Applications of differentiation

Use derivative signs and changes to interpret graphs, optimization, and motion.

Meaning

Derivative signs help examine whether functions increase or decrease and are used to interpret extrema, optimization, and motion. Absolute maxima and minima also require attention to the domain and endpoints.

Example

For the derivative is negative below 4 and positive above 4. On the real numbers, the minimum value is therefore 2 at Here, without differentiation, (x−4)²≥0 gives f(x)≥2, with equality at x=4. Vertex form makes this shorter method available; it agrees with the derivative's change from decreasing to increasing.A point where the derivative is zero need not be an extremum. The function keeps increasing even though its derivative is zero at 0.

Practice for this concept

Scope and references

Topics

Tangents, increasing and decreasing functions, local and absolute extrema, graph analysis, velocity and acceleration

Learning goal

Use derivative signs and changes to interpret graphs, optimization, and motion.

Helpful background

Review how the derivative relates to increases and decreases in the original function.

Applications of derivatives

These ordinary choice questions provide limited practice in this concept. They do not establish independent solution, full-concept performance, or mastery.

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