Calculus is the mathematics of change and accumulation. Derivatives answer “how fast?” Integrals answer “how much in total?” Limits hold the two ideas together by making precise what happens as we zoom in or add finer and finer pieces.
What you should be able to do
After a solid first pass, you should be able to:
- Explain a limit in words and check simple ones algebraically or numerically.
- Compute derivatives of polynomials, exponentials, sine and cosine, and products/quotients/compositions.
- Interpret $f'(a)$ as an instantaneous rate and as the slope of a tangent line.
- Set up and evaluate definite integrals as signed area / net accumulation.
- Use the Fundamental Theorem of Calculus to connect derivatives and integrals.
Limits without mysticism
Saying $\lim_{x \to a} f(x) = L$ means: values of $f(x)$ can be forced as close to $L$ as we like by taking $x$ sufficiently close to $a$ (but not necessarily equal to $a$). Continuity at $a$ adds that $f(a)$ exists and equals that limit. Removable gaps, jumps, and vertical blow-ups are the classic ways continuity fails.
Derivatives as linear approximation
The derivative $f'(a)$ is the unique number (when it exists) such that near $a$, $f(x) \approx f(a) + f'(a)(x-a)$. That linear approximation is why derivatives show up in optimization, related rates, and numerical methods. For a walkthrough of the meaning, see What a derivative actually measures. To estimate slopes numerically, try the numerical derivative calculator.
Integrals as accumulation
A definite integral $\int_a^b f(x)\,dx$ measures net signed area under $y=f(x)$ from $a$ to $b$. Riemann sums make the idea concrete: rectangles of height $f(x_i^*)$ and width $\Delta x$ approximate the total; the integral is the limit of those sums as the mesh goes to zero (when the limit exists).