You will be able to
- Evaluate a limit by factoring, cancelling, and direct substitution, and explain what it means for a function to approach a value.
- Differentiate polynomial functions with the power rule and interpret the derivative as the slope of a tangent line.
- Estimate a tangent slope with secant lines and confirm it with the limit definition of the derivative.
- Find an antiderivative with the power rule and state the difference between indefinite and definite integrals.
- Compute the net change of a quantity from a rate of change using a definite integral or the area under a graph.
- Interpret derivatives as instantaneous rates of change and integrals as accumulated totals in word problems.
Good to know first
- Basic algebra, including factoring differences of squares and cubes
- Evaluating functions and exponents, including square and cube roots
- Computing slope as rise over run between two points
- Basic geometry: areas of rectangles and triangles
Syllabus
8 lessons and a final project
- 1
Overview: limits, derivatives, and integrals Free
This lesson gives a 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 tool to reach for later.
- 2
Limits: what a function approaches near a value Beyond Pass
A limit describes the value a function gets closer and closer to as x approaches a number, even when the function is undefined at that number. You can find it by testing nearby values or by simplifying the expression and substituting.
2 exercises3 questions - 3
The power rule and tangent lines Beyond Pass
You will learn the power rule, which finds the derivative of x raised to a constant power in one step. You will also see how the derivative gives the slope of a tangent line at a point.
3 build steps4 exercises3 questions - 4
Slope from secant lines and the limit definition Beyond Pass
We estimate the slope of a tangent line at x = 2 using secant lines through points on either side of 2, watch the estimates get closer to 12, and then get the exact value with a limit. The limit of the difference quotient is the same idea that leads to the derivative.
2 exercises3 questions - 5
Integration as the antiderivative Beyond Pass
Integration is the reverse of differentiation: you find a function whose derivative is the one you started with. You learn the power rule for integrals, why a constant C is always added, and how integrals (areas) differ from derivatives (slopes).
2 formulas3 build steps3 exercises3 questions - 6
Rates of change: finding how fast a water tank fills with a derivative Beyond Pass
You learn to use a derivative to find how fast a quantity changes at one instant, using a water-tank function. You also check the answer with a secant-line slope and a table of values.
1 code4 build steps4 exercises4 questions - 7
Accumulation: the definite integral of a water flow rate Beyond Pass
A definite integral of a rate gives the total change over an interval, here the gallons of water added to a tank. You can check the integral by finding the area under the rate graph with geometry.
3 formulas3 exercises4 questions - 8
Review and next steps: limits, derivatives and integration Beyond Pass
This review shows how the three big ideas of calculus fit together: limits, derivatives and integration. You will learn what each one measures and how the slope of a tangent line connects to the slope of a secant line.
2 build steps3 exercises3 questions
Final project
The Water Tank Report: One Tank, Three Big Ideas
Build a short analysis of a water tank with volume gallons after minutes. You will use limits, derivatives and integrals on the same tank so you can see how they fit together. The result is a one-page report with a small Python check.
Its 7 tasks open with Beyond Pass.
Free preview · Lesson 1 in full
Overview: limits, derivatives, and integrals
This lesson gives a 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 tool to reach for later.
Draw and write next to the video.
On your scratchpad
Draw a simple curve. Mark a point x = 2 on the horizontal axis. Draw a small dot at the spot where the curve should be, then draw a short line (tangent) touching the curve there. Label one arrow "limit: what f approaches" and the tangent li
Lessons 2–8 and the final project
They open with Beyond Pass: every lesson of every course, in full. One payment: $14.99 for 90 days or $29 for a year. No subscription.
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