1 2 3 4 5 6 7 8

0:008 lessons across the whole video36:21

You will be able to

Evaluate a limit by factoring, canceling, and direct substitution
Apply the power rule to find the derivative of a polynomial term
Explain how a derivative gives the slope of a tangent line and estimate it with a secant line
Derive the slope of a tangent line using the limit definition of the derivative
Find an antiderivative and evaluate a definite integral to get accumulated change
Interpret derivatives as instantaneous rates of change and integrals as area under a curve

Good to know first

Basic algebra, including evaluating functions and simplifying expressions
Factoring, including difference of squares
Slope as rise over run
Exponent rules, such as raising a power to a power and x^0 = 1

Syllabus

8 lessons and a final project

1
The three big ideas of calculus: limits, derivatives, integrals Free
0:01–2:292 min of video
This lesson introduces the three core ideas of calculus: limits, derivatives, and integrals. You will see what each one measures and how derivatives and integrals undo each other.
3 formulas2 exercises4 questions
2
Evaluating limits by factoring and substitution Beyond Pass
2:29–5:423 min of video
You learn what a limit is: the value a function approaches as x gets close to a number, even when the function is undefined there. You learn to evaluate limits by factoring away the problem term and then substituting.
3
The Power Rule and the Slope of a Tangent Line Beyond Pass
5:42–8:443 min of video
You will learn the power rule for finding the derivative of x raised to a power, and how the derivative gives the slope of a tangent line at a point. You will practise both by applying the rule and reading the slope as a rate of change.
3 build steps3 exercises4 questions
4
Estimating and computing a tangent slope with secants and limits Beyond Pass
8:44–15:487 min of video
You will estimate the slope of a tangent line by computing slopes of secant lines between points that get closer and closer to the point of interest. Then you will compute the exact tangent slope with a limit, using factoring to cancel the problem term.
1 code3 build steps2 exercises3 questions
5
Integration as the Antiderivative Beyond Pass
15:48–18:223 min of video
Integration is the reverse of differentiation: you start from a derivative and find the function that produced it. The power rule for integrals, with its "+ C", lets you do this for any power of x.
2 exercises3 questions
6
Instantaneous rate of change in a water tank Beyond Pass
18:22–26:158 min of video
You learn how to find the instantaneous rate of change of a quantity by taking the derivative, using the power rule and the rule for constants. You then check the answer by approximating the tangent slope with a secant slope from a table of values.
2 exercises3 questions
7
Definite integrals: finding accumulated change and area under a curve Beyond Pass
26:15–34:258 min of video
A definite integral with a lower and upper limit gives a single number: the total accumulated change over an interval. Geometrically, that number is the area under the rate curve between those limits, which you can check by splitting the area into a rectangle and a triangle.
6 build steps2 exercises4 questions
8
Recap: Limits, Derivatives, and Integrals Beyond Pass
34:25–36:212 min of video
This recap ties together the three pillars of calculus: limits, derivatives, and integrals. You will see how each one answers a different question about how a function behaves.
2 exercises3 questions

Final project

The Filling Tank: Limits, Derivatives and Integrals in One Model

A tank holds $W (t) = 0.03 t^{2} + 0.2 t + 50$ gallons after $t$ minutes. You will study it with all three ideas of calculus. Use limits and secants to estimate the fill rate, the power rule to get it exactly, and a definite integral to recover the total change in water.

Its 7 tasks open with Beyond Pass.

Free preview · Lesson 1 in full

The three big ideas of calculus: limits, derivatives, integrals

This lesson introduces the three core ideas of calculus: limits, derivatives, and integrals. You will see what each one measures and how derivatives and integrals undo each other.

0:01 to 2:29 in the video

Scratchpad

Draw and write next to the video.

The idea

Calculus is built on three core ideas. The first is the limit. A limit tells you what a function is heading toward as x gets close to some value, even when the function is undefined right at that value or hard to evaluate directly. Instead of asking "what is f(2)?", a limit asks "what happens to f(x) as x approaches 2?"

The second idea is the derivative. A derivative is itself a function, written f'(x) (read "f prime of x"), made from an original function f(x). The value of f'(x) at any x gives the slope of the tangent line to f(x) at that point. Because slope measures how fast one quantity changes with another, derivatives are the tool for calculating rates of change.

The third idea is integration. Integration is the opposite of differentiation: instead of finding slopes, it finds the area under a curve. Integration is useful for figuring out how much something accumulates over time, such as the total distance travelled when you know your speed at every moment.

The key link is that derivatives and integrals undo each other. If you take the derivative of f(x) you get f'(x). If you then integrate f'(x), you get back f(x) (plus a constant, which the video does not discuss here). So integration can be thought of as finding the antiderivative, the function whose derivative is the one you started with.

