Week 4: Introduction to Calculus

CALCULUS

The Concept of a Limit

Calculus is the study of change. At its core lies the Limit.

Approaching... but never touching.

💡 Definition

We say $\lim_{x \to c} f(x) = L$ if we can make $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $c$ .

Imagine walking towards a door. You take steps that are half the distance to the door.

You get infinitely close, but theoretically, you never hit the door. That’s a limit.

A function is continuous if you can draw its graph without lifting your pen.

A function $f(x)$ is continuous at $x=c$ if:

The derivative measures the instantaneous rate of change.

SLOPE OF THE TANGENT

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

If $s(t)$ is position, then $s'(t)$ is velocity.

If $v(t)$ is velocity, then $v'(t)$ is acceleration.

You have now completed the first module of Mathematics 1. We’ve covered Logic, Sets, Functions, and the basics of Calculus.

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