametric Curves

3. Calculus with Parametric Curves

4. Polar Coordinates

5. Calculus in Polar Coordinates

6. Now We Encounter Infinity

7. Singularities

8. Infinite Series

9. How Can Computers Calculate So Accurately?

10. Convergence

11. Power Series

12. Taylor Series

13. Taylor Polynomials

14. Why Approximation Is So Important

15. Calculus Becomes a Language of Approximation

16. The Three-Part Calculus Journey

Calculus 1A — Differentiation

Calculus 1B — Integration

Calculus 1C — Coordinate Systems & Infinite Series

17. From Newton to Modern Computing

18. The Most Important Lesson

Explore the MITx Course

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Like this:

DavidsonNext: AP® Calculus: Challenging Concepts from Calculus AB & Calculus BC

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Calculus 1A: Differentiation

Calculus 1B: Integration

Calculus 1C: Coordinate Systems & Infinite Series