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MIT 18.01 Single Variable Calculus, lecture 1, explained in plain English

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Part 1 · 0:00–10:00

This part explains what a derivative is by finding the tangent line to a curve at one point, and says the real challenge is finding that line's slope with a method, not by eye.

Key ideas

The main problem is finding the tangent line to a graph of a function at a chosen point. A tangent line is a straight line that touches the curve at that point. 3:50
Any straight line through a point can be written as y minus y0 equals m times (x minus x0). Here m is the slope, which tells how steep the line is. 6:23
To write the tangent line's equation, you need two things: the point on the curve, which is the height f(x0), and the slope m. 7:07
The derivative at x0, written f'(x0), is defined as the slope of the tangent line at that point. 8:19
Drawing the line by eye is not enough. The lecture says the task is to find the slope in a way a machine could follow too. 9:01

Words to know

derivative: A number that tells you how steep a function's graph is at one specific point. It is the slope of the tangent line there.
tangent line: A straight line that just touches a curve at one point and goes in the same direction as the curve at that point.
slope: How steep a line is. It tells you how much the line goes up or down for each step sideways.
analytically: Solving a problem with a set of written steps or formulas, rather than by drawing or guessing from a picture.

Example

Imagine a ruler laid on a curved hill so that it touches the hill at just one spot and points the same way the hill slopes there. The ruler is like a tangent line, and how tilted the ruler is at that spot is like the slope, or derivative. The lecture does not use this example; it is only a way to picture the idea.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part shows how to find the slope of a curve at one point by drawing a line through two nearby points and then making those points closer together.

Key ideas

A tangent line touches a curve at one point. Crossing the curve twice does not make a line a tangent, so we need a different way to find it. 10:00
Take a line through two points P and Q on the curve, called a secant line. As Q moves closer to P, the slope of the secant gets closer to the slope of the tangent. 10:19
Measure the slope as the change in height divided by the change in horizontal distance. Write the horizontal distance as delta x and the height change as delta f. Then let delta x shrink toward 0. 12:42
Write the points with formulas: P is (x0, f(x0)) and Q is (x0 + delta x, f(x0 + delta x)). Plug these into the slope formula to get the derivative formula, which the lecture calls the most important formula today. 15:05
For the example f(x) = 1/x, plug the formula into the slope ratio and simplify with algebra. This ratio is called a difference quotient. 18:14

Words to know

secant line: A straight line that cuts through a curve at two points.
slope: How steep a line is. It is the rise divided by the run, or how much the height changes for each step to the side.
difference quotient: A fraction where the top is a difference in heights and the bottom is a difference in positions. It gives the slope between two points.
delta x: A small horizontal distance, read as 'change in x'. The Greek letter delta means a change.

Example

Imagine walking up a hill and wanting to know how steep it is right where you stand. You check the slope between your spot and a point a few steps ahead, then a point closer, then closer still. The steepness you get as the second point gets closer to you is the steepness at your exact spot.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part shows how to find the exact slope of the curve y = 1/x at any point, and then starts a word problem about triangles formed by a tangent line.

Key ideas

After some cancelling, the difference quotient becomes -1 over (x_0 + delta x) times x_0. Cancelling is what lets us avoid dividing zero by zero. 20:17
Once delta x is cancelled out of the problem, we can set delta x to 0 and get the slope: f'(x_0) = -1/x_0^2. The word 'limit' means this value is what we get as delta x gets closer and closer to 0. 21:03
We can check the answer with the graph. The slope is always negative, meaning the curve always slopes down. As x gets larger, the slope gets closer to 0, so the curve gets flatter. 21:55
To test the limit idea with a number, the teacher uses x_0 = 3. Setting delta x to 0 gives -1/9, which matches the general formula. 23:52
The word problem asks for the area of a triangle made by the two axes and a tangent line to the curve y = 1/x. Finding the tangent line needs calculus, but the teacher says the calculus part is the easy part. 28:12

Words to know

limit: The value a expression gets closer and closer to as a small number, like delta x, gets closer and closer to 0.
numerator: The top number in a fraction. In this part, the top of the fraction is the change in height over the change in x.
tangent line: A straight line that just touches a curve at one point and has the same slope as the curve at that point.
hyperbola: A curve with two smooth halves that get closer and closer to the axes without touching them. The graph of y = 1/x is one example.

Example

Think of driving up or down a hill. The slope tells you how steep the road is at one exact spot, not over a long stretch. A limit is like zooming in closer and closer on that spot until the road looks like a straight line.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part shows how the lecturer finds where a tangent line crosses the two axes, and why the letters in the algebra can be easy to mix up.

Key ideas

Finding the formula for the tangent line is the only real calculus step in this problem. After that, the rest is algebra using the slope. 30:21
To find the x-intercept, where the tangent line crosses the horizontal axis, you set y = 0 and solve. The lecturer gets the point x = 2x_0. 33:39
The y-intercept can be found with a shortcut. Because the curve y = 1/x is symmetric, you can swap x and y in the formulas, which gives 2y_0 for the y-intercept. 34:27
Letters like y are used loosely in this kind of work. The same letter can mean different things in different steps, so you must check where each formula sits on the diagram. 39:25

Words to know

tangent line: A straight line that touches a curve at one point and has the same slope as the curve at that point.
x-intercept: The point where a line or curve crosses the horizontal x-axis. At that point, y is 0.
symmetry: A balance or matching shape. Here, swapping x and y in a formula gives another correct formula, like a mirror image across a line.
variable: A letter, such as x or y, that stands for a number that can change or that we do not know yet.

Example

Think of the word 'bank', which can mean a place for money or the side of a river. The sentence tells you which meaning is correct. In this lesson, the letter y can mean the height of the curve in one step and the height of the horizontal line y = 0 in another, so you have to check the diagram to know which one is meant.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–51:32

This part shows how to find the derivative of x to the power n, using a simple algebra trick, and how to extend that rule to polynomials.

Key ideas

The lecturer introduces other ways to write a derivative. Newton's f prime, Leibniz's df/dx and dy/dx all mean the same thing, and delta y means the same as delta f. 41:51
Some notations leave out information, such as the base point x_0. You have to fill in that missing detail from context. 43:03
To find the derivative of x^n, you put x fixed and let delta x move. Then you expand (x + delta x)^n using the binomial theorem, which gives x^n plus n times x^(n-1) times delta x, plus smaller junk terms. 45:40
Dividing by delta x leaves n times x^(n-1) plus junk that still has at least one factor of delta x. As delta x goes to 0, the junk disappears, so the derivative of x^n is n times x^(n-1). 49:10
The power rule works term by term on polynomials. For example, the derivative of x^3 + 5x^10 is 3x^2 + 50x^9. 49:53

Words to know

derivative: The rate at which a function changes. In this lesson it is found by forming delta f divided by delta x and seeing what happens as delta x gets very small.
binomial theorem: A formula for expanding something like (x + delta x) raised to the power n into separate terms. The lecturer only needs the first two terms.
big O: A short notation for the leftover 'junk' terms. O(delta x)^2 means terms with delta x squared or higher powers, which become negligible.
limit: The value something gets closer and closer to as a small number, here delta x, shrinks toward 0.

Example

Think of a car's distance over time. Its speed at one instant is found by looking at how much distance changes over a tiny stretch of time, then letting that stretch shrink toward zero. This is the same idea as delta f divided by delta x shrinking toward zero.

3 checked practice questions for this part are in the app.

Lecture 3: Derivatives →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

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