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Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007

The video “Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007” by MIT OpenCourseWare as 6 lessons, with the idea explained, steps to try in the practice terminal, practice problems with solutions and a quiz that leads back to the moment.

6lessons
9formulas
14exercises
18questions
52min of video
Open the lessons →Lesson 1 is free · Beyond Pass opens all
Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007, video by MIT OpenCourseWare▶ 51:32 · MIT OpenCourseWare
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0:006 lessons across the whole video51:32

You will be able to

  • Explain calculus as the study of rates of change and accumulation
  • Compute the slope of a secant line between two points on a curve
  • Approximate the slope of a tangent line using a limit of secant slopes
  • Write the limit definition of the derivative and apply it to simple functions
  • Interpret a derivative as an instantaneous rate of change

Good to know first

  • Basic algebra, including simplifying expressions and solving equations
  • Functions and their graphs
  • The slope of a straight line

Syllabus

6 lessons and a final project

  1. 1

    Course overview: what a derivative is and why it matters Free

    0:00–5:005 min of video

    This chapter introduces the course's first topic, the derivative, and gives its two main interpretations: geometric (slope of a tangent line) and physical (rate of change). It also shows why derivatives matter across science, engineering, economics and political science.

    2 exercises3 questions
  2. 2

    Change and accumulation: the two big ideas of calculus Beyond Pass

    5:00–13:208 min of video

    Calculus studies two linked questions: how fast something changes (the derivative, seen as the slope of a tangent line) and how much accumulates (the integral). This lesson introduces the geometric picture of the tangent line and the idea that it is the limit of secant lines.

    3 build steps2 exercises3 questions
  3. 3

    Slope of a secant line and the derivative of 1/x Beyond Pass

    13:20–23:2010 min of video

    The slope of a secant line between two points on a graph is the change in f divided by the change in x. Letting the gap Δx shrink to zero turns this secant slope into the slope of the tangent line, and we work this out for f(x) = 1/x.

    3 formulas3 exercises3 questions
  4. 4

    Tangent lines and the limit idea Beyond Pass

    23:20–33:2010 min of video

    A tangent line touches a curve at one point and has the slope the curve has there. We get it by taking the slope of a secant line through two points and letting the second point slide toward the first, which leads to the limit definition of the derivative.

    3 exercises3 questions
  5. 5

    The derivative as a slope: tangent lines and notation Beyond Pass

    33:20–43:2010 min of video

    The derivative of a function at a point is the slope of the tangent line there, found by a limit of secant slopes. This lesson works through the tangent to y = 1/x, uses symmetry as a shortcut to find where the tangent meets an axis, and introduces the notations f', dy/dx and df/dx.

    2 formulas2 exercises3 questions
  6. 6

    Summary and preview: the derivative of xⁿ and what comes next Beyond Pass

    43:20–51:328 min of video

    This lesson wraps up the first lecture of MIT 18.01 by recalling the geometric meaning of the derivative as a tangent-line slope. It then shows, using the binomial theorem, why the derivative of xⁿ is n·x^(n-1), and how the rule extends to polynomials.

    4 formulas2 exercises3 questions

Final project

Chasing the Tangent: A Derivative Investigation

You will study the derivative of a function from several angles. You will use secant slopes, the limit definition, and a rule, then check that they agree. You will do this for f(x)=1/x and for f(x)=xn, and write a short report that explains what the derivative means.

Its 6 tasks open with Beyond Pass.

Free preview · Lesson 1 in full

Course overview: what a derivative is and why it matters

This chapter introduces the course's first topic, the derivative, and gives its two main interpretations: geometric (slope of a tangent line) and physical (rate of change). It also shows why derivatives matter across science, engineering, economics and political science.

0:00 to 5:00 in the video

Scratchpad

Draw and write next to the video.

The idea

This opening chapter of MIT 18.01 Single Variable Calculus sets the stage. The course is organised into units, and Unit I is about differentiation. The lecturer writes the heading '18.01 UNIT I DIFFERENTIATION' on the board, then asks the first question of the unit: 'What is a derivative?'

