This section introduces two fundamental problems that motivated the development of calculus: finding tangent lines and instantaneous velocities.
| Skill | Description | Difficulty |
|---|---|---|
| Tangent Problem Introduction | The geometric motivation for derivatives | Beginner |
| Velocity Problem Introduction | The physical motivation for derivatives | Beginner |
| Secant Lines and the Difference Quotient | Approximating tangent slopes | Intermediate |
| Secant to Tangent | Limit process for tangent lines | Intermediate |
| Tangent Slope via Limits | Computing tangent slopes | Intermediate |
| Instantaneous Velocity | Finding velocity at a moment | Intermediate |
Both problems lead to the same mathematical expression:
$$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$
This is the derivative of $f$ at $x = a$.
graph TD
A["Tangent Problem Intro"] --> C["Secant Lines"]
B["Velocity Problem Intro"] --> C
C --> D["Secant to Tangent"]
D --> E["Tangent Slope via Limits"]
D --> F["Instantaneous Velocity"]
click A "tangent-problem-intro.html"
click B "velocity-problem-intro.html"
click C "secant-lines-difference-quotient.html"
click D "secant-to-tangent.html"
click E "tangent-slope-via-limits.html"
click F "../ch2-sec1/instantaneous-velocity.html"
For a deeper exploration of how these two problems connect, see:
| ← Section 1.3 | Skills Index | Section 1.5 → |