To remember the chapter in one line: limits look at what happens near a point, derivatives measure slope and rate of change, and integrals measure area and accumulation.

Key points

A limit asks what a function approaches as x gets close to a value, which is useful when the function is undefined at that exact value. 0:12
A derivative is a function, written f'(x), that gives the slope of the original function at each x. 0:45
Derivatives are the tool for rates of change, because slope tells how fast something changes. 0:47
Integration finds the area under a curve and measures how much something accumulates over time. 1:29
Derivatives and integrals are opposites: integrating f'(x) gives back f(x), so integration finds the antiderivative. 2:14

Formulas

f^{'} (x) = \frac{d}{d x} f (x)

The derivative of f at x is the function f prime of x, which gives the slope of the tangent line of f at x.

$f (x)$: the original function
$f^{'} (x)$: the derivative of f, a function giving the slope at each x
$x$: the input value

\int f^{'} (x) d x = f (x)

The integral of the derivative of f gives back f, so integration undoes differentiation.

$\int$: the integral sign, meaning integrate
$f^{'} (x)$: the derivative being integrated
$d x$: the variable we integrate with respect to
$f (x)$: the antiderivative that comes back

\lim_{x \to 2} f (x)

The limit of f(x) as x approaches 2: the value f(x) gets close to as x gets close to 2.

$\lim$: the limit operator
$x \to 2$: x approaches 2 (it need not equal 2)
$f (x)$: the function whose behaviour we study

Practice

Explain Solution checked

In your own words, explain the difference between a derivative and an integral, using the words slope and area.

Solution

A derivative takes a function f(x) and gives a new function f'(x) whose value at each x is the slope of f at that point.
An integral takes a function and gives the area under its curve.
Because they are opposites, integrating f'(x) returns f(x).

A derivative finds the slope (rate of change) of a function at each point, while an integral finds the area under a curve; they undo each other, so integrating a derivative gives back the original function.

Explain Solution checked

Why might a limit be needed to study f(x) at x = 2 when f(2) cannot be computed directly? Give a model answer in your own words.

Solution

A limit looks at the values of f(x) for x near 2, not at x = 2 itself.
If those values settle toward one number, that number is the limit, and it describes what f does around 2.

A limit lets us see what f(x) approaches as x gets close to 2, even if f is undefined exactly at 2, so we still learn how the function behaves near that point.

On your scratchpad

Draw a simple curve going up, mark one point on it, and draw a short straight line touching the curve at that point. Label the touching line as the slope (derivative) and shade the region under the curve between two x values as the area (in

Quiz

What does a limit ask you to find?
1. The exact value of f at x = 2, even if f is undefined there
2. What value f(x) approaches as x gets close to 2
3. The area under the curve of f from 0 to 2
4. The slope of f at x = 2 only
Show the answer
B. What value f(x) approaches as x gets close to 2 · The video says a limit shows what happens as x approaches 2, which is useful when the function is undefined or hard to evaluate directly.
0:19 watch the moment
What does the derivative f'(x) give you?
1. The area under f(x)
2. The slope of the tangent line to f(x) at x
3. The limit of f(x) as x goes to infinity
4. The value of f(0)
Show the answer
B. The slope of the tangent line to f(x) at x · The video says the derivative function gives the slope of the tangent line at some x value, which is used for rates of change.
0:56 watch the moment
What is the relationship between integration and derivatives?
1. They are the same operation
2. Integration is the opposite: integrating f'(x) gives back f(x)
3. Integration gives slopes and derivatives give areas
4. They have no connection
Show the answer
B. Integration is the opposite: integrating f'(x) gives back f(x) · The video says derivatives and integration are opposites; the integral of f prime of x is f of x, so integration finds the antiderivative.
2:14 watch the moment
What does integration measure?
1. The slope of a curve
2. The limit of a function
3. The area under the curve, which relates to accumulation
4. The value of a function at a single point
Show the answer
C. The area under the curve, which relates to accumulation · The video says integration is finding the area under the curve and is useful for calculating accumulation over time.
1:42 watch the moment

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Lessons 2–8 and the final project

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Made by Beyond from the video “Understand Calculus in 35 Minutes” by The Organic Chemistry Tutor. The video plays in YouTube's own player. Watch it on YouTube · All lessons

8 lessons and a final project

The three big ideas of calculus: limits, derivatives, integrals Free

Evaluating limits by factoring and substitution Beyond Pass

The Power Rule and the Slope of a Tangent Line Beyond Pass

Estimating and computing a tangent slope with secants and limits Beyond Pass

Integration as the Antiderivative Beyond Pass

Instantaneous rate of change in a water tank Beyond Pass

Definite integrals: finding accumulated change and area under a curve Beyond Pass

Recap: Limits, Derivatives, and Integrals Beyond Pass