Teaching / Calculus · Open
Study in sequence
Move through these units in order. Each idea depends on the previous one; being able to explain why a procedure works matters more than applying it quickly.
Read a function as a relationship between varying quantities. Move between formulas, graphs and verbal descriptions.
Describe what a function approaches, including cases where direct substitution does not reveal the behavior.
Measure instantaneous change, connect slopes to rates and use derivatives to study the shape of a function.
Accumulate infinitely many small contributions and connect total change back to the derivative.
Three essential ideas
A limit describes the behavior of f(x) as x gets arbitrarily close to a value. The function may be undefined at that exact point and still have a limit there.
Why? For x ≠ 2, factor the numerator: (x − 2)(x + 2)/(x − 2) = x + 2. Values near 2 therefore approach 4.
The derivative is the limit of average rates of change over intervals that shrink toward zero. Geometrically, it is the slope of the tangent line.
Example. If f(x) = x², expanding (x+h)² and simplifying leaves 2x + h. As h approaches 0, f′(x) = 2x.
A definite integral is the limit of sums of many thin contributions. The Fundamental Theorem of Calculus says differentiation and integration undo one another.
Example. ∫03 2x dx = [x²]03 = 9. It is both the area under y = 2x and the accumulated change of x².
Try before revealing
Write the relevant definition or theorem before computing. The answer is less valuable than a solution another person can follow.
Factor x² − 9 = (x − 3)(x + 3). For x ≠ 3 the expression equals x + 3, whose limit at 3 is 6.
For x ≠ 1, f(x) = x + 1. Its limit as x→1 is 2, so continuity requires k = 2.
Apply linearity and the power rule: f′(x) = 9x² − 5. The constant contributes zero.
f′(x)=2x−4, so the only critical point is x=2. Since f″(x)=2>0, it is a minimum. Its value is f(2)=−3.
An antiderivative of x1/2 is (2/3)x3/2. Evaluation gives (2/3)(8−1) = 14/3.
Displacement is ∫03(3t²−6t)dt = [t³−3t²]03 = 27−27 = 0. It moved, but ended at its initial position.
Use after understanding