DERIVADAS (II) | @laprofedematemola

Summary

The lesson introduces applications of derivatives, focusing on monotonicity, relative extrema, concavity and convexity, and inflection points. Hôpital’s rule, graphing functions, and optimization are mentioned as topics in the broader unit but are not developed in detail.

1. Increasing and decreasing functions

A function is increasing on an interval when its derivative is nonnegative there, and decreasing when its derivative is nonpositive. If the derivative is strictly positive throughout an interval, the function is strictly increasing; if it is strictly negative, the function is strictly decreasing.

To study monotonicity:

1. Identify the function’s domain, including any discontinuities.
2. Calculate the first derivative.
3. Find where the derivative is zero or undefined within the domain.
4. Use those values to divide the domain into intervals.
5. Test the sign of the derivative on each interval.
6. Interpret positive signs as increasing and negative signs as decreasing.

A derivative equal to zero does not automatically mean there is a maximum or minimum. For example, (x^3) increases on both sides of (x=0), so it has no extremum there even though its derivative is zero.

2. Relative maxima and minima

A relative maximum is a point where the function’s value is at least as large as nearby values. A relative minimum is a point where its value is at least as small as nearby values.

For a differentiable function, a relative extremum at an interior point requires a horizontal tangent, so the derivative there is zero. This is a necessary condition, but it does not by itself prove that an extremum exists.

Changes in the sign of the first derivative help identify extrema:

• Increasing then decreasing: relative maximum.
• Decreasing then increasing: relative minimum.
• No change in direction: no relative extremum.

The lesson also presents the second-derivative test. At a stationary point (a), where (f’(a)=0):

• If (f’‘(a)>0), there is a relative minimum.
• If (f’‘(a)<0), there is a relative maximum.
• If (f’‘(a)=0), the test is inconclusive.

Relative extrema are local; they need not be the greatest or least values over the function’s entire domain. Worked examples involving polynomial, quotient, exponential-form, and logarithmic functions emphasize the importance of accounting for discontinuities and domain restrictions when dividing the domain into intervals.

3. Concavity, convexity, and curvature

The lesson uses the following convention:

• If (f’‘(x)>0), the function is concave.
• If (f’‘(x)<0), the function is convex.

To study curvature:

1. Calculate the second derivative.
2. Find where it is zero or undefined within the function’s domain.
3. Divide the domain into intervals using those values.
4. Test the sign of the second derivative on each interval.
5. Label each interval as concave or convex according to the convention above.

The instructor also offers a memory aid for distinguishing the shapes and shows how curvature information can help sketch a graph.

4. Inflection points

An inflection point is a point where the function changes curvature—from concave to convex or from convex to concave.

To find possible inflection points, solve (f’‘(x)=0) and check relevant points where the second derivative is undefined. A candidate is an inflection point only if the curvature actually changes across it.

The lesson also presents a derivative-based test: if (f’‘(a)=0) and (f’‘’(a)\ne 0), then (a) is confirmed as an inflection point by this test, assuming the required derivatives exist.

Main takeaway

Derivative applications rely on accurate differentiation and systematic sign analysis. Checking domains and testing signs interval by interval helps identify where a function rises or falls, locate relative extrema, determine curvature, and find inflection points.

Speakers and sources

• Instructor/presenter: The teacher explaining the material; not named in the subtitles.
• No other speakers are identified in the subtitles.