You will be able to
- Evaluate limits by direct substitution and by factoring to remove a 0/0 form
- Apply the power rule to differentiate polynomial functions
- Interpret a derivative as the slope of a tangent line and an instantaneous rate of change
- Estimate a tangent slope with a secant line and confirm it with the limit definition of the derivative
- Compute antiderivatives and definite integrals to find accumulated change
- Calculate accumulated change as the area under a rate-of-change graph using geometry
Good to know first
- Basic algebra, including factoring differences of squares and cubes
- Working with exponents and evaluating polynomials
- Computing the slope of a line from two points (rise over run)
- Basic geometry formulas for rectangles and triangles
Syllabus
8 lessons and a final project
- 1
Overview of Limits, Derivatives, and Integrals
This lesson gives a first map of calculus: the three main areas are limits, derivatives, and integrals. You will learn what each one asks about a function, so you know which question to use for which job.
3 exercises3 questions - 2
Evaluating limits and handling 0/0 by factoring
You learn why plugging in a value can give 0/0 and that the function may still have a limit there. You learn to factor, cancel the common factor, and then substitute to find the limit.
4 build steps3 exercises3 questions - 3
The power rule and the tangent line as a slope
You learn the power rule for differentiating x raised to a constant power. You also learn that the derivative gives the slope of the tangent line, and you check this with f(x) = x^3 at x = 2.
3 build steps3 exercises3 questions - 4
Secant approximations and the limit definition of the derivative
You will see how the slope of a secant line between two points close to x = 2 approximates the slope of the tangent line there. Then you will turn that idea into a limit that gives the exact slope, using f(x) = x^3 as the example.
3 formulas3 exercises4 questions - 5
Integration as the Antiderivative
Integration is the reverse of differentiation: it finds a function whose derivative is the given one. You learn the power rule for integrals and why the constant of integration +C is always needed.
2 formulas4 exercises3 questions - 6
Finding a rate of change for a water tank
You learn how to read a water-tank function A(t) as an amount over time, and how to use its derivative A'(t) to find how fast the water is changing at one moment. You also check the answer with a secant line between two nearby times.
4 build steps3 exercises3 questions - 7
Definite integrals: accumulated change under a curve
A definite integral with lower and upper limits gives a number: the net change accumulated over an interval, which equals the area under a rate-of-change curve. You can compute it with an antiderivative or check it with geometry.
2 formulas3 exercises3 questions - 8
The three core ideas of calculus: limits, derivatives, integration
This review recaps the three pillars of calculus: limits, derivatives and integrals. It shows how each one answers a different question, and how together they make up calculus.
4 exercises3 questions
Final project
Water Tank Calculus Report
Model a water tank with a function and use all three ideas of calculus on it: limits, derivatives and integrals. You will write a short report that shows how fast the tank is filling at a moment and how much water it gains over a period. You will check each answer in two ways.
Free preview · Lesson 1 in full
Overview of Limits, Derivatives, and Integrals
This lesson gives a first map of calculus: the three main areas are limits, derivatives, and integrals. You will learn what each one asks about a function, so you know which question to use for which job.
Draw and write next to the video.
The idea
Calculus is the mathematics of change. It is built on three main areas: limits, derivatives, and integrals. Think of them as three different questions you can ask about a function f(x).
The first area is limits. Sometimes you want f(2), but the function is undefined at 2, so you cannot compute it directly. A limit lets you see what f(x) gets closer and closer to as x gets closer and closer to 2. So a limit tells you what a function approaches near a value.
The second area is derivatives. A derivative is itself a function. If you start with f(x), its derivative, written f'(x) and read 'f prime of x', gives the slope of f at any x value. That slope is the slope of the tangent line, the line that just touches the curve at that point.
Slope measures how fast something changes. So derivatives are the tool for rates of change, such as how fast a car's position changes at one moment.
The third area is integrals. An integral adds up small pieces of a quantity over an interval, so it tells you how much has built up in total. You will meet the details later in the course.
In short: limits tell you what a function approaches, derivatives tell you how fast it changes at a point, and integrals tell you how much accumulates. Keep these three questions in mind as you learn the details.
Key points
- Calculus has three main areas: limits, derivatives, and integrals. 0:01
- A limit shows what f(x) approaches as x gets close to a value, even when f is undefined there. 0:19
- A derivative f'(x) is a function that gives the slope of f(x) at each x. 0:56
- The slope of f at a point is the slope of the tangent line at that point. 1:08
- Derivatives are used for rates of change. 1:21
- Integrals are the third area: they add up quantities over an interval to give a total.
Words to know
- Limit
- The value that f(x) gets closer and closer to as x gets close to a given number.
- Derivative
- A function f'(x) that gives the slope of f(x) at each x; it measures the rate of change.
- Tangent line
- A line that just touches a curve at one point and has the same slope as the curve there.
- Integral
- A tool that adds up small pieces of a quantity over an interval to find the total amount.
Practice
In your own words, say which of the three areas (limits, derivatives, integrals) you would use to find how fast a car's position is changing at one moment, and why.
Hint
Think about which area gives the slope of a function at a value.
Solution
- The question asks for a rate of change at a moment.
- A rate of change at a point is a slope, which is what a derivative gives.
- So use derivatives.
Derivatives, because a derivative gives the slope of the position graph at a moment, which is the rate of change (speed) at that moment.
A function f(x) = (x^2 - 4)/(x - 2) is undefined at x = 2. Which area of calculus helps you see what value the function approaches near x = 2, and what is the name for that idea?
Hint
Look for the area that handles 'what happens near a value' rather than 'at a value'.
Solution
- The function cannot be evaluated directly at x = 2.
- Limits show what a function approaches as x gets close to a value.
- So the area is limits.
Limits: the limit of f(x) as x approaches 2 shows the value the function approaches near 2.
You know a car's speed at every moment during a 10-second trip. Which area would you use to find the total distance it travelled? Which area would you use to find how quickly its position is changing at second 4?
Hint
One question is about a total that builds up; the other is about a slope at one moment.
Solution
- A total that builds up over an interval is what integrals find.
- How fast something changes at one moment is a slope, which is what derivatives find.
Total distance over the 10 seconds: integrals. How quickly position changes at second 4: derivatives.
On your scratchpad
Draw a simple curve f(x) and mark a point on it. Draw the tangent line touching the curve at that point, and label its slope. Then mark a gap or hole in the curve and write 'limit' there.
Quiz
What does a limit help you find?
- The exact value of f(x) at every x
- What f(x) approaches as x gets close to a value
- The area under the curve
- The slope of a line at x = 0 only
Show the answer
B. What f(x) approaches as x gets close to a value · A limit shows what happens to the function as x approaches a value, even if the function is undefined there. (Video at 0:19-0:45.)
0:19 watch the momentWhat is f'(x), the derivative of f(x)?
- The area under f(x)
- A function that gives the slope of f(x) at each x
- The value of f(0)
- The limit of f(x) as x goes to infinity
Show the answer
B. A function that gives the slope of f(x) at each x · The derivative is a function that gives the slope of the original function (the tangent line slope) at any x. (Video at 0:51-1:21.)
0:56 watch the momentWhy are derivatives useful?
- They compute areas
- They calculate rates of change
- They remove undefined points
- They always give whole numbers
Show the answer
B. They calculate rates of change · Since a derivative is a slope, it measures how fast something changes, so it is used for rates of change. (Video at 1:15-1:25.)
1:21 watch the moment
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