Calculus Basics

The sequence is organized around one idea: describe change locally, then accumulate it globally.

1. Limits establish local behavior and continuity.
2. Derivatives measure instantaneous change.
3. Integrals accumulate quantities and connect back to derivatives.
4. Gradients and Gradient Descent extend change to several variables.
5. Optimization turns derivatives into decisions.
6. Optimization in Neural Networks applies the machinery to learning systems.

Before starting

Be comfortable rearranging equations, reading function domains and graphs, and using powers, logarithms, and trigonometric functions (with angles in radians). Single-variable calculus is the starting point; vectors and linear algebra become useful when moving to gradients and many parameters.

One calculation connecting the topics

Let position be $s(t)=t^2$ metres, with $t$ measured in seconds. The difference quotient is

$$\frac{s(t+h)-s(t)}h=2t+h\quad(h\ne0).$$

Taking $h\to0$ gives velocity $s'(t)=2t$ metres per second. Over the interval from 1 to 3 seconds, integrating this velocity gives displacement $\int_1^3 2t\,dt=9-1=8$ metres. This is signed displacement, not necessarily total distance; distance would integrate $|s'(t)|$ if direction changed.

The fundamental theorem connects these operations: if $f$ is continuous on $[a,b]$, then $F(x)=\int_a^x f(t)\,dt$ has derivative $f(x)$ in the interior, and any antiderivative $G$ gives $\int_a^b f(x)\,dx=G(b)-G(a)$. Continuity is a sufficient condition here, not a claim that every function is differentiable or integrable.

For optimization, a zero derivative is only a candidate: $f(x)=x^3$ has $f'(0)=0$ but no extremum there. On a closed interval, also check endpoints and points where a derivative fails to exist.

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