The video says a derivative has two interpretations you should keep in mind from the start. The first is the geometric interpretation: the derivative of a function tells you the slope of its graph, and specifically the slope of the tangent line at a point. The second is the physical interpretation: the derivative tells you how fast a quantity is changing. Both views describe the same object, seen from two angles.

Why does it matter? The lecturer points out that derivatives are important to all measurements in science, engineering, economics and political science. Whenever something changes and you want to measure how fast, you are doing differentiation.

Here is the geometric idea in the form the lecture starts with. Suppose you have a curve y=f(x) and a point P=(x0,y0) on it. The question is: find the tangent line to the curve at P. A tangent line is the straight line that just touches the curve at P and has the same direction as the curve there. Its slope is the derivative at x0.

Keep the two ideas together. On a graph, the derivative is a slope: steep means changing fast, flat means changing slowly, zero means momentarily not changing. In a physical situation, the same number is a rate: how many litres per minute, how many dollars per year, how many metres per second. Learning to move between these two pictures is the core skill of Unit I.

Key points

  • The course is organised into units; Unit I is Differentiation, and the first topic is 'What is a derivative?' 1:00
  • A derivative has two interpretations: geometric (slope of the tangent line) and physical (rate of change). 2:31
  • Derivatives matter for all measurements in science, engineering, economics and political science. 2:31
  • Geometric interpretation: to find the tangent line to y = f(x) at a point P = (x0, y0) on the curve. 4:00
  • The tangent line touches the curve at P and has the curve's direction there; its slope is the derivative at x0. 4:41

Practice

Calculate Solution checked

A tangent line passes through the points (1, 2) and (3, 6). What is its slope?

Hint

Slope is rise over run: change in y divided by change in x.

Solution
  1. Slope = (6 - 2) / (3 - 1).
  2. = 4 / 2 = 2.

2

Explain Solution checked

In your own words, explain why the derivative can be read as both a slope and a rate of change.

Hint

Think about a graph of distance against time, and what a steep part of the graph means.

Solution
  1. On a graph of y = f(x), the derivative at a point is the slope of the tangent there, so it measures how steeply the graph rises or falls.
  2. If x is time and y is a quantity, then slope = change in y per change in x, which is a rate: how fast y changes.
  3. So the same number is a geometric slope and a physical rate.

The derivative is the slope of the tangent line, and when the axes are time and some quantity, that slope is the rate at which the quantity changes.

On your scratchpad

On your scratchpad, draw a curve y = f(x), mark a point P on it, and sketch the tangent line touching the curve at P. Label the slope and write one sentence saying what that slope means physically.

Quiz

  1. According to the chapter, what are the two interpretations of a derivative?

    1. Geometric (slope of the tangent) and physical (rate of change)
    2. Algebraic and numerical
    3. Integral and area
    4. Discrete and continuous
    Show the answer

    A. Geometric (slope of the tangent) and physical (rate of change) · The board lists 'geom. interp.' and 'physical interp.' under 'What is a derivative?'

    2:31 watch the moment

  2. At the point P = (x0, y0) on y = f(x), what does the geometric interpretation ask you to find?

    1. The area under the curve
    2. The tangent line to the curve at P
    3. The x-intercept of the curve
    4. The maximum value of f
    Show the answer

    B. The tangent line to the curve at P · The board reads 'Find the tangent line to y = f(x) at P = (x0, y0)'.

    4:41 watch the moment

  3. Which fields does the lecturer say derivatives are important to?

    1. Only physics
    2. Science, engineering, economics and political science
    3. Only mathematics majors
    4. Only computer graphics
    Show the answer

    B. Science, engineering, economics and political science · The slide lists sci, eng, econ, polit sci under 'importance to all measurements'.

    2:31 watch the moment

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Lessons 2–6 and the final project